This file is indexed.

/usr/share/gap/doc/ref/chap41.html is in gap-doc 4r6p5-3.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

   1
   2
   3
   4
   5
   6
   7
   8
   9
  10
  11
  12
  13
  14
  15
  16
  17
  18
  19
  20
  21
  22
  23
  24
  25
  26
  27
  28
  29
  30
  31
  32
  33
  34
  35
  36
  37
  38
  39
  40
  41
  42
  43
  44
  45
  46
  47
  48
  49
  50
  51
  52
  53
  54
  55
  56
  57
  58
  59
  60
  61
  62
  63
  64
  65
  66
  67
  68
  69
  70
  71
  72
  73
  74
  75
  76
  77
  78
  79
  80
  81
  82
  83
  84
  85
  86
  87
  88
  89
  90
  91
  92
  93
  94
  95
  96
  97
  98
  99
 100
 101
 102
 103
 104
 105
 106
 107
 108
 109
 110
 111
 112
 113
 114
 115
 116
 117
 118
 119
 120
 121
 122
 123
 124
 125
 126
 127
 128
 129
 130
 131
 132
 133
 134
 135
 136
 137
 138
 139
 140
 141
 142
 143
 144
 145
 146
 147
 148
 149
 150
 151
 152
 153
 154
 155
 156
 157
 158
 159
 160
 161
 162
 163
 164
 165
 166
 167
 168
 169
 170
 171
 172
 173
 174
 175
 176
 177
 178
 179
 180
 181
 182
 183
 184
 185
 186
 187
 188
 189
 190
 191
 192
 193
 194
 195
 196
 197
 198
 199
 200
 201
 202
 203
 204
 205
 206
 207
 208
 209
 210
 211
 212
 213
 214
 215
 216
 217
 218
 219
 220
 221
 222
 223
 224
 225
 226
 227
 228
 229
 230
 231
 232
 233
 234
 235
 236
 237
 238
 239
 240
 241
 242
 243
 244
 245
 246
 247
 248
 249
 250
 251
 252
 253
 254
 255
 256
 257
 258
 259
 260
 261
 262
 263
 264
 265
 266
 267
 268
 269
 270
 271
 272
 273
 274
 275
 276
 277
 278
 279
 280
 281
 282
 283
 284
 285
 286
 287
 288
 289
 290
 291
 292
 293
 294
 295
 296
 297
 298
 299
 300
 301
 302
 303
 304
 305
 306
 307
 308
 309
 310
 311
 312
 313
 314
 315
 316
 317
 318
 319
 320
 321
 322
 323
 324
 325
 326
 327
 328
 329
 330
 331
 332
 333
 334
 335
 336
 337
 338
 339
 340
 341
 342
 343
 344
 345
 346
 347
 348
 349
 350
 351
 352
 353
 354
 355
 356
 357
 358
 359
 360
 361
 362
 363
 364
 365
 366
 367
 368
 369
 370
 371
 372
 373
 374
 375
 376
 377
 378
 379
 380
 381
 382
 383
 384
 385
 386
 387
 388
 389
 390
 391
 392
 393
 394
 395
 396
 397
 398
 399
 400
 401
 402
 403
 404
 405
 406
 407
 408
 409
 410
 411
 412
 413
 414
 415
 416
 417
 418
 419
 420
 421
 422
 423
 424
 425
 426
 427
 428
 429
 430
 431
 432
 433
 434
 435
 436
 437
 438
 439
 440
 441
 442
 443
 444
 445
 446
 447
 448
 449
 450
 451
 452
 453
 454
 455
 456
 457
 458
 459
 460
 461
 462
 463
 464
 465
 466
 467
 468
 469
 470
 471
 472
 473
 474
 475
 476
 477
 478
 479
 480
 481
 482
 483
 484
 485
 486
 487
 488
 489
 490
 491
 492
 493
 494
 495
 496
 497
 498
 499
 500
 501
 502
 503
 504
 505
 506
 507
 508
 509
 510
 511
 512
 513
 514
 515
 516
 517
 518
 519
 520
 521
 522
 523
 524
 525
 526
 527
 528
 529
 530
 531
 532
 533
 534
 535
 536
 537
 538
 539
 540
 541
 542
 543
 544
 545
 546
 547
 548
 549
 550
 551
 552
 553
 554
 555
 556
 557
 558
 559
 560
 561
 562
 563
 564
 565
 566
 567
 568
 569
 570
 571
 572
 573
 574
 575
 576
 577
 578
 579
 580
 581
 582
 583
 584
 585
 586
 587
 588
 589
 590
 591
 592
 593
 594
 595
 596
 597
 598
 599
 600
 601
 602
 603
 604
 605
 606
 607
 608
 609
 610
 611
 612
 613
 614
 615
 616
 617
 618
 619
 620
 621
 622
 623
 624
 625
 626
 627
 628
 629
 630
 631
 632
 633
 634
 635
 636
 637
 638
 639
 640
 641
 642
 643
 644
 645
 646
 647
 648
 649
 650
 651
 652
 653
 654
 655
 656
 657
 658
 659
 660
 661
 662
 663
 664
 665
 666
 667
 668
 669
 670
 671
 672
 673
 674
 675
 676
 677
 678
 679
 680
 681
 682
 683
 684
 685
 686
 687
 688
 689
 690
 691
 692
 693
 694
 695
 696
 697
 698
 699
 700
 701
 702
 703
 704
 705
 706
 707
 708
 709
 710
 711
 712
 713
 714
 715
 716
 717
 718
 719
 720
 721
 722
 723
 724
 725
 726
 727
 728
 729
 730
 731
 732
 733
 734
 735
 736
 737
 738
 739
 740
 741
 742
 743
 744
 745
 746
 747
 748
 749
 750
 751
 752
 753
 754
 755
 756
 757
 758
 759
 760
 761
 762
 763
 764
 765
 766
 767
 768
 769
 770
 771
 772
 773
 774
 775
 776
 777
 778
 779
 780
 781
 782
 783
 784
 785
 786
 787
 788
 789
 790
 791
 792
 793
 794
 795
 796
 797
 798
 799
 800
 801
 802
 803
 804
 805
 806
 807
 808
 809
 810
 811
 812
 813
 814
 815
 816
 817
 818
 819
 820
 821
 822
 823
 824
 825
 826
 827
 828
 829
 830
 831
 832
 833
 834
 835
 836
 837
 838
 839
 840
 841
 842
 843
 844
 845
 846
 847
 848
 849
 850
 851
 852
 853
 854
 855
 856
 857
 858
 859
 860
 861
 862
 863
 864
 865
 866
 867
 868
 869
 870
 871
 872
 873
 874
 875
 876
 877
 878
 879
 880
 881
 882
 883
 884
 885
 886
 887
 888
 889
 890
 891
 892
 893
 894
 895
 896
 897
 898
 899
 900
 901
 902
 903
 904
 905
 906
 907
 908
 909
 910
 911
 912
 913
 914
 915
 916
 917
 918
 919
 920
 921
 922
 923
 924
 925
 926
 927
 928
 929
 930
 931
 932
 933
 934
 935
 936
 937
 938
 939
 940
 941
 942
 943
 944
 945
 946
 947
 948
 949
 950
 951
 952
 953
 954
 955
 956
 957
 958
 959
 960
 961
 962
 963
 964
 965
 966
 967
 968
 969
 970
 971
 972
 973
 974
 975
 976
 977
 978
 979
 980
 981
 982
 983
 984
 985
 986
 987
 988
 989
 990
 991
 992
 993
 994
 995
 996
 997
 998
 999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (ref) - Chapter 41: Group Actions</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap41"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chap16.html">16</a>  <a href="chap17.html">17</a>  <a href="chap18.html">18</a>  <a href="chap19.html">19</a>  <a href="chap20.html">20</a>  <a href="chap21.html">21</a>  <a href="chap22.html">22</a>  <a href="chap23.html">23</a>  <a href="chap24.html">24</a>  <a href="chap25.html">25</a>  <a href="chap26.html">26</a>  <a href="chap27.html">27</a>  <a href="chap28.html">28</a>  <a href="chap29.html">29</a>  <a href="chap30.html">30</a>  <a href="chap31.html">31</a>  <a href="chap32.html">32</a>  <a href="chap33.html">33</a>  <a href="chap34.html">34</a>  <a href="chap35.html">35</a>  <a href="chap36.html">36</a>  <a href="chap37.html">37</a>  <a href="chap38.html">38</a>  <a href="chap39.html">39</a>  <a href="chap40.html">40</a>  <a href="chap41.html">41</a>  <a href="chap42.html">42</a>  <a href="chap43.html">43</a>  <a href="chap44.html">44</a>  <a href="chap45.html">45</a>  <a href="chap46.html">46</a>  <a href="chap47.html">47</a>  <a href="chap48.html">48</a>  <a href="chap49.html">49</a>  <a href="chap50.html">50</a>  <a href="chap51.html">51</a>  <a href="chap52.html">52</a>  <a href="chap53.html">53</a>  <a href="chap54.html">54</a>  <a href="chap55.html">55</a>  <a href="chap56.html">56</a>  <a href="chap57.html">57</a>  <a href="chap58.html">58</a>  <a href="chap59.html">59</a>  <a href="chap60.html">60</a>  <a href="chap61.html">61</a>  <a href="chap62.html">62</a>  <a href="chap63.html">63</a>  <a href="chap64.html">64</a>  <a href="chap65.html">65</a>  <a href="chap66.html">66</a>  <a href="chap67.html">67</a>  <a href="chap68.html">68</a>  <a href="chap69.html">69</a>  <a href="chap70.html">70</a>  <a href="chap71.html">71</a>  <a href="chap72.html">72</a>  <a href="chap73.html">73</a>  <a href="chap74.html">74</a>  <a href="chap75.html">75</a>  <a href="chap76.html">76</a>  <a href="chap77.html">77</a>  <a href="chap78.html">78</a>  <a href="chap79.html">79</a>  <a href="chap80.html">80</a>  <a href="chap81.html">81</a>  <a href="chap82.html">82</a>  <a href="chap83.html">83</a>  <a href="chap84.html">84</a>  <a href="chap85.html">85</a>  <a href="chap86.html">86</a>  <a href="chap87.html">87</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<div class="chlinkprevnexttop">&nbsp;<a href="chap0.html">[Top of Book]</a>&nbsp;  <a href="chap0.html#contents">[Contents]</a>&nbsp;  &nbsp;<a href="chap40.html">[Previous Chapter]</a>&nbsp;  &nbsp;<a href="chap42.html">[Next Chapter]</a>&nbsp;  </div>

<p id="mathjaxlink" class="pcenter"><a href="chap41_mj.html">[MathJax on]</a></p>
<p><a id="X87115591851FB7F4" name="X87115591851FB7F4"></a></p>
<div class="ChapSects"><a href="chap41.html#X87115591851FB7F4">41 <span class="Heading">Group Actions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X83661AFD7B7BD1D9">41.1 <span class="Heading">About Group Actions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X81B8F9CD868CD953">41.2 <span class="Heading">Basic Actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7FE417DD837987B4">41.2-1 OnPoints</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7960924D84B5B18F">41.2-2 OnRight</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X832DF5327ECA0E44">41.2-3 OnLeftInverse</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X85AA04347CD117F9">41.2-4 OnSets</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X832CC5F87EEA4A7E">41.2-5 OnTuples</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X80DAA1D2855B1456">41.2-6 OnPairs</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7C10492081D72376">41.2-7 OnSetsSets</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7E23686E7A9D3A20">41.2-8 OnSetsDisjointSets</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7ADE244E819035FF">41.2-9 OnSetsTuples</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7FF556CD7E6739A9">41.2-10 OnTuplesSets</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X844E902382EB4151">41.2-11 OnTuplesTuples</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X86DC2DD5829CAD9A">41.2-12 OnLines</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7FA394D27E721E2B">41.2-13 OnIndeterminates</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7BA8D76586F1F06E">41.2-14 Permuted</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X85124D197F0F9C4D">41.2-15 OnSubspacesByCanonicalBasis</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X82181CA07A5B2056">41.3 <span class="Heading">Action on canonical representatives</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X81E0FF0587C54543">41.4 <span class="Heading">Orbits</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X80E0234E7BD79409">41.4-1 Orbit</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X86BCAE17869BBEAA">41.4-2 Orbits</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X86BC8B958123F953">41.4-3 <span class="Heading">OrbitsDomain</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X799910CF832EDC45">41.4-4 OrbitLength</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X8032F73078DF2DDB">41.4-5 <span class="Heading">OrbitLengths</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X8520E2487F7E98AF">41.4-6 <span class="Heading">OrbitLengthsDomain</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X797BD60E7ACEF1B1">41.5 <span class="Heading">Stabilizers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7C34EC437EF598BF">41.5-1 OrbitStabilizer</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X86FB962786397E02">41.5-2 Stabilizer</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X78C3A8568414BC44">41.5-3 OrbitStabilizerAlgorithm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X7A9389097BAF670D">41.6 <span class="Heading">Elements with Prescribed Images</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X857DC7B085EB0539">41.6-1 RepresentativeAction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X87F73CCA7921DE65">41.7 <span class="Heading">The Permutation Image of an Action</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X78E6A002835288A4">41.7-1 <span class="Heading">ActionHomomorphism</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X85A8E93D786C3C9C">41.7-2 Action</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X86FF54A383B73967">41.7-3 SparseActionHomomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X7FED50ED7ACA5FB2">41.8 <span class="Heading">Action of a group on itself</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X78C37C4C7B2BDC44">41.8-1 FactorCosetAction</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X8561DEBA79E01ABD">41.8-2 RegularActionHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X835317A7847477D4">41.8-3 AbelianSubfactorAction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X807AA91E841D132B">41.9 <span class="Heading">Permutations Induced by Elements and Cycles</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7807A33381DCAB26">41.9-1 <span class="Heading">Permutation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X81D4EA42810974A0">41.9-2 PermutationCycle</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X80AF6E0683CA7F14">41.9-3 Cycle</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7F559E897B333758">41.9-4 CycleLength</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7F3B387A7FD8AE5E">41.9-5 Cycles</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X83040A6080C2C6C6">41.9-6 CycleLengths</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X87FDA6838065CDCB">41.9-7 <span class="Heading">CycleIndex</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X850A84618421392A">41.10 <span class="Heading">Tests for Actions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X79B15750851828CB">41.10-1 <span class="Heading">IsTransitive</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X8295D733796B7A37">41.10-2 <span class="Heading">Transitivity</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X8166A6A17C8D6E73">41.10-3 <span class="Heading">RankAction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7B77040F8543CD6E">41.10-4 <span class="Heading">IsSemiRegular</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7CF02C4785F0EAB5">41.10-5 <span class="Heading">IsRegular</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7CB1D74280F92AFC">41.10-6 <span class="Heading">Earns</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X84C19AD68247B760">41.10-7 <span class="Heading">IsPrimitive</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X7E9D3D0B7A9A8572">41.11 <span class="Heading">Block Systems</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X84FE699F85371643">41.11-1 <span class="Heading">Blocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X79936EB97AAD1144">41.11-2 <span class="Heading">MaximalBlocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7941DB6380B74510">41.11-3 <span class="Heading">RepresentativesMinimalBlocks</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X835658B07B28EF3B">41.11-4 AllBlocks</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap41.html#X7FD3D2D2788709B7">41.12 <span class="Heading">External Sets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X8264C3C479FF0A8B">41.12-1 IsExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7C90F648793E47DD">41.12-2 ExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7B9DB15D80CE28B4">41.12-3 ActingDomain</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X86153CB087394DC1">41.12-4 FunctionAction</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X86A0CC1479A5932A">41.12-5 HomeEnumerator</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X879DE63C7858453C">41.12-6 IsExternalSubset</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X87D1EA1486D86233">41.12-7 ExternalSubset</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7E081F568407317F">41.12-8 IsExternalOrbit</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7FB656AE7A066C35">41.12-9 ExternalOrbit</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7BAFF02B7D6DF9F2">41.12-10 StabilizerOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X867262FA82FDD592">41.12-11 <span class="Heading">ExternalOrbits</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7A64EF807CE8893E">41.12-12 <span class="Heading">ExternalOrbitsStabilizers</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X8048AE727A7F1A2F">41.12-13 CanonicalRepresentativeOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X8071A8D784DC8325">41.12-14 CanonicalRepresentativeDeterminatorOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X85E9A6A77B8D00B8">41.12-15 ActorOfExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X8190A8247F29A5C7">41.12-16 UnderlyingExternalSet</a></span>
<span class="ContSS"><br /><span class="nocss">&nbsp;&nbsp;</span><a href="chap41.html#X7A3D87DE809FBFD4">41.12-17 SurjectiveActionHomomorphismAttr</a></span>
</div></div>
</div>

<h3>41 <span class="Heading">Group Actions</span></h3>

<p>A <em>group action</em> is a triple <span class="SimpleMath">(G, Ω, μ)</span>, where <span class="SimpleMath">G</span> is a group, <span class="SimpleMath">Ω</span> a set and <span class="SimpleMath">μ : Ω × G → Ω</span> a function that is compatible with the group arithmetic. We call <span class="SimpleMath">Ω</span> the <em>domain</em> of the action.</p>

<p>In <strong class="pkg">GAP</strong>, <span class="SimpleMath">Ω</span> can be a duplicate-free collection (an object that permits access to its elements via the <span class="SimpleMath">Ω[n]</span> operation, for example a list), it does not need to be sorted (see <code class="func">IsSet</code> (<a href="chap21.html#X80CDAF45782E8DCB"><span class="RefLink">21.17-4</span></a>)).</p>

<p>The acting function <span class="SimpleMath">μ</span> is a binary <strong class="pkg">GAP</strong> function that returns the image <span class="SimpleMath">μ( x, g )</span> for a point <span class="SimpleMath">x ∈ Ω</span> and a group element <span class="SimpleMath">g ∈ G</span>.</p>

<p>In <strong class="pkg">GAP</strong>, groups always act from the right, that is <span class="SimpleMath">μ( μ( x, g ), h ) = μ( x, gh )</span>.</p>

<p><strong class="pkg">GAP</strong> does not test whether the acting function <span class="SimpleMath">μ</span> satisfies the conditions for a group operation but silently assumes that is does. (If it does not, results are unpredictable.)</p>

<p>The first section of this chapter, <a href="chap41.html#X83661AFD7B7BD1D9"><span class="RefLink">41.1</span></a>, describes the various ways how operations for group actions can be called.</p>

<p>Functions for several commonly used action are already built into <strong class="pkg">GAP</strong>. These are listed in section <a href="chap41.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a>.</p>

<p>The sections <a href="chap41.html#X87F73CCA7921DE65"><span class="RefLink">41.7</span></a> and <a href="chap41.html#X7FED50ED7ACA5FB2"><span class="RefLink">41.8</span></a> describe homomorphisms and mappings associated to group actions as well as the permutation group image of an action.</p>

<p>The other sections then describe operations to compute orbits, stabilizers, as well as properties of actions.</p>

<p>Finally section <a href="chap41.html#X7FD3D2D2788709B7"><span class="RefLink">41.12</span></a> describes the concept of "external sets" which represent the concept of a <em><span class="SimpleMath">G</span>-set</em> and underly the actions mechanism.</p>

<p><a id="X83661AFD7B7BD1D9" name="X83661AFD7B7BD1D9"></a></p>

<h4>41.1 <span class="Heading">About Group Actions</span></h4>

<p>The syntax which is used by the operations for group actions is quite flexible. For example we can call the operation <code class="func">OrbitsDomain</code> (<a href="chap41.html#X86BC8B958123F953"><span class="RefLink">41.4-3</span></a>) for the orbits of the group <var class="Arg">G</var> on the domain <var class="Arg">Omega</var> in the following ways:</p>


<dl>
<dt><strong class="Mark"><code class="code">OrbitsDomain</code><span class="SimpleMath">( G, Ω[, μ] )</span></strong></dt>
<dd><p>The acting function <span class="SimpleMath">μ</span> is optional. If it is not given, the built-in action <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) (which defines an action via the caret operator <code class="code">^</code>) is used as a default.</p>

</dd>
<dt><strong class="Mark"><code class="code">OrbitsDomain</code><span class="SimpleMath">( G, Ω, gens, acts[, μ] )</span></strong></dt>
<dd><p>This second version of <code class="func">OrbitsDomain</code> (<a href="chap41.html#X86BC8B958123F953"><span class="RefLink">41.4-3</span></a>) permits one to implement an action induced by a homomorphism: If the group <span class="SimpleMath">H</span> acts on <span class="SimpleMath">Ω</span> via <span class="SimpleMath">μ</span> and <span class="SimpleMath">φ : G → H</span> is a homomorphism, <span class="SimpleMath">G</span> acts on <span class="SimpleMath">Ω</span> via the induced action <span class="SimpleMath">μ'( x, g ) = μ( x, g^φ )</span>.</p>

<p>Here <span class="SimpleMath">gens</span> must be a set of generators of <span class="SimpleMath">G</span> and <span class="SimpleMath">acts</span> the images of <span class="SimpleMath">gens</span> under <span class="SimpleMath">φ</span>. <span class="SimpleMath">μ</span> is the acting function for <span class="SimpleMath">H</span>. Again, the function <span class="SimpleMath">μ</span> is optional and <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) is used as a default.</p>

<p>The advantage of this notation is that <strong class="pkg">GAP</strong> does not need to construct this homomorphism <span class="SimpleMath">φ</span> and the range group <var class="Arg">H</var> as <strong class="pkg">GAP</strong> objects. (If a small group <span class="SimpleMath">G</span> acts via complicated objects <span class="SimpleMath">acts</span> this otherwise could lead to performance problems.)</p>

<p><strong class="pkg">GAP</strong> does not test whether the mapping <span class="SimpleMath">gens ↦ acts</span> actually induces a homomorphism and the results are unpredictable if this is not the case.</p>

</dd>
<dt><strong class="Mark"><code class="code">OrbitsDomain</code><span class="SimpleMath">( extset )</span></strong></dt>
<dd><p>A third variant is to call the operation with an external set, which then provides <span class="SimpleMath">G</span>, <span class="SimpleMath">Ω</span> and <span class="SimpleMath">μ</span>. You will find more about external sets in Section <a href="chap41.html#X7FD3D2D2788709B7"><span class="RefLink">41.12</span></a>.</p>

</dd>
</dl>
<p>For operations like <code class="func">Stabilizer</code> (<a href="chap41.html#X86FB962786397E02"><span class="RefLink">41.5-2</span></a>) of course the domain must be replaced by an element of the domain of the action.</p>

<p><a id="X81B8F9CD868CD953" name="X81B8F9CD868CD953"></a></p>

<h4>41.2 <span class="Heading">Basic Actions</span></h4>

<p><strong class="pkg">GAP</strong> already provides acting functions for the more common actions of a group. For built-in operations such as <code class="func">Stabilizer</code> (<a href="chap41.html#X86FB962786397E02"><span class="RefLink">41.5-2</span></a>) special methods are available for many of these actions.</p>

<p>If one needs an action for which no acting function is provided by the library it can be implemented via a <strong class="pkg">GAP</strong> function that conforms to the syntax</p>

<p><code class="code">actfun( omega, g )</code></p>

<p>where <code class="code">omega</code> is an element of the action domain, <code class="code">g</code> is an element of the acting group, and the return value is the image of <code class="code">omega</code> under <code class="code">g</code>.</p>

<p>For example one could define the following function that acts on pairs of polynomials via <code class="func">OnIndeterminates</code> (<a href="chap41.html#X7FA394D27E721E2B"><span class="RefLink">41.2-13</span></a>):</p>


<div class="example"><pre>
OnIndeterminatesPairs:= function( polypair, g )
  return [ OnIndeterminates( polypair[1], g ),
           OnIndeterminates( polypair[2], g ) ];
end;
</pre></div>

<p>Note that this function <em>must</em> implement a group action from the <em>right</em>. This is not verified by <strong class="pkg">GAP</strong> and results are unpredictable otherwise.</p>

<p><a id="X7FE417DD837987B4" name="X7FE417DD837987B4"></a></p>

<h5>41.2-1 OnPoints</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnPoints</code>( <var class="Arg">pnt</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <code class="code"><var class="Arg">pnt</var> ^ <var class="Arg">g</var></code>. This is for example the action of a permutation group on points, or the action of a group on its elements via conjugation. The action of a matrix group on vectors from the right is described by both <code class="func">OnPoints</code> and <code class="func">OnRight</code> (<a href="chap41.html#X7960924D84B5B18F"><span class="RefLink">41.2-2</span></a>).</p>

<p><a id="X7960924D84B5B18F" name="X7960924D84B5B18F"></a></p>

<h5>41.2-2 OnRight</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnRight</code>( <var class="Arg">pnt</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <code class="code"><var class="Arg">pnt</var> * <var class="Arg">g</var></code>. This is for example the action of a group on its elements via right multiplication, or the action of a group on the cosets of a subgroup. The action of a matrix group on vectors from the right is described by both <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) and <code class="func">OnRight</code>.</p>

<p><a id="X832DF5327ECA0E44" name="X832DF5327ECA0E44"></a></p>

<h5>41.2-3 OnLeftInverse</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnLeftInverse</code>( <var class="Arg">pnt</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <span class="SimpleMath"><var class="Arg">g</var>^{-1}</span> <code class="code">* <var class="Arg">pnt</var></code>. Forming the inverse is necessary to make this a proper action, as in <strong class="pkg">GAP</strong> groups always act from the right.</p>

<p><code class="func">OnLeftInverse</code> is used for example in the representation of a right coset as an external set (see <a href="chap41.html#X7FD3D2D2788709B7"><span class="RefLink">41.12</span></a>), that is, a right coset <span class="SimpleMath">Ug</span> is an external set for the group <span class="SimpleMath">U</span> acting on it via <code class="func">OnLeftInverse</code>.)</p>

<p><a id="X85AA04347CD117F9" name="X85AA04347CD117F9"></a></p>

<h5>41.2-4 OnSets</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnSets</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">set</var> be a proper set (see <a href="chap21.html#X80ABC25582343910"><span class="RefLink">21.19</span></a>). <code class="func">OnSets</code> returns the proper set formed by the images of all points <span class="SimpleMath">x</span> of <var class="Arg">set</var> via the action function <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>), applied to <span class="SimpleMath">x</span> and <var class="Arg">g</var>.</p>

<p><code class="func">OnSets</code> is for example used to compute the action of a permutation group on blocks.</p>

<p>(<code class="func">OnTuples</code> (<a href="chap41.html#X832CC5F87EEA4A7E"><span class="RefLink">41.2-5</span></a>) is an action on lists that preserves the ordering of entries.)</p>

<p><a id="X832CC5F87EEA4A7E" name="X832CC5F87EEA4A7E"></a></p>

<h5>41.2-5 OnTuples</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnTuples</code>( <var class="Arg">tup</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">tup</var> be a list. <code class="func">OnTuples</code> returns the list formed by the images of all points <span class="SimpleMath">x</span> of <var class="Arg">tup</var> via the action function <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>), applied to <span class="SimpleMath">x</span> and <var class="Arg">g</var>.</p>

<p>(<code class="func">OnSets</code> (<a href="chap41.html#X85AA04347CD117F9"><span class="RefLink">41.2-4</span></a>) is an action on lists that additionally sorts the entries of the result.)</p>

<p><a id="X80DAA1D2855B1456" name="X80DAA1D2855B1456"></a></p>

<h5>41.2-6 OnPairs</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnPairs</code>( <var class="Arg">tup</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a special case of <code class="func">OnTuples</code> (<a href="chap41.html#X832CC5F87EEA4A7E"><span class="RefLink">41.2-5</span></a>) for lists <var class="Arg">tup</var> of length 2.</p>

<p><a id="X7C10492081D72376" name="X7C10492081D72376"></a></p>

<h5>41.2-7 OnSetsSets</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnSetsSets</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>implements the action on sets of sets. For the special case that the sets are pairwise disjoint, it is possible to use <code class="func">OnSetsDisjointSets</code> (<a href="chap41.html#X7E23686E7A9D3A20"><span class="RefLink">41.2-8</span></a>). <var class="Arg">set</var> must be a sorted list whose entries are again sorted lists, otherwise an error is triggered (see <a href="chap41.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>

<p><a id="X7E23686E7A9D3A20" name="X7E23686E7A9D3A20"></a></p>

<h5>41.2-8 OnSetsDisjointSets</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnSetsDisjointSets</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>implements the action on sets of pairwise disjoint sets (see also <code class="func">OnSetsSets</code> (<a href="chap41.html#X7C10492081D72376"><span class="RefLink">41.2-7</span></a>)). <var class="Arg">set</var> must be a sorted list whose entries are again sorted lists, otherwise an error is triggered (see <a href="chap41.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>

<p><a id="X7ADE244E819035FF" name="X7ADE244E819035FF"></a></p>

<h5>41.2-9 OnSetsTuples</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnSetsTuples</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>implements the action on sets of tuples. <var class="Arg">set</var> must be a sorted list, otherwise an error is triggered (see <a href="chap41.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>

<p><a id="X7FF556CD7E6739A9" name="X7FF556CD7E6739A9"></a></p>

<h5>41.2-10 OnTuplesSets</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnTuplesSets</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>implements the action on tuples of sets. <var class="Arg">set</var> must be a list whose entries are again sorted lists, otherwise an error is triggered (see <a href="chap41.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>

<p><a id="X844E902382EB4151" name="X844E902382EB4151"></a></p>

<h5>41.2-11 OnTuplesTuples</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnTuplesTuples</code>( <var class="Arg">set</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>implements the action on tuples of tuples.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3),(2,3,4));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(g,1,OnPoints);</span>
[ 1, 2, 3, 4 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(g,(),OnRight);</span>
[ (), (1,2,3), (2,3,4), (1,3,2), (1,3)(2,4), (1,2)(3,4), (2,4,3), 
  (1,4,2), (1,4,3), (1,3,4), (1,2,4), (1,4)(2,3) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(g,[1,2],OnPairs);</span>
[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 1 ], [ 3, 4 ], [ 2, 1 ], 
  [ 1, 4 ], [ 4, 1 ], [ 4, 2 ], [ 3, 2 ], [ 2, 4 ], [ 4, 3 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(g,[1,2],OnSets);</span>
[ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 4 ], [ 1, 4 ], [ 2, 4 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(g,[[1,2],[3,4]],OnSetsSets);</span>
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ], 
  [ [ 1, 3 ], [ 2, 4 ] ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(g,[[1,2],[3,4]],OnTuplesSets);</span>
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ], 
  [ [ 1, 3 ], [ 2, 4 ] ], [ [ 3, 4 ], [ 1, 2 ] ], 
  [ [ 1, 4 ], [ 2, 3 ] ], [ [ 2, 4 ], [ 1, 3 ] ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(g,[[1,2],[3,4]],OnSetsTuples);</span>
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ], 
  [ [ 1, 3 ], [ 4, 2 ] ], [ [ 2, 4 ], [ 3, 1 ] ], 
  [ [ 2, 1 ], [ 4, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(g,[[1,2],[3,4]],OnTuplesTuples);</span>
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ], 
  [ [ 1, 3 ], [ 4, 2 ] ], [ [ 3, 1 ], [ 2, 4 ] ], 
  [ [ 3, 4 ], [ 1, 2 ] ], [ [ 2, 1 ], [ 4, 3 ] ], 
  [ [ 1, 4 ], [ 2, 3 ] ], [ [ 4, 1 ], [ 3, 2 ] ], 
  [ [ 4, 2 ], [ 1, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ], 
  [ [ 2, 4 ], [ 3, 1 ] ], [ [ 4, 3 ], [ 2, 1 ] ] ]
</pre></div>

<p><a id="X86DC2DD5829CAD9A" name="X86DC2DD5829CAD9A"></a></p>

<h5>41.2-12 OnLines</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnLines</code>( <var class="Arg">vec</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">vec</var> be a <em>normed</em> row vector, that is, its first nonzero entry is normed to the identity of the relevant field, see <code class="func">NormedRowVector</code> (<a href="chap23.html#X785DC60D8482695D"><span class="RefLink">23.2-1</span></a>). The function <code class="func">OnLines</code> returns the row vector obtained from first multiplying <var class="Arg">vec</var> from the right with <var class="Arg">g</var> (via <code class="func">OnRight</code> (<a href="chap41.html#X7960924D84B5B18F"><span class="RefLink">41.2-2</span></a>)) and then normalizing the resulting row vector by scalar multiplication from the left.</p>

<p>This action corresponds to the projective action of a matrix group on one-dimensional subspaces.</p>

<p>If <var class="Arg">vec</var> is a zero vector or is not normed then an error is triggered (see <a href="chap41.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">gl:=GL(2,5);;v:=[1,0]*Z(5)^0;</span>
[ Z(5)^0, 0*Z(5) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h:=Action(gl,Orbit(gl,v,OnLines),OnLines);</span>
Group([ (2,3,5,6), (1,2,4)(3,6,5) ])
</pre></div>

<p><a id="X7FA394D27E721E2B" name="X7FA394D27E721E2B"></a></p>

<h5>41.2-13 OnIndeterminates</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnIndeterminates</code>( <var class="Arg">poly</var>, <var class="Arg">perm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>A permutation <var class="Arg">perm</var> acts on the multivariate polynomial <var class="Arg">poly</var> by permuting the indeterminates as it permutes points.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">x:=Indeterminate(Rationals,1);; y:=Indeterminate(Rationals,2);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">OnIndeterminates(x^7*y+x*y^4,(1,17)(2,28));</span>
x_17^7*x_28+x_17*x_28^4
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Stabilizer(Group((1,2,3,4),(1,2)),x*y,OnIndeterminates);</span>
Group([ (1,2), (3,4) ])
</pre></div>

<p><a id="X7BA8D76586F1F06E" name="X7BA8D76586F1F06E"></a></p>

<h5>41.2-14 Permuted</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Permuted</code>( <var class="Arg">list</var>, <var class="Arg">perm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The following example demonstrates <code class="func">Permuted</code> (<a href="chap21.html#X7B5A19098406347A"><span class="RefLink">21.20-18</span></a>) being used to implement a permutation action on a domain:</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">dom:=[ "a", "b", "c" ];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(g,dom,Permuted);</span>
[ [ "a", "b", "c" ], [ "c", "a", "b" ], [ "b", "a", "c" ], 
  [ "b", "c", "a" ], [ "a", "c", "b" ], [ "c", "b", "a" ] ]
</pre></div>

<p><a id="X85124D197F0F9C4D" name="X85124D197F0F9C4D"></a></p>

<h5>41.2-15 OnSubspacesByCanonicalBasis</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnSubspacesByCanonicalBasis</code>( <var class="Arg">bas</var>, <var class="Arg">mat</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OnSubspacesByCanonicalBasisConcatenations</code>( <var class="Arg">basvec</var>, <var class="Arg">mat</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>implements the operation of a matrix group on subspaces of a vector space. <var class="Arg">bas</var> must be a list of (linearly independent) vectors which forms a basis of the subspace in Hermite normal form. <var class="Arg">mat</var> is an element of the acting matrix group. The function returns a mutable matrix which gives the basis of the image of the subspace in Hermite normal form. (In other words: it triangulizes the product of <var class="Arg">bas</var> with <var class="Arg">mat</var>.)</p>

<p><var class="Arg">bas</var> must be given in Hermite normal form, otherwise an error is triggered (see <a href="chap41.html#X82181CA07A5B2056"><span class="RefLink">41.3</span></a>).</p>

<p><a id="X82181CA07A5B2056" name="X82181CA07A5B2056"></a></p>

<h4>41.3 <span class="Heading">Action on canonical representatives</span></h4>

<p>A variety of action functions assumes that the objects on which it acts are given in a particular form, for example canonical representatives. Affected actions are for example <code class="func">OnSetsSets</code> (<a href="chap41.html#X7C10492081D72376"><span class="RefLink">41.2-7</span></a>), <code class="func">OnSetsDisjointSets</code> (<a href="chap41.html#X7E23686E7A9D3A20"><span class="RefLink">41.2-8</span></a>), <code class="func">OnSetsTuples</code> (<a href="chap41.html#X7ADE244E819035FF"><span class="RefLink">41.2-9</span></a>), <code class="func">OnTuplesSets</code> (<a href="chap41.html#X7FF556CD7E6739A9"><span class="RefLink">41.2-10</span></a>), <code class="func">OnLines</code> (<a href="chap41.html#X86DC2DD5829CAD9A"><span class="RefLink">41.2-12</span></a>) and <code class="func">OnSubspacesByCanonicalBasis</code> (<a href="chap41.html#X85124D197F0F9C4D"><span class="RefLink">41.2-15</span></a>).</p>

<p>If orbit seeds or domain elements are not given in the required form <strong class="pkg">GAP</strong> will issue an error message:</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(SymmetricGroup(5),[[2,4],[1,3]],OnSetsSets);</span>
Error, Action not well-defined. See the manual section
``Action on canonical representatives''.
</pre></div>

<p>In this case the affected domain elements have to be brought in canonical form, as documented for the respective action function. For interactive use this is most easily done by acting with the identity element of the group.</p>

<p>(A similar error could arise if a user-defined action function is used which actually does not implement an action from the right.)</p>

<p><a id="X81E0FF0587C54543" name="X81E0FF0587C54543"></a></p>

<h4>41.4 <span class="Heading">Orbits</span></h4>

<p>If a group <span class="SimpleMath">G</span> acts on a set <span class="SimpleMath">Ω</span>, the set of all images of <span class="SimpleMath">x ∈ Ω</span> under elements of <span class="SimpleMath">G</span> is called the <em>orbit</em> of <span class="SimpleMath">x</span>. The set of orbits of <span class="SimpleMath">G</span> is a partition of <span class="SimpleMath">Ω</span>.</p>

<p><a id="X80E0234E7BD79409" name="X80E0234E7BD79409"></a></p>

<h5>41.4-1 Orbit</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Orbit</code>( <var class="Arg">G</var>[, <var class="Arg">Omega</var>], <var class="Arg">pnt</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The orbit of the point <var class="Arg">pnt</var> is the list of all images of <var class="Arg">pnt</var> under the action of the group <var class="Arg">G</var> w.r.t. the action function <var class="Arg">act</var> or <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) if no action function is given.</p>

<p>(Note that the arrangement of points in this list is not defined by the operation.)</p>

<p>The orbit of <var class="Arg">pnt</var> will always contain one element that is <em>equal</em> to <var class="Arg">pnt</var>, however for performance reasons this element is not necessarily <em>identical</em> to <var class="Arg">pnt</var>, in particular if <var class="Arg">pnt</var> is mutable.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,3,2),(2,4,3));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(g,1);</span>
[ 1, 3, 2, 4 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbit(g,[1,2],OnSets);</span>
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 3, 4 ], [ 2, 4 ] ]
</pre></div>

<p>(See Section <a href="chap41.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>

<p><a id="X86BCAE17869BBEAA" name="X86BCAE17869BBEAA"></a></p>

<h5>41.4-2 Orbits</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Orbits</code>( <var class="Arg">G</var>, <var class="Arg">seeds</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Orbits</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a duplicate-free list of the orbits of the elements in <var class="Arg">seeds</var> under the action <var class="Arg">act</var> of <var class="Arg">G</var> or under <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) if no action function is given.</p>

<p>(Note that the arrangement of orbits or of points within one orbit is not defined by the operation.)</p>

<p><a id="X86BC8B958123F953" name="X86BC8B958123F953"></a></p>

<h5>41.4-3 <span class="Heading">OrbitsDomain</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OrbitsDomain</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OrbitsDomain</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of the orbits of <var class="Arg">G</var> on the domain <var class="Arg">Omega</var> (given as lists) under the action <var class="Arg">act</var> or under <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) if no action function is given.</p>

<p>This operation is often faster than <code class="func">Orbits</code> (<a href="chap41.html#X86BCAE17869BBEAA"><span class="RefLink">41.4-2</span></a>). The domain <var class="Arg">Omega</var> must be closed under the action of <var class="Arg">G</var>, otherwise an error can occur.</p>

<p>(Note that the arrangement of orbits or of points within one orbit is not defined by the operation.)</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,3,2),(2,4,3));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbits(g,[1..5]);</span>
[ [ 1, 3, 2, 4 ], [ 5 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">OrbitsDomain(g,Arrangements([1..4],3),OnTuples);</span>
[ [ [ 1, 2, 3 ], [ 3, 1, 2 ], [ 1, 4, 2 ], [ 2, 3, 1 ], [ 2, 1, 4 ], 
      [ 3, 4, 1 ], [ 1, 3, 4 ], [ 4, 2, 1 ], [ 4, 1, 3 ], 
      [ 2, 4, 3 ], [ 3, 2, 4 ], [ 4, 3, 2 ] ], 
  [ [ 1, 2, 4 ], [ 3, 1, 4 ], [ 1, 4, 3 ], [ 2, 3, 4 ], [ 2, 1, 3 ], 
      [ 3, 4, 2 ], [ 1, 3, 2 ], [ 4, 2, 3 ], [ 4, 1, 2 ], 
      [ 2, 4, 1 ], [ 3, 2, 1 ], [ 4, 3, 1 ] ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">OrbitsDomain(g,GF(2)^2,[(1,2,3),(1,4)(2,3)],</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">[[[Z(2)^0,Z(2)^0],[Z(2)^0,0*Z(2)]],[[Z(2)^0,0*Z(2)],[0*Z(2),Z(2)^0]]]);</span>
[ [ &lt;an immutable GF2 vector of length 2&gt; ], 
  [ &lt;an immutable GF2 vector of length 2&gt;, 
      &lt;an immutable GF2 vector of length 2&gt;, 
      &lt;an immutable GF2 vector of length 2&gt; ] ]
</pre></div>

<p>(See Section <a href="chap41.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>

<p><a id="X799910CF832EDC45" name="X799910CF832EDC45"></a></p>

<h5>41.4-4 OrbitLength</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OrbitLength</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>, <var class="Arg">pnt</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes the length of the orbit of <var class="Arg">pnt</var> under the action function <var class="Arg">act</var> or <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>) if no action function is given.</p>

<p><a id="X8032F73078DF2DDB" name="X8032F73078DF2DDB"></a></p>

<h5>41.4-5 <span class="Heading">OrbitLengths</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OrbitLengths</code>( <var class="Arg">G</var>, <var class="Arg">seeds</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OrbitLengths</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the lengths of all the orbits of the elements in <var class="Arg">seeds</var> under the action <var class="Arg">act</var> of <var class="Arg">G</var>.</p>

<p><a id="X8520E2487F7E98AF" name="X8520E2487F7E98AF"></a></p>

<h5>41.4-6 <span class="Heading">OrbitLengthsDomain</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OrbitLengthsDomain</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OrbitLengthsDomain</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the lengths of all the orbits of <var class="Arg">G</var> on <var class="Arg">Omega</var>.</p>

<p>This operation is often faster than <code class="func">OrbitLengths</code> (<a href="chap41.html#X8032F73078DF2DDB"><span class="RefLink">41.4-5</span></a>). The domain <var class="Arg">Omega</var> must be closed under the action of <var class="Arg">G</var>, otherwise an error can occur.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,3,2),(2,4,3));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">OrbitLength(g,[1,2,3,4],OnTuples);</span>
12
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">OrbitLengths(g,Arrangements([1..4],4),OnTuples);</span>
[ 12, 12 ]
</pre></div>

<p><a id="X797BD60E7ACEF1B1" name="X797BD60E7ACEF1B1"></a></p>

<h4>41.5 <span class="Heading">Stabilizers</span></h4>

<p>The <em>stabilizer</em> of a point <span class="SimpleMath">x</span> under the action of a group <span class="SimpleMath">G</span> is the set of all those elements in <span class="SimpleMath">G</span> which fix <span class="SimpleMath">x</span>.</p>

<p><a id="X7C34EC437EF598BF" name="X7C34EC437EF598BF"></a></p>

<h5>41.5-1 OrbitStabilizer</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OrbitStabilizer</code>( <var class="Arg">G</var>[, <var class="Arg">Omega</var>], <var class="Arg">pnt</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>], <var class="Arg">act</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes the orbit and the stabilizer of <var class="Arg">pnt</var> simultaneously in a single orbit-stabilizer algorithm.</p>

<p>The stabilizer will have <var class="Arg">G</var> as its parent.</p>

<p><a id="X86FB962786397E02" name="X86FB962786397E02"></a></p>

<h5>41.5-2 Stabilizer</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Stabilizer</code>( <var class="Arg">G</var>[, <var class="Arg">Omega</var>], <var class="Arg">pnt</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the stabilizer in <var class="Arg">G</var> of the point <var class="Arg">pnt</var>, that is the subgroup of those elements of <var class="Arg">G</var> that fix <var class="Arg">pnt</var>. The stabilizer will have <var class="Arg">G</var> as its parent.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,3,2),(2,4,3));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Stabilizer(g,4);</span>
Group([ (1,3,2) ])
</pre></div>

<p>The stabilizer of a set or tuple of points can be computed by specifying an action of sets or tuples of points.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Stabilizer(g,[1,2],OnSets);</span>
Group([ (1,2)(3,4) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Stabilizer(g,[1,2],OnTuples);</span>
Group(())
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">OrbitStabilizer(g,[1,2],OnSets);</span>
rec( 
  orbit := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 3, 4 ], 
      [ 2, 4 ] ], stabilizer := Group([ (1,2)(3,4) ]) )
</pre></div>

<p>(See Section <a href="chap41.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>

<p>The standard methods for all these actions are an orbit-stabilizer algorithm. For permutation groups backtrack algorithms are used. For solvable groups an orbit-stabilizer algorithm for solvable groups, which uses the fact that the orbits of a normal subgroup form a block system (see <a href="chapBib.html#biBSOGOS">[LNS84]</a>) is used.</p>

<p><a id="X78C3A8568414BC44" name="X78C3A8568414BC44"></a></p>

<h5>41.5-3 OrbitStabilizerAlgorithm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; OrbitStabilizerAlgorithm</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>, <var class="Arg">blist</var>, <var class="Arg">gens</var>, <var class="Arg">acts</var>, <var class="Arg">pntact</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This operation should not be called by a user. It is documented however for purposes to extend or maintain the group actions package (the word "package" here refers to the <strong class="pkg">GAP</strong> functionality for group actions, not to a <strong class="pkg">GAP</strong> package).</p>

<p><code class="func">OrbitStabilizerAlgorithm</code> performs an orbit stabilizer algorithm for the group <var class="Arg">G</var> acting with the generators <var class="Arg">gens</var> via the generator images <var class="Arg">gens</var> and the group action <var class="Arg">act</var> on the element <var class="Arg">pnt</var>. (For technical reasons <var class="Arg">pnt</var> and <var class="Arg">act</var> are put in one record with components <code class="code">pnt</code> and <code class="code">act</code> respectively.)</p>

<p>The <var class="Arg">pntact</var> record may carry a component <var class="Arg">stabsub</var>. If given, this must be a subgroup stabilizing <em>all</em> points in the domain and can be used to abbreviate stabilizer calculations.</p>

<p>The <var class="Arg">pntact</var> component also may contain the boolean entry <code class="code">onlystab</code> set to <code class="keyw">true</code>. In this case the <code class="code">orbit</code> component may be omitted from the result.</p>

<p>The argument <var class="Arg">Omega</var> (which may be replaced by <code class="keyw">false</code> to be ignored) is the set within which the orbit is computed (once the orbit is the full domain, the orbit calculation may stop). If <var class="Arg">blist</var> is given it must be a bit list corresponding to <var class="Arg">Omega</var> in which elements which have been found already will be "ticked off" with <code class="keyw">true</code>. (In particular, the entries for the orbit of <var class="Arg">pnt</var> still must be all set to <code class="keyw">false</code>). Again the remaining action domain (the bits set initially to <code class="keyw">false</code>) can be used to stop if the orbit cannot grow any longer. Another use of the bit list is if <var class="Arg">Omega</var> is an enumerator which can determine <code class="func">PositionCanonical</code> (<a href="chap21.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>) values very quickly. In this situation it can be worth to search images not in the orbit found so far, but via their position in <var class="Arg">Omega</var> and use a the bit list to keep track whether the element is in the orbit found so far.</p>

<p><a id="X7A9389097BAF670D" name="X7A9389097BAF670D"></a></p>

<h4>41.6 <span class="Heading">Elements with Prescribed Images</span></h4>

<p><a id="X857DC7B085EB0539" name="X857DC7B085EB0539"></a></p>

<h5>41.6-1 RepresentativeAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RepresentativeAction</code>( <var class="Arg">G</var>[, <var class="Arg">Omega</var>], <var class="Arg">d</var>, <var class="Arg">e</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes an element of <var class="Arg">G</var> that maps <var class="Arg">d</var> to <var class="Arg">e</var> under the given action and returns <code class="keyw">fail</code> if no such element exists.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,3,2),(2,4,3));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RepresentativeAction(g,1,3);</span>
(1,3)(2,4)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RepresentativeAction(g,1,3,OnPoints);</span>
(1,3)(2,4)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RepresentativeAction(g,(1,2,3),(2,4,3));</span>
(1,2,4)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RepresentativeAction(g,(1,2,3),(2,3,4));</span>
fail
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RepresentativeAction(g,Group((1,2,3)),Group((2,3,4)));</span>
(1,2,4)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput"> RepresentativeAction(g,[1,2,3],[1,2,4],OnSets);</span>
(2,4,3)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput"> RepresentativeAction(g,[1,2,3],[1,2,4],OnTuples);</span>
fail
</pre></div>

<p>(See Section <a href="chap41.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>

<p>Again the standard method for <code class="func">RepresentativeAction</code> is an orbit-stabilizer algorithm, for permutation groups and standard actions a backtrack algorithm is used.</p>

<p><a id="X87F73CCA7921DE65" name="X87F73CCA7921DE65"></a></p>

<h4>41.7 <span class="Heading">The Permutation Image of an Action</span></h4>

<p>When a group <span class="SimpleMath">G</span> acts on a domain <span class="SimpleMath">Ω</span>, an enumeration of <span class="SimpleMath">Omega</span> yields a homomorphism from <span class="SimpleMath">G</span> into the symmetric group on <span class="SimpleMath">{ 1, ..., |Ω| }</span>. In <strong class="pkg">GAP</strong>, the enumeration of <span class="SimpleMath">Ω</span> is provided by the <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) value of <span class="SimpleMath">Ω</span> which of course is <span class="SimpleMath">Ω</span> itself if it is a list.</p>

<p>For an action homomorphism, the operation <code class="func">UnderlyingExternalSet</code> (<a href="chap41.html#X8190A8247F29A5C7"><span class="RefLink">41.12-16</span></a>) will return the external set on <span class="SimpleMath">Ω</span> which affords the action.</p>

<p><a id="X78E6A002835288A4" name="X78E6A002835288A4"></a></p>

<h5>41.7-1 <span class="Heading">ActionHomomorphism</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ActionHomomorphism</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>][, <var class="Arg">"surjective"</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ActionHomomorphism</code>( <var class="Arg">xset</var>[, <var class="Arg">"surjective"</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ActionHomomorphism</code>( <var class="Arg">action</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes a homomorphism from <var class="Arg">G</var> into the symmetric group on <span class="SimpleMath">|<var class="Arg">Omega</var>|</span> points that gives the permutation action of <var class="Arg">G</var> on <var class="Arg">Omega</var>.</p>

<p>By default the homomorphism returned by <code class="func">ActionHomomorphism</code> is not necessarily surjective (its <code class="func">Range</code> (<a href="chap32.html#X7B6FD7277CDE9FCB"><span class="RefLink">32.3-7</span></a>) value is the full symmetric group) to avoid unnecessary computation of the image. If the optional string argument <code class="code">"surjective"</code> is given, a surjective homomorphism is created.</p>

<p>The third version (which is supported only for <strong class="pkg">GAP</strong>3 compatibility) returns the action homomorphism that belongs to the image obtained via <code class="func">Action</code> (<a href="chap41.html#X85A8E93D786C3C9C"><span class="RefLink">41.7-2</span></a>).</p>

<p>(See Section <a href="chap41.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples);</span>
&lt;action homomorphism&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Image(hom);</span>
Group(
[ (1,9,13)(2,10,14)(3,7,15)(4,8,16)(5,12,17)(6,11,18)(19,22,23)(20,21,
    24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,15)(14,16)(17,18)(19,
    21)(20,22)(23,24) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Range(hom));Size(Image(hom));</span>
620448401733239439360000
6
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples,</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">"surjective");;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(Range(hom));</span>
6
</pre></div>

<p>When acting on a domain, the operation <code class="func">PositionCanonical</code> (<a href="chap21.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>) is used to determine the position of elements in the domain. This can be used to act on a domain given by a list of representatives for which <code class="func">PositionCanonical</code> (<a href="chap21.html#X7B4B10AE81602D4E"><span class="RefLink">21.16-3</span></a>) is implemented, for example the return value of <code class="func">RightTransversal</code> (<a href="chap39.html#X85C65D06822E716F"><span class="RefLink">39.8-1</span></a>).</p>

<p><a id="X85A8E93D786C3C9C" name="X85A8E93D786C3C9C"></a></p>

<h5>41.7-2 Action</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Action</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Action</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the image group of <code class="func">ActionHomomorphism</code> (<a href="chap41.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>) called with the same parameters.</p>

<p>Note that (for compatibility reasons to be able to get the action homomorphism) this image group internally stores the action homomorphism. If <var class="Arg">G</var> or <var class="Arg">Omega</var> are extremely big, this can cause memory problems. In this case compute only generator images and form the image group yourself.</p>

<p>(See Section <a href="chap41.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>

<p>The following code shows for example how to create the regular action of a group.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Action(g,AsList(g),OnRight);</span>
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
</pre></div>

<p><a id="X86FF54A383B73967" name="X86FF54A383B73967"></a></p>

<h5>41.7-3 SparseActionHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SparseActionHomomorphism</code>( <var class="Arg">G</var>, <var class="Arg">start</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SortedSparseActionHomomorphism</code>( <var class="Arg">G</var>, <var class="Arg">start</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">SparseActionHomomorphism</code> computes the action homomorphism (see <code class="func">ActionHomomorphism</code> (<a href="chap41.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>)) with arguments <var class="Arg">G</var>, <span class="SimpleMath">D</span>, and the optional arguments given, where <span class="SimpleMath">D</span> is the union of the <var class="Arg">G</var>-orbits of all points in <var class="Arg">start</var>. In the <code class="func">Orbit</code> (<a href="chap41.html#X80E0234E7BD79409"><span class="RefLink">41.4-1</span></a>) calls that are used to create <span class="SimpleMath">D</span>, again the optional arguments given are entered.)</p>

<p>If <var class="Arg">G</var> acts on a very large domain not surjectively this may yield a permutation image of substantially smaller degree than by action on the whole domain.</p>

<p>The operation <code class="func">SparseActionHomomorphism</code> will only use <code class="func">\=</code> (<a href="chap31.html#X7EF67D047F03CA6F"><span class="RefLink">31.11-1</span></a>) comparisons of points in the orbit. Therefore it can be used even if no good <code class="func">\&lt;</code> (<a href="chap31.html#X7EF67D047F03CA6F"><span class="RefLink">31.11-1</span></a>) comparison method for these points is available. However the image group will depend on the generators <var class="Arg">gens</var> of <var class="Arg">G</var>.</p>

<p>The operation <code class="func">SortedSparseActionHomomorphism</code> in contrast will sort the orbit and thus produce an image group which does not depend on these generators.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">h:=Group(Z(3)*[[[1,1],[0,1]]]);</span>
Group([ [ [ Z(3), Z(3) ], [ 0*Z(3), Z(3) ] ] ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">hom:=ActionHomomorphism(h,GF(3)^2,OnRight);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Image(hom);</span>
Group([ (2,3)(4,9,6,7,5,8) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">hom:=SparseActionHomomorphism(h,[Z(3)*[1,0]],OnRight);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Image(hom);</span>
Group([ (1,2,3,4,5,6) ])
</pre></div>

<p><a id="X7FED50ED7ACA5FB2" name="X7FED50ED7ACA5FB2"></a></p>

<h4>41.8 <span class="Heading">Action of a group on itself</span></h4>

<p>Of particular importance is the action of a group on its elements or cosets of a subgroup. These actions can be obtained by using <code class="func">ActionHomomorphism</code> (<a href="chap41.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>) for a suitable domain (for example a list of subgroups). For the following (frequently used) types of actions however special (often particularly efficient) functions are provided. A special case is the regular action on all elements.</p>

<p><a id="X78C37C4C7B2BDC44" name="X78C37C4C7B2BDC44"></a></p>

<h5>41.8-1 FactorCosetAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FactorCosetAction</code>( <var class="Arg">G</var>, <var class="Arg">U</var>[, <var class="Arg">N</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This command computes the action of the group <var class="Arg">G</var> on the right cosets of the subgroup <var class="Arg">U</var>. If a normal subgroup <var class="Arg">N</var> of <var class="Arg">G</var> is given, it is stored as kernel of this action.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4,5),(1,2));;u:=SylowSubgroup(g,2);;Index(g,u);</span>
15
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">FactorCosetAction(g,u);</span>
&lt;action epimorphism&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Range(last);</span>
Group([ (1,9,13,10,4)(2,8,14,11,5)(3,7,15,12,6), 
  (1,7)(2,8)(3,9)(5,6)(10,11)(14,15) ])
</pre></div>

<p><a id="X8561DEBA79E01ABD" name="X8561DEBA79E01ABD"></a></p>

<h5>41.8-2 RegularActionHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RegularActionHomomorphism</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an isomorphism from <var class="Arg">G</var> onto the regular permutation representation of <var class="Arg">G</var>.</p>

<p><a id="X835317A7847477D4" name="X835317A7847477D4"></a></p>

<h5>41.8-3 AbelianSubfactorAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AbelianSubfactorAction</code>( <var class="Arg">G</var>, <var class="Arg">M</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a group and <span class="SimpleMath"><var class="Arg">M</var> ≥ <var class="Arg">N</var></span> be subgroups of a common parent that are normal under <var class="Arg">G</var>, such that the subfactor <span class="SimpleMath"><var class="Arg">M</var>/<var class="Arg">N</var></span> is elementary abelian. The operation <code class="func">AbelianSubfactorAction</code> returns a list <code class="code">[ <var class="Arg">phi</var>, <var class="Arg">alpha</var>, <var class="Arg">bas</var> ]</code> where <var class="Arg">bas</var> is a list of elements of <var class="Arg">M</var> which are representatives for a basis of <span class="SimpleMath"><var class="Arg">M</var>/<var class="Arg">N</var></span>, <var class="Arg">alpha</var> is a map from <var class="Arg">M</var> into a <span class="SimpleMath">n</span>-dimensional row space over <span class="SimpleMath">GF(p)</span> where <span class="SimpleMath">[<var class="Arg">M</var>:<var class="Arg">N</var>] = p^n</span> that is the natural homomorphism of <var class="Arg">M</var> by <var class="Arg">N</var> with the quotient represented as an additive group. Finally <var class="Arg">phi</var> is a homomorphism from <var class="Arg">G</var> into <span class="SimpleMath">GL_n(p)</span> that represents the action of <var class="Arg">G</var> on the factor <span class="SimpleMath"><var class="Arg">M</var>/<var class="Arg">N</var></span>.</p>

<p>Note: If only matrices for the action are needed, <code class="func">LinearActionLayer</code> (<a href="chap45.html#X7C2135B98732BBC3"><span class="RefLink">45.14-3</span></a>) might be faster.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,8,10,7,3,5)(2,4,12,9,11,6),</span>
<span class="GAPprompt">&gt;</span> <span class="GAPinput">            (1,9,5,6,3,10)(2,11,12,8,4,7));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">c:=ChiefSeries(g);;List(c,Size);</span>
[ 96, 48, 16, 4, 1 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HasElementaryAbelianFactorGroup(c[3],c[4]);</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">SetName(c[3],"my_group");;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">a:=AbelianSubfactorAction(g,c[3],c[4]);</span>
[ [ (1,8,10,7,3,5)(2,4,12,9,11,6), (1,9,5,6,3,10)(2,11,12,8,4,7) ] -&gt; 
    [ &lt;an immutable 2x2 matrix over GF2&gt;, 
      &lt;an immutable 2x2 matrix over GF2&gt; ], 
  MappingByFunction( my_group, ( GF(2)^
    2 ), function( e ) ... end, function( r ) ... end ), 
  Pcgs([ (2,9,3,8)(4,11,5,10), (1,6,12,7)(4,10,5,11) ]) ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">mat:=Image(a[1],g);</span>
Group([ &lt;an immutable 2x2 matrix over GF2&gt;, 
  &lt;an immutable 2x2 matrix over GF2&gt; ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Size(mat);</span>
3
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">e:=PreImagesRepresentative(a[2],[Z(2),0*Z(2)]);</span>
(2,9,3,8)(4,11,5,10)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">e in c[3];e in c[4];</span>
true
false
</pre></div>

<p><a id="X807AA91E841D132B" name="X807AA91E841D132B"></a></p>

<h4>41.9 <span class="Heading">Permutations Induced by Elements and Cycles</span></h4>

<p>If only the permutation image of a single element is needed, it might not be worth to create the action homomorphism, the following operations yield the permutation image and cycles of a single element.</p>

<p><a id="X7807A33381DCAB26" name="X7807A33381DCAB26"></a></p>

<h5>41.9-1 <span class="Heading">Permutation</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Permutation</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Permutation</code>( <var class="Arg">g</var>, <var class="Arg">xset</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the permutation that corresponds to the action of <var class="Arg">g</var> on the permutation domain <var class="Arg">Omega</var> (a list of objects that are permuted). If an external set <var class="Arg">xset</var> is given, the permutation domain is the <code class="func">HomeEnumerator</code> (<a href="chap41.html#X86A0CC1479A5932A"><span class="RefLink">41.12-5</span></a>) value of this external set (see Section <a href="chap41.html#X7FD3D2D2788709B7"><span class="RefLink">41.12</span></a>). Note that the points of the returned permutation refer to the positions in <var class="Arg">Omega</var>, even if <var class="Arg">Omega</var> itself consists of integers.</p>

<p>If <var class="Arg">g</var> does not leave the domain invariant, or does not map the domain injectively then <code class="keyw">fail</code> is returned.</p>

<p><a id="X81D4EA42810974A0" name="X81D4EA42810974A0"></a></p>

<h5>41.9-2 PermutationCycle</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; PermutationCycle</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>, <var class="Arg">pnt</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes the permutation that represents the cycle of <var class="Arg">pnt</var> under the action of the element <var class="Arg">g</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Permutation([[Z(3),-Z(3)],[Z(3),0*Z(3)]],AsList(GF(3)^2));</span>
(2,7,6)(3,4,8)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Permutation((1,2,3)(4,5)(6,7),[4..7]);</span>
(1,2)(3,4)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">PermutationCycle((1,2,3)(4,5)(6,7),[4..7],4);</span>
(1,2)
</pre></div>

<p><a id="X80AF6E0683CA7F14" name="X80AF6E0683CA7F14"></a></p>

<h5>41.9-3 Cycle</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Cycle</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>, <var class="Arg">pnt</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a list of the points in the cycle of <var class="Arg">pnt</var> under the action of the element <var class="Arg">g</var>.</p>

<p><a id="X7F559E897B333758" name="X7F559E897B333758"></a></p>

<h5>41.9-4 CycleLength</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CycleLength</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>, <var class="Arg">pnt</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the length of the cycle of <var class="Arg">pnt</var> under the action of the element <var class="Arg">g</var>.</p>

<p><a id="X7F3B387A7FD8AE5E" name="X7F3B387A7FD8AE5E"></a></p>

<h5>41.9-5 Cycles</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Cycles</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns a list of the cycles (as lists of points) of the action of the element <var class="Arg">g</var>.</p>

<p><a id="X83040A6080C2C6C6" name="X83040A6080C2C6C6"></a></p>

<h5>41.9-6 CycleLengths</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CycleLengths</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the lengths of all the cycles under the action of the element <var class="Arg">g</var> on <var class="Arg">Omega</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Cycle((1,2,3)(4,5)(6,7),[4..7],4);</span>
[ 4, 5 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CycleLength((1,2,3)(4,5)(6,7),[4..7],4);</span>
2
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Cycles((1,2,3)(4,5)(6,7),[4..7]);</span>
[ [ 4, 5 ], [ 6, 7 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CycleLengths((1,2,3)(4,5)(6,7),[4..7]);</span>
[ 2, 2 ]
</pre></div>

<p><a id="X87FDA6838065CDCB" name="X87FDA6838065CDCB"></a></p>

<h5>41.9-7 <span class="Heading">CycleIndex</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CycleIndex</code>( <var class="Arg">g</var>, <var class="Arg">Omega</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CycleIndex</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">act</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>The <em>cycle index</em> of a permutation <var class="Arg">g</var> acting on <var class="Arg">Omega</var> is defined as</p>

<p class="pcenter">z(<var class="Arg">g</var>) = s_1^{c_1} s_2^{c_2} ⋯ s_n^{c_n}</p>

<p>where <span class="SimpleMath">c_k</span> is the number of <span class="SimpleMath">k</span>-cycles in the cycle decomposition of <var class="Arg">g</var> and the <span class="SimpleMath">s_i</span> are indeterminates.</p>

<p>The <em>cycle index</em> of a group <var class="Arg">G</var> is defined as</p>

<p class="pcenter">Z(<var class="Arg">G</var>) = ( ∑_{g ∈ <var class="Arg">G</var>} z(g) ) / |<var class="Arg">G</var>| .</p>

<p>The indeterminates used by <code class="func">CycleIndex</code> are the indeterminates <span class="SimpleMath">1</span> to <span class="SimpleMath">n</span> over the rationals (see <code class="func">Indeterminate</code> (<a href="chap66.html#X79D0380D7FA39F7D"><span class="RefLink">66.1-1</span></a>)).</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=TransitiveGroup(6,8);</span>
S_4(6c) = 1/2[2^3]S(3)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CycleIndex(g);</span>
1/24*x_1^6+1/8*x_1^2*x_2^2+1/4*x_1^2*x_4+1/4*x_2^3+1/3*x_3^2
</pre></div>

<p><a id="X850A84618421392A" name="X850A84618421392A"></a></p>

<h4>41.10 <span class="Heading">Tests for Actions</span></h4>

<p><a id="X79B15750851828CB" name="X79B15750851828CB"></a></p>

<h5>41.10-1 <span class="Heading">IsTransitive</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsTransitive</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsTransitive</code>( <var class="Arg">xset</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the action implied by the arguments is transitive, or <code class="keyw">false</code> otherwise.</p>

<p>We say that a group <var class="Arg">G</var> acts <em>transitively</em> on a domain <span class="SimpleMath">D</span> if and only if for every pair of points <span class="SimpleMath">d, e ∈ D</span> there is an element <span class="SimpleMath">g</span> in <var class="Arg">G</var> such that <span class="SimpleMath">d^g = e</span>.</p>

<p><a id="X8295D733796B7A37" name="X8295D733796B7A37"></a></p>

<h5>41.10-2 <span class="Heading">Transitivity</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Transitivity</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Transitivity</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the degree <span class="SimpleMath">k</span> (a non-negative integer) of transitivity of the action implied by the arguments, i.e. the largest integer <span class="SimpleMath">k</span> such that the action is <span class="SimpleMath">k</span>-transitive. If the action is not transitive <code class="code">0</code> is returned.</p>

<p>An action is <em><span class="SimpleMath">k</span>-transitive</em> if every <span class="SimpleMath">k</span>-tuple of points can be mapped simultaneously to every other <span class="SimpleMath">k</span>-tuple.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,3,2),(2,4,3));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsTransitive(g,[1..5]);</span>
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Transitivity(g,[1..4]);</span>
2
</pre></div>

<p><em>Note:</em> For permutation groups, the syntax <code class="code">IsTransitive(<var class="Arg">G</var>)</code> is also permitted and tests whether the group is transitive on the points moved by it, that is the group <span class="SimpleMath">⟨ (2,3,4),(2,3) ⟩</span> is transitive (on 3 points).</p>

<p><a id="X8166A6A17C8D6E73" name="X8166A6A17C8D6E73"></a></p>

<h5>41.10-3 <span class="Heading">RankAction</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RankAction</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RankAction</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the rank of a transitive action, i.e. the number of orbits of the point stabilizer.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RankAction(g,Combinations([1..4],2),OnSets);</span>
4
</pre></div>

<p><a id="X7B77040F8543CD6E" name="X7B77040F8543CD6E"></a></p>

<h5>41.10-4 <span class="Heading">IsSemiRegular</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsSemiRegular</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsSemiRegular</code>( <var class="Arg">xset</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the action implied by the arguments is semiregular, or <code class="keyw">false</code> otherwise.</p>

<p>An action is <em>semiregular</em> is the stabilizer of each point is the identity.</p>

<p><a id="X7CF02C4785F0EAB5" name="X7CF02C4785F0EAB5"></a></p>

<h5>41.10-5 <span class="Heading">IsRegular</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsRegular</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsRegular</code>( <var class="Arg">xset</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the action implied by the arguments is regular, or <code class="keyw">false</code> otherwise.</p>

<p>An action is <em>regular</em> if it is both semiregular (see <code class="func">IsSemiRegular</code> (<a href="chap41.html#X7B77040F8543CD6E"><span class="RefLink">41.10-4</span></a>)) and transitive (see <code class="func">IsTransitive</code> (<a href="chap41.html#X79B15750851828CB"><span class="RefLink">41.10-1</span></a>)). In this case every point <var class="Arg">pnt</var> of <var class="Arg">Omega</var> defines a one-to-one correspondence between <var class="Arg">G</var> and <var class="Arg">Omega</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsSemiRegular(g,Arrangements([1..4],3),OnTuples);</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsRegular(g,Arrangements([1..4],3),OnTuples);</span>
false
</pre></div>

<p><a id="X7CB1D74280F92AFC" name="X7CB1D74280F92AFC"></a></p>

<h5>41.10-6 <span class="Heading">Earns</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Earns</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Earns</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of the elementary abelian regular (when acting on <var class="Arg">Omega</var>) normal subgroups of <var class="Arg">G</var>.</p>

<p>At the moment only methods for a primitive group <var class="Arg">G</var> are implemented.</p>

<p><a id="X84C19AD68247B760" name="X84C19AD68247B760"></a></p>

<h5>41.10-7 <span class="Heading">IsPrimitive</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsPrimitive</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsPrimitive</code>( <var class="Arg">xset</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the action implied by the arguments is primitive, or <code class="keyw">false</code> otherwise.</p>

<p>An action is <em>primitive</em> if it is transitive and the action admits no nontrivial block systems. See <a href="chap41.html#X7E9D3D0B7A9A8572"><span class="RefLink">41.11</span></a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">IsPrimitive(g,Orbit(g,(1,2)(3,4)));</span>
true
</pre></div>

<p><a id="X7E9D3D0B7A9A8572" name="X7E9D3D0B7A9A8572"></a></p>

<h4>41.11 <span class="Heading">Block Systems</span></h4>

<p>A <em>block system</em> (system of imprimitivity) for the action of a group <span class="SimpleMath">G</span> on an action domain <span class="SimpleMath">Ω</span> is a partition of <span class="SimpleMath">Ω</span> which –as a partition– remains invariant under the action of <span class="SimpleMath">G</span>.</p>

<p><a id="X84FE699F85371643" name="X84FE699F85371643"></a></p>

<h5>41.11-1 <span class="Heading">Blocks</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Blocks</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">seed</var>][, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; Blocks</code>( <var class="Arg">xset</var>[, <var class="Arg">seed</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes a block system for the action. If <var class="Arg">seed</var> is not given and the action is imprimitive, a minimal nontrivial block system will be found. If <var class="Arg">seed</var> is given, a block system in which <var class="Arg">seed</var> is the subset of one block is computed. The action must be transitive.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=TransitiveGroup(8,3);</span>
E(8)=2[x]2[x]2
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Blocks(g,[1..8]);</span>
[ [ 1, 8 ], [ 2, 3 ], [ 4, 5 ], [ 6, 7 ] ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Blocks(g,[1..8],[1,4]);</span>
[ [ 1, 4 ], [ 2, 7 ], [ 3, 6 ], [ 5, 8 ] ]
</pre></div>

<p>(See Section <a href="chap41.html#X81B8F9CD868CD953"><span class="RefLink">41.2</span></a> for information about specific actions.)</p>

<p><a id="X79936EB97AAD1144" name="X79936EB97AAD1144"></a></p>

<h5>41.11-2 <span class="Heading">MaximalBlocks</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MaximalBlocks</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">seed</var>][, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; MaximalBlocks</code>( <var class="Arg">xset</var>[, <var class="Arg">seed</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a block system that is maximal (i.e., blocks are maximal with respect to inclusion) for the action of <var class="Arg">G</var> on <var class="Arg">Omega</var>. If <var class="Arg">seed</var> is given, a block system is computed in which <var class="Arg">seed</var> is a subset of one block.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">MaximalBlocks(g,[1..8]);</span>
[ [ 1, 2, 3, 8 ], [ 4, 5, 6, 7 ] ]
</pre></div>

<p><a id="X7941DB6380B74510" name="X7941DB6380B74510"></a></p>

<h5>41.11-3 <span class="Heading">RepresentativesMinimalBlocks</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RepresentativesMinimalBlocks</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; RepresentativesMinimalBlocks</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes a list of block representatives for all minimal (i.e blocks are minimal with respect to inclusion) nontrivial block systems for the action.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">RepresentativesMinimalBlocks(g,[1..8]);</span>
[ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], 
  [ 1, 8 ] ]
</pre></div>

<p><a id="X835658B07B28EF3B" name="X835658B07B28EF3B"></a></p>

<h5>41.11-4 AllBlocks</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; AllBlocks</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes a list of representatives of all block systems for a permutation group <var class="Arg">G</var> acting transitively on the points moved by the group.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">AllBlocks(g);</span>
[ [ 1, 8 ], [ 1, 2, 3, 8 ], [ 1, 4, 5, 8 ], [ 1, 6, 7, 8 ], [ 1, 3 ], 
  [ 1, 3, 5, 7 ], [ 1, 3, 4, 6 ], [ 1, 5 ], [ 1, 2, 5, 6 ], [ 1, 2 ], 
  [ 1, 2, 4, 7 ], [ 1, 4 ], [ 1, 7 ], [ 1, 6 ] ]
</pre></div>

<p>The stabilizer of a block can be computed via the action <code class="func">OnSets</code> (<a href="chap41.html#X85AA04347CD117F9"><span class="RefLink">41.2-4</span></a>):</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Stabilizer(g,[1,8],OnSets);</span>
Group([ (1,8)(2,3)(4,5)(6,7) ])
</pre></div>

<p>If <code class="code">bs</code> is a partition of the action domain, given as a set of sets, the stabilizer under the action <code class="func">OnSetsDisjointSets</code> (<a href="chap41.html#X7E23686E7A9D3A20"><span class="RefLink">41.2-8</span></a>) returns the largest subgroup which preserves <code class="code">bs</code> as a block system.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3,4,5,6,7,8),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">bs:=[[1,2,3,4],[5,6,7,8]];;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Stabilizer(g,bs,OnSetsDisjointSets);</span>
Group([ (6,7), (5,6), (5,8), (2,3), (3,4)(5,7), (1,4), 
  (1,5,4,8)(2,6,3,7) ])
</pre></div>

<p><a id="X7FD3D2D2788709B7" name="X7FD3D2D2788709B7"></a></p>

<h4>41.12 <span class="Heading">External Sets</span></h4>

<p>When considering group actions, sometimes the concept of a <em><span class="SimpleMath">G</span>-set</em> is used. This is a set <span class="SimpleMath">Ω</span> endowed with an action of <span class="SimpleMath">G</span>. The elements of the <span class="SimpleMath">G</span>-set are the same as those of <span class="SimpleMath">Ω</span>, however concepts like equality and equivalence of <span class="SimpleMath">G</span>-sets do not only consider the underlying domain <span class="SimpleMath">Ω</span> but the group action as well.</p>

<p>This concept is implemented in <strong class="pkg">GAP</strong> via <em>external sets</em>.</p>

<p>The constituents of an external set are stored in the attributes <code class="func">ActingDomain</code> (<a href="chap41.html#X7B9DB15D80CE28B4"><span class="RefLink">41.12-3</span></a>), <code class="func">FunctionAction</code> (<a href="chap41.html#X86153CB087394DC1"><span class="RefLink">41.12-4</span></a>) and <code class="func">HomeEnumerator</code> (<a href="chap41.html#X86A0CC1479A5932A"><span class="RefLink">41.12-5</span></a>).</p>

<p>Most operations for actions are applicable as an attribute for an external set.</p>

<p>The most prominent external subsets are orbits, see <code class="func">ExternalOrbit</code> (<a href="chap41.html#X7FB656AE7A066C35"><span class="RefLink">41.12-9</span></a>).</p>

<p>Many subsets of a group, such as conjugacy classes or cosets (see <code class="func">ConjugacyClass</code> (<a href="chap39.html#X7B2F207F7F85F5B8"><span class="RefLink">39.10-1</span></a>) and <code class="func">RightCoset</code> (<a href="chap39.html#X8412ABD57986B9FC"><span class="RefLink">39.7-1</span></a>)) are implemented as external orbits.</p>

<p>External sets also are implicitly underlying action homomorphisms, see <code class="func">UnderlyingExternalSet</code> (<a href="chap41.html#X8190A8247F29A5C7"><span class="RefLink">41.12-16</span></a>) and <code class="func">SurjectiveActionHomomorphismAttr</code> (<a href="chap41.html#X7A3D87DE809FBFD4"><span class="RefLink">41.12-17</span></a>).</p>

<p><a id="X8264C3C479FF0A8B" name="X8264C3C479FF0A8B"></a></p>

<h5>41.12-1 IsExternalSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsExternalSet</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>An <em>external set</em> specifies a group action <span class="SimpleMath">μ: Ω × G ↦ Ω</span> of a group <span class="SimpleMath">G</span> on a domain <span class="SimpleMath">Ω</span>. The external set knows the group, the domain and the actual acting function. Mathematically, an external set is the set <span class="SimpleMath">Ω</span>, which is endowed with the action of a group <span class="SimpleMath">G</span> via the group action <span class="SimpleMath">μ</span>. For this reason <strong class="pkg">GAP</strong> treats an external set as a domain whose elements are the elements of <span class="SimpleMath">Ω</span>. An external set is always a union of orbits. Currently the domain <span class="SimpleMath">Ω</span> must always be finite. If <span class="SimpleMath">Ω</span> is not a list, an enumerator for <span class="SimpleMath">Ω</span> is automatically chosen, see <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>).</p>

<p><a id="X7C90F648793E47DD" name="X7C90F648793E47DD"></a></p>

<h5>41.12-2 ExternalSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ExternalSet</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>creates the external set for the action <var class="Arg">act</var> of <var class="Arg">G</var> on <var class="Arg">Omega</var>. <var class="Arg">Omega</var> can be either a proper set, or a domain which is represented as described in <a href="chap12.html#X7BAF69417BB925F6"><span class="RefLink">12.4</span></a> and <a href="chap30.html#X8050A8037984E5B6"><span class="RefLink">30</span></a>, or (to use less memory but with a slower performance) an enumerator (see <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) ) of this domain.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3),(2,3,4));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">e:=ExternalSet(g,[1..4]);</span>
&lt;xset:[ 1, 2, 3, 4 ]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">e:=ExternalSet(g,g,OnRight);</span>
&lt;xset:[ (), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2), 
  (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3) ]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Orbits(e);</span>
[ [ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,4,2), 
      (1,2,3), (1,3,4), (2,3,4), (1,3,2), (1,4,3), (1,2,4) ] ]
</pre></div>

<p><a id="X7B9DB15D80CE28B4" name="X7B9DB15D80CE28B4"></a></p>

<h5>41.12-3 ActingDomain</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ActingDomain</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute returns the group with which the external set <var class="Arg">xset</var> was defined.</p>

<p><a id="X86153CB087394DC1" name="X86153CB087394DC1"></a></p>

<h5>41.12-4 FunctionAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; FunctionAction</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the acting function with which the external set <var class="Arg">xset</var> was defined.</p>

<p><a id="X86A0CC1479A5932A" name="X86A0CC1479A5932A"></a></p>

<h5>41.12-5 HomeEnumerator</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; HomeEnumerator</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an enumerator of the action domain with which the external set <var class="Arg">xset</var> was defined. For external subsets, this is in general different from the <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) value of <var class="Arg">xset</var>, which enumerates only the subset.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ActingDomain(e);</span>
Group([ (1,2,3), (2,3,4) ])
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">FunctionAction(e)=OnRight;</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HomeEnumerator(e);</span>
[ (), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2), 
  (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3) ]
</pre></div>

<p><a id="X879DE63C7858453C" name="X879DE63C7858453C"></a></p>

<h5>41.12-6 IsExternalSubset</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsExternalSubset</code>( <var class="Arg">obj</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>An external subset is the restriction of an external set to a subset of the domain (which must be invariant under the action). It is again an external set.</p>

<p><a id="X87D1EA1486D86233" name="X87D1EA1486D86233"></a></p>

<h5>41.12-7 ExternalSubset</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ExternalSubset</code>( <var class="Arg">G</var>, <var class="Arg">xset</var>, <var class="Arg">start</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>], <var class="Arg">act</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>constructs the external subset of <var class="Arg">xset</var> on the union of orbits of the points in <var class="Arg">start</var>.</p>

<p><a id="X7E081F568407317F" name="X7E081F568407317F"></a></p>

<h5>41.12-8 IsExternalOrbit</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsExternalOrbit</code>( <var class="Arg">obj</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<p>An external orbit is an external subset consisting of one orbit.</p>

<p><a id="X7FB656AE7A066C35" name="X7FB656AE7A066C35"></a></p>

<h5>41.12-9 ExternalOrbit</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ExternalOrbit</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>, <var class="Arg">pnt</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>], <var class="Arg">act</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>constructs the external subset on the orbit of <var class="Arg">pnt</var>. The <code class="func">Representative</code> (<a href="chap30.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>) value of this external set is <var class="Arg">pnt</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">e:=ExternalOrbit(g,g,(1,2,3));</span>
(1,2,3)^G
</pre></div>

<p><a id="X7BAFF02B7D6DF9F2" name="X7BAFF02B7D6DF9F2"></a></p>

<h5>41.12-10 StabilizerOfExternalSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; StabilizerOfExternalSet</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the stabilizer of the <code class="func">Representative</code> (<a href="chap30.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>) value of the external set <var class="Arg">xset</var>. The stabilizer will have the acting group of <var class="Arg">xset</var> as its parent.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Representative(e);</span>
(1,2,3)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">StabilizerOfExternalSet(e);</span>
Group([ (1,2,3) ])
</pre></div>

<p><a id="X867262FA82FDD592" name="X867262FA82FDD592"></a></p>

<h5>41.12-11 <span class="Heading">ExternalOrbits</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ExternalOrbits</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ExternalOrbits</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes a list of external orbits that give the orbits of <var class="Arg">G</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ExternalOrbits(g,AsList(g));</span>
[ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]
</pre></div>

<p><a id="X7A64EF807CE8893E" name="X7A64EF807CE8893E"></a></p>

<h5>41.12-12 <span class="Heading">ExternalOrbitsStabilizers</span></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ExternalOrbitsStabilizers</code>( <var class="Arg">G</var>, <var class="Arg">Omega</var>[, <var class="Arg">gens</var>, <var class="Arg">acts</var>][, <var class="Arg">act</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ExternalOrbitsStabilizers</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>In addition to <code class="func">ExternalOrbits</code> (<a href="chap41.html#X867262FA82FDD592"><span class="RefLink">41.12-11</span></a>), this operation also computes the stabilizers of the representatives of the external orbits at the same time. (This can be quicker than computing the <code class="func">ExternalOrbits</code> (<a href="chap41.html#X867262FA82FDD592"><span class="RefLink">41.12-11</span></a>) value first and the stabilizers afterwards.)</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">e:=ExternalOrbitsStabilizers(g,AsList(g));</span>
[ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">HasStabilizerOfExternalSet(e[3]);</span>
true
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">StabilizerOfExternalSet(e[3]);</span>
Group([ (2,4,3) ])
</pre></div>

<p><a id="X8048AE727A7F1A2F" name="X8048AE727A7F1A2F"></a></p>

<h5>41.12-13 CanonicalRepresentativeOfExternalSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanonicalRepresentativeOfExternalSet</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The canonical representative of an external set <var class="Arg">xset</var> may only depend on the defining attributes <var class="Arg">G</var>, <var class="Arg">Omega</var>, <var class="Arg">act</var> of <var class="Arg">xset</var> and (in the case of external subsets) <code class="code">Enumerator( <var class="Arg">xset</var> )</code>. It must <em>not</em> depend, e.g., on the representative of an external orbit. <strong class="pkg">GAP</strong> does not know methods for arbitrary external sets to compute a canonical representative, see <code class="func">CanonicalRepresentativeDeterminatorOfExternalSet</code> (<a href="chap41.html#X8071A8D784DC8325"><span class="RefLink">41.12-14</span></a>).</p>

<p><a id="X8071A8D784DC8325" name="X8071A8D784DC8325"></a></p>

<h5>41.12-14 CanonicalRepresentativeDeterminatorOfExternalSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; CanonicalRepresentativeDeterminatorOfExternalSet</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a function that takes as its arguments the acting group and a point. This function returns a list of length 1 or 3, the first entry being the canonical representative and the other entries (if bound) being the stabilizer of the canonical representative and a conjugating element, respectively. An external set is only guaranteed to be able to compute a canonical representative if it has a <code class="func">CanonicalRepresentativeDeterminatorOfExternalSet</code>.</p>

<p><a id="X85E9A6A77B8D00B8" name="X85E9A6A77B8D00B8"></a></p>

<h5>41.12-15 ActorOfExternalSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ActorOfExternalSet</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an element mapping <code class="code">Representative(<var class="Arg">xset</var>)</code> to <code class="code">CanonicalRepresentativeOfExternalSet(<var class="Arg">xset</var>)</code> under the given action.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">u:=Subgroup(g,[(1,2,3)]);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">e:=RightCoset(u,(1,2)(3,4));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">CanonicalRepresentativeOfExternalSet(e);</span>
(2,4,3)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">ActorOfExternalSet(e);</span>
(1,3,2)
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">FunctionAction(e)((1,2)(3,4),last);</span>
(2,4,3)
</pre></div>

<p><a id="X8190A8247F29A5C7" name="X8190A8247F29A5C7"></a></p>

<h5>41.12-16 UnderlyingExternalSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; UnderlyingExternalSet</code>( <var class="Arg">acthom</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The underlying set of an action homomorphism <var class="Arg">acthom</var> is the external set on which it was defined.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">g:=Group((1,2,3),(1,2));;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples);;</span>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">s:=UnderlyingExternalSet(hom);</span>
&lt;xset:[[ 1, 2, 3 ],[ 1, 2, 4 ],[ 1, 3, 2 ],[ 1, 3, 4 ],[ 1, 4, 2 ],
[ 1, 4, 3 ],[ 2, 1, 3 ],[ 2, 1, 4 ],[ 2, 3, 1 ],[ 2, 3, 4 ],
[ 2, 4, 1 ],[ 2, 4, 3 ],[ 3, 1, 2 ],[ 3, 1, 4 ],[ 3, 2, 1 ], ...]&gt;
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Print(s,"\n");</span>
[ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 2 ], [ 1, 3, 4 ], [ 1, 4, 2 ], 
  [ 1, 4, 3 ], [ 2, 1, 3 ], [ 2, 1, 4 ], [ 2, 3, 1 ], [ 2, 3, 4 ], 
  [ 2, 4, 1 ], [ 2, 4, 3 ], [ 3, 1, 2 ], [ 3, 1, 4 ], [ 3, 2, 1 ], 
  [ 3, 2, 4 ], [ 3, 4, 1 ], [ 3, 4, 2 ], [ 4, 1, 2 ], [ 4, 1, 3 ], 
  [ 4, 2, 1 ], [ 4, 2, 3 ], [ 4, 3, 1 ], [ 4, 3, 2 ] ]
</pre></div>

<p><a id="X7A3D87DE809FBFD4" name="X7A3D87DE809FBFD4"></a></p>

<h5>41.12-17 SurjectiveActionHomomorphismAttr</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; SurjectiveActionHomomorphismAttr</code>( <var class="Arg">xset</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an action homomorphism for the external set <var class="Arg">xset</var> which is surjective. (As the <code class="func">Image</code> (<a href="chap32.html#X87F4D35A826599C6"><span class="RefLink">32.4-6</span></a>) value of this homomorphism has to be computed to obtain the range, this may take substantially longer than <code class="func">ActionHomomorphism</code> (<a href="chap41.html#X78E6A002835288A4"><span class="RefLink">41.7-1</span></a>).)</p>


<div class="chlinkprevnextbot">&nbsp;<a href="chap0.html">[Top of Book]</a>&nbsp;  <a href="chap0.html#contents">[Contents]</a>&nbsp;  &nbsp;<a href="chap40.html">[Previous Chapter]</a>&nbsp;  &nbsp;<a href="chap42.html">[Next Chapter]</a>&nbsp;  </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chap16.html">16</a>  <a href="chap17.html">17</a>  <a href="chap18.html">18</a>  <a href="chap19.html">19</a>  <a href="chap20.html">20</a>  <a href="chap21.html">21</a>  <a href="chap22.html">22</a>  <a href="chap23.html">23</a>  <a href="chap24.html">24</a>  <a href="chap25.html">25</a>  <a href="chap26.html">26</a>  <a href="chap27.html">27</a>  <a href="chap28.html">28</a>  <a href="chap29.html">29</a>  <a href="chap30.html">30</a>  <a href="chap31.html">31</a>  <a href="chap32.html">32</a>  <a href="chap33.html">33</a>  <a href="chap34.html">34</a>  <a href="chap35.html">35</a>  <a href="chap36.html">36</a>  <a href="chap37.html">37</a>  <a href="chap38.html">38</a>  <a href="chap39.html">39</a>  <a href="chap40.html">40</a>  <a href="chap41.html">41</a>  <a href="chap42.html">42</a>  <a href="chap43.html">43</a>  <a href="chap44.html">44</a>  <a href="chap45.html">45</a>  <a href="chap46.html">46</a>  <a href="chap47.html">47</a>  <a href="chap48.html">48</a>  <a href="chap49.html">49</a>  <a href="chap50.html">50</a>  <a href="chap51.html">51</a>  <a href="chap52.html">52</a>  <a href="chap53.html">53</a>  <a href="chap54.html">54</a>  <a href="chap55.html">55</a>  <a href="chap56.html">56</a>  <a href="chap57.html">57</a>  <a href="chap58.html">58</a>  <a href="chap59.html">59</a>  <a href="chap60.html">60</a>  <a href="chap61.html">61</a>  <a href="chap62.html">62</a>  <a href="chap63.html">63</a>  <a href="chap64.html">64</a>  <a href="chap65.html">65</a>  <a href="chap66.html">66</a>  <a href="chap67.html">67</a>  <a href="chap68.html">68</a>  <a href="chap69.html">69</a>  <a href="chap70.html">70</a>  <a href="chap71.html">71</a>  <a href="chap72.html">72</a>  <a href="chap73.html">73</a>  <a href="chap74.html">74</a>  <a href="chap75.html">75</a>  <a href="chap76.html">76</a>  <a href="chap77.html">77</a>  <a href="chap78.html">78</a>  <a href="chap79.html">79</a>  <a href="chap80.html">80</a>  <a href="chap81.html">81</a>  <a href="chap82.html">82</a>  <a href="chap83.html">83</a>  <a href="chap84.html">84</a>  <a href="chap85.html">85</a>  <a href="chap86.html">86</a>  <a href="chap87.html">87</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>