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++ Authors: M. Weller, G. Schneider, J. Grabmeier
++ Date Created: 20 February 1990
++ Date Last Updated: 09 June 1990
++ References:
++ J. Conway, R. Curtis, S. Norton, R. Parker, R. Wilson:
++ Atlas of Finite Groups, Oxford, Clarendon Press, 1987
++ Description:
++ PermutationGroupExamples provides permutation groups for
++ some classes of groups: symmetric, alternating, dihedral, cyclic,
++ direct products of cyclic, which are in fact the finite abelian groups
++ of symmetric groups called Young subgroups.
++ Furthermore, Rubik's group as permutation group of 48 integers and a list
++ of sporadic simple groups derived from the atlas of finite groups.
PermutationGroupExamples() : SIG == CODE where
L ==> List
I ==> Integer
PI ==> PositiveInteger
NNI ==> NonNegativeInteger
PERM ==> Permutation
PERMGRP ==> PermutationGroup
SIG ==> with
symmetricGroup : PI -> PERMGRP I
++ symmetricGroup(n) constructs the symmetric group Sn
++ acting on the integers 1,...,n, generators are the
++ n-cycle (1,...,n) and the 2-cycle (1,2).
symmetricGroup : L I -> PERMGRP I
++ symmetricGroup(li) constructs the symmetric group acting on
++ the integers in the list li, generators are the
++ cycle given by li and the 2-cycle (li.1,li.2).
++ Note that duplicates in the list will be removed.
alternatingGroup : PI -> PERMGRP I
++ alternatingGroup(n) constructs the alternating group An
++ acting on the integers 1,...,n, generators are in general the
++ n-2-cycle (3,...,n) and the 3-cycle (1,2,3)
++ if n is odd and the product of the 2-cycle (1,2) with
++ n-2-cycle (3,...,n) and the 3-cycle (1,2,3)
++ if n is even.
alternatingGroup : L I -> PERMGRP I
++ alternatingGroup(li) constructs the alternating group acting
++ on the integers in the list li, generators are in general the
++ n-2-cycle (li.3,...,li.n) and the 3-cycle
++ (li.1,li.2,li.3), if n is odd and
++ product of the 2-cycle (li.1,li.2) with
++ n-2-cycle (li.3,...,li.n) and the 3-cycle
++ (li.1,li.2,li.3), if n is even.
++ Note that duplicates in the list will be removed.
abelianGroup : L PI -> PERMGRP I
++ abelianGroup([n1,...,nk]) constructs the abelian group that
++ is the direct product of cyclic groups with order ni.
cyclicGroup : PI -> PERMGRP I
++ cyclicGroup(n) constructs the cyclic group of order n acting
++ on the integers 1,...,n.
cyclicGroup : L I -> PERMGRP I
++ cyclicGroup([i1,...,ik]) constructs the cyclic group of
++ order k acting on the integers i1,...,ik.
++ Note that duplicates in the list will be removed.
dihedralGroup : PI -> PERMGRP I
++ dihedralGroup(n) constructs the dihedral group of order 2n
++ acting on integers 1,...,N.
dihedralGroup : L I -> PERMGRP I
++ dihedralGroup([i1,...,ik]) constructs the dihedral group of
++ order 2k acting on the integers out of i1,...,ik.
++ Note that duplicates in the list will be removed.
mathieu11 : L I -> PERMGRP I
++ mathieu11(li) constructs the mathieu group acting on the 11
++ integers given in the list li.
++ Note that duplicates in the list will be removed.
++ error, if li has less or more than 11 different entries.
mathieu11 : () -> PERMGRP I
++ mathieu11 constructs the mathieu group acting on the
++ integers 1,...,11.
mathieu12 : L I -> PERMGRP I
++ mathieu12(li) constructs the mathieu group acting on the 12
++ integers given in the list li.
++ Note that duplicates in the list will be removed
++ Error: if li has less or more than 12 different entries.
mathieu12 : () -> PERMGRP I
++ mathieu12 constructs the mathieu group acting on the
++ integers 1,...,12.
mathieu22 : L I -> PERMGRP I
++ mathieu22(li) constructs the mathieu group acting on the 22
++ integers given in the list li.
++ Note that duplicates in the list will be removed.
++ Error: if li has less or more than 22 different entries.
mathieu22 : () -> PERMGRP I
++ mathieu22 constructs the mathieu group acting on the
++ integers 1,...,22.
mathieu23 : L I -> PERMGRP I
++ mathieu23(li) constructs the mathieu group acting on the 23
++ integers given in the list li.
++ Note that duplicates in the list will be removed.
++ Error: if li has less or more than 23 different entries.
mathieu23 : () -> PERMGRP I
++ mathieu23 constructs the mathieu group acting on the
++ integers 1,...,23.
mathieu24 : L I -> PERMGRP I
++ mathieu24(li) constructs the mathieu group acting on the 24
++ integers given in the list li.
++ Note that duplicates in the list will be removed.
++ Error: if li has less or more than 24 different entries.
mathieu24 : () -> PERMGRP I
++ mathieu24 constructs the mathieu group acting on the
++ integers 1,...,24.
janko2 : L I -> PERMGRP I
++ janko2(li) constructs the janko group acting on the 100
++ integers given in the list li.
++ Note that duplicates in the list will be removed.
++ Error: if li has less or more than 100 different entries
janko2 : () -> PERMGRP I
++ janko2 constructs the janko group acting on the
++ integers 1,...,100.
rubiksGroup : () -> PERMGRP I
++ rubiksGroup constructs the permutation group representing
++ Rubic's Cube acting on integers 10*i+j for
++ 1 <= i <= 6, 1 <= j <= 8.
++ The faces of Rubik's Cube are labelled in the obvious way
++ Front, Right, Up, Down, Left, Back and numbered from 1 to 6
++ in this given ordering, the pieces on each face
++ (except the unmoveable center piece) are clockwise numbered
++ from 1 to 8 starting with the piece in the upper left
++ corner. The moves of the cube are represented as permutations
++ on these pieces, represented as a two digit
++ integer ij where i is the numer of theface (1 to 6)
++ and j is the number of the piece on this face.
++ The remaining ambiguities are resolved by looking
++ at the 6 generators, which represent a 90 degree turns of the
++ faces, or from the following pictorial description.
++ Permutation group representing Rubic's Cube acting on integers
++ 10*i+j for 1 <= i <= 6, 1 <= j <=8.
++
++ \begin{verbatim}
++ Rubik's Cube: +-----+ +-- B where: marks Side # :
++ / U /|/
++ / / | F(ront) <-> 1
++ L --> +-----+ R| R(ight) <-> 2
++ | | + U(p) <-> 3
++ | F | / D(own) <-> 4
++ | |/ L(eft) <-> 5
++ +-----+ B(ack) <-> 6
++ ^
++ |
++ D
++
++ The Cube's surface:
++ The pieces on each side
++ +---+ (except the unmoveable center
++ |567| piece) are clockwise numbered
++ |4U8| from 1 to 8 starting with the
++ |321| piece in the upper left
++ +---+---+---+ corner (see figure on the
++ |781|123|345| left). The moves of the cube
++ |6L2|8F4|2R6| are represented as
++ |543|765|187| permutations on these pieces.
++ +---+---+---+ Each of the pieces is
++ |123| represented as a two digit
++ |8D4| integer ij where i is the
++ |765| # of the side ( 1 to 6 for
++ +---+ F to B (see table above ))
++ |567| and j is the # of the piece.
++ |4B8|
++ |321|
++ +---+
++ \end{verbatim}
youngGroup : L I -> PERMGRP I
++ youngGroup([n1,...,nk]) constructs the direct product of the
++ symmetric groups Sn1,...,Snk.
youngGroup : Partition -> PERMGRP I
++ youngGroup(lambda) constructs the direct product of the symmetric
++ groups given by the parts of the partition lambda.
CODE ==> add
-- import the permutation and permutation group domains:
import PERM I
import PERMGRP I
-- import the needed map function:
import ListFunctions2(L L I,PERM I)
-- the internal functions:
llli2gp(l:L L L I):PERMGRP I ==
--++ Converts an list of permutations each represented by a list
--++ of cycles ( each of them represented as a list of Integers )
--++ to the permutation group generated by these permutations.
(map(cycles,l))::PERMGRP I
li1n(n:I):L I ==
--++ constructs the list of integers from 1 to n
[i for i in 1..n]
-- definition of the exported functions:
youngGroup(l:L I):PERMGRP I ==
gens:= nil()$(L L L I)
element:I:= 1
for n in l | n > 1 repeat
gens:=cons(list [i for i in element..(element+n-1)], gens)
if n >= 3 then gens := cons([[element,element+1]],gens)
element:=element+n
llli2gp
#gens = 0 => [[[1]]]
gens
youngGroup(lambda : Partition):PERMGRP I ==
youngGroup(convert(lambda)$Partition)
rubiksGroup():PERMGRP I ==
-- each generator represents a 90 degree turn of the appropriate
-- side.
f:L L I:=
[[11,13,15,17],[12,14,16,18],[51,31,21,41],_
[53,33,23,43],[52,32,22,42]]
r:L L I:=
[[21,23,25,27],[22,24,26,28],[13,37,67,43],_
[15,31,61,45],[14,38,68,44]]
u:L L I:=
[[31,33,35,37],[32,34,36,38],[13,51,63,25],_
[11,57,61,23],[12,58,62,24]]
d:L L I:=
[[41,43,45,47],[42,44,46,48],[17,21,67,55],_
[15,27,65,53],[16,28,66,54]]
l:L L I:=
[[51,53,55,57],[52,54,56,58],[11,41,65,35],_
[17,47,63,33],[18,48,64,34]]
b:L L I:=
[[61,63,65,67],[62,64,66,68],[45,25,35,55],_
[47,27,37,57],[46,26,36,56]]
llli2gp [f,r,u,d,l,b]
mathieu11(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ^= 11 => error "Exactly 11 integers for mathieu11 needed !"
a:L L I:=[[l.1,l.10],[l.2,l.8],[l.3,l.11],[l.5,l.7]]
llli2gp [a,[[l.1,l.4,l.7,l.6],[l.2,l.11,l.10,l.9]]]
mathieu11():PERMGRP I == mathieu11 li1n 11
mathieu12(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ^= 12 => error "Exactly 12 integers for mathieu12 needed !"
a:L L I:=
[[l.1,l.2,l.3,l.4,l.5,l.6,l.7,l.8,l.9,l.10,l.11]]
llli2gp [a,[[l.1,l.6,l.5,l.8,l.3,l.7,l.4,l.2,l.9,l.10],[l.11,l.12]]]
mathieu12():PERMGRP I == mathieu12 li1n 12
mathieu22(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ^= 22 => error "Exactly 22 integers for mathieu22 needed !"
a:L L I:=[[l.1,l.2,l.4,l.8,l.16,l.9,l.18,l.13,l.3,l.6,l.12], _
[l.5,l.10,l.20,l.17,l.11,l.22,l.21,l.19,l.15,l.7,l.14]]
b:L L I:= [[l.1,l.2,l.6,l.18],[l.3,l.15],[l.5,l.8,l.21,l.13], _
[l.7,l.9,l.20,l.12],[l.10,l.16],[l.11,l.19,l.14,l.22]]
llli2gp [a,b]
mathieu22():PERMGRP I == mathieu22 li1n 22
mathieu23(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ^= 23 => error "Exactly 23 integers for mathieu23 needed !"
a:L L I:= [[l.1,l.2,l.3,l.4,l.5,l.6,l.7,l.8,l.9,l.10,_
l.11,l.12,l.13,l.14,_
l.15,l.16,l.17,l.18,l.19,l.20,l.21,l.22,l.23]]
b:L L I:= [[l.2,l.16,l.9,l.6,l.8],[l.3,l.12,l.13,l.18,l.4], _
[l.7,l.17,l.10,l.11,l.22],[l.14,l.19,l.21,l.20,l.15]]
llli2gp [a,b]
mathieu23():PERMGRP I == mathieu23 li1n 23
mathieu24(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ^= 24 => error "Exactly 24 integers for mathieu24 needed !"
a:L L I:=[[l.1,l.16,l.10,l.22,l.24],[l.2,l.12,l.18,l.21,l.7], _
[l.4,l.5,l.8,l.6,l.17],[l.9,l.11,l.13,l.19,l.15]]
b:L L I:=[[l.1,l.22,l.13,l.14,l.6,l.20,l.3,l.21,l.8,l.11],[l.2,l.10], _
[l.4,l.15,l.18,l.17,l.16,l.5,l.9,l.19,l.12,l.7],[l.23,l.24]]
llli2gp [a,b]
mathieu24():PERMGRP I == mathieu24 li1n 24
janko2(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ^= 100 => error "Exactly 100 integers for janko2 needed !"
a:L L I:=[ _
[l.2,l.3,l.4,l.5,l.6,l.7,l.8], _
[l.9,l.10,l.11,l.12,l.13,l.14,l.15], _
[l.16,l.17,l.18,l.19,l.20,l.21,l.22], _
[l.23,l.24,l.25,l.26,l.27,l.28,l.29], _
[l.30,l.31,l.32,l.33,l.34,l.35,l.36], _
[l.37,l.38,l.39,l.40,l.41,l.42,l.43], _
[l.44,l.45,l.46,l.47,l.48,l.49,l.50], _
[l.51,l.52,l.53,l.54,l.55,l.56,l.57], _
[l.58,l.59,l.60,l.61,l.62,l.63,l.64], _
[l.65,l.66,l.67,l.68,l.69,l.70,l.71], _
[l.72,l.73,l.74,l.75,l.76,l.77,l.78], _
[l.79,l.80,l.81,l.82,l.83,l.84,l.85], _
[l.86,l.87,l.88,l.89,l.90,l.91,l.92], _
[l.93,l.94,l.95,l.96,l.97,l.98,l.99] ]
b:L L I:=[
[l.1,l.74,l.83,l.21,l.36,l.77,l.44,l.80,l.64,_
l.2,l.34,l.75,l.48,l.17,l.100],_
[l.3,l.15,l.31,l.52,l.19,l.11,l.73,l.79,l.26,_
l.56,l.41,l.99,l.39,l.84,l.90],_
[l.4,l.57,l.86,l.63,l.85,l.95,l.82,l.97,l.98,_
l.81,l.8,l.69,l.38,l.43,l.58],_
[l.5,l.66,l.49,l.59,l.61],_
[l.6,l.68,l.89,l.94,l.92,l.20,l.13,l.54,l.24,_
l.51,l.87,l.27,l.76,l.23,l.67],_
[l.7,l.72,l.22,l.35,l.30,l.70,l.47,l.62,l.45,_
l.46,l.40,l.28,l.65,l.93,l.42],_
[l.9,l.71,l.37,l.91,l.18,l.55,l.96,l.60,l.16,_
l.53,l.50,l.25,l.32,l.14,l.33],_
[l.10,l.78,l.88,l.29,l.12] ]
llli2gp [a,b]
janko2():PERMGRP I == janko2 li1n 100
abelianGroup(l:L PI):PERMGRP I ==
gens:= nil()$(L L L I)
element:I:= 1
for n in l | n > 1 repeat
gens:=cons( list [i for i in element..(element+n-1) ], gens )
element:=element+n
llli2gp
#gens = 0 => [[[1]]]
gens
alternatingGroup(l:L I):PERMGRP I ==
l:=removeDuplicates l
#l = 0 =>
error "Cannot construct alternating group on empty set"
#l < 3 => llli2gp [[[l.1]]]
#l = 3 => llli2gp [[[l.1,l.2,l.3]]]
tmp:= [l.i for i in 3..(#l)]
gens:L L L I:=[[tmp],[[l.1,l.2,l.3]]]
odd?(#l) => llli2gp gens
gens.1 := cons([l.1,l.2],gens.1)
llli2gp gens
alternatingGroup(n:PI):PERMGRP I == alternatingGroup li1n n
symmetricGroup(l:L I):PERMGRP I ==
l:=removeDuplicates l
#l = 0 => error "Cannot construct symmetric group on empty set !"
#l < 3 => llli2gp [[l]]
llli2gp [[l],[[l.1,l.2]]]
symmetricGroup(n:PI):PERMGRP I == symmetricGroup li1n n
cyclicGroup(l:L I):PERMGRP I ==
l:=removeDuplicates l
#l = 0 => error "Cannot construct cyclic group on empty set"
llli2gp [[l]]
cyclicGroup(n:PI):PERMGRP I == cyclicGroup li1n n
dihedralGroup(l:L I):PERMGRP I ==
l:=removeDuplicates l
#l < 3 => error "in dihedralGroup: Minimum of 3 elements needed !"
tmp := [[l.i, l.(#l-i+1) ] for i in 1..(#l quo 2)]
llli2gp [ [ l ], tmp ]
dihedralGroup(n:PI):PERMGRP I ==
n = 1 => symmetricGroup (2::PI)
n = 2 => llli2gp [[[1,2]],[[3,4]]]
dihedralGroup li1n n
|