/usr/share/gap/lib/grpcompl.gi is in gap-libs 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 | #############################################################################
##
#W grpcompl.gi GAP Library Alexander Hulpke
##
##
#Y Copyright (C) 1997
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains the operations for the computation of complements in
## 'white box groups'
##
BindGlobal("COCohomologyAction",function(oc,actgrp,auts,orbs)
local b, mats, orb, com, stabilizer, i,coc,u;
if not IsBound(oc.complement) then
return [];
fi;
oc.zero:=Zero(LeftActingDomain(oc.oneCocycles));
b:=BaseSteinitzVectors(BasisVectors(Basis(oc.oneCocycles)),
BasisVectors(Basis(oc.oneCoboundaries)));
if Length(b.factorspace)=0 then
u:=rec(com:=[rec(cocycle:=Zero(oc.oneCocycles),stabilizer:=actgrp)],
bas:=b);
if orbs then
u.com[1].orbit:=[Zero(oc.oneCocycles)];
fi;
return u;
fi;
Info(InfoComplement,2,"fuse ",
Characteristic(oc.zero)^Length(b.factorspace)," classes");
if Length(auts)=0 then
auts:=[One(actgrp)];
fi;
mats:=COAffineCohomologyAction(oc,oc.complementGens,auts,b);
orb:=COAffineBlocks(actgrp,auts,mats,orbs);
com:=[];
for i in orb do
coc:=i.vector*b.factorspace;
#u:=oc.cocycleToComplement(coc);
u:=rec(cocycle:=coc,
#complement:=u,
stabilizer:=i.stabilizer);
if orbs then u.orbit:=i.orbit;fi;
Add(com,u);
od;
Info(InfoComplement,1,"obtain ",Length(com)," orbits");
return rec(com:=com,bas:=b,mats:=mats);
end);
ComplementClassesRepresentativesSolvableWBG:=function(arg)
local G,N,K,s, h, q, fpi, factorpres, com, comgens, cen, ocrels, fpcgs, ncom,
ncomgens, ncen, nlcom, nlcomgens, nlcen, ocr, generators, modulePcgs,
l, complement, k, v, afu, i, j, jj;
G:=arg[1];
N:=arg[2];
# compute a series through N
s:=ChiefSeriesUnderAction(G,N);
if Length(arg)=2 then
K:=fail;
else
K:=arg[3];
# build a series only down to K
h:=List(s,x->ClosureGroup(K,x));
s:=[h[1]];
for i in h{[2..Length(h)]} do
if Size(i)<Size(s[Length(s)]) then
Add(s,i);
fi;
od;
fi;
Info(InfoComplement,1,"Series of factors:",
List([1..Length(s)-1],i->Size(s[i])/Size(s[i+1])));
# #T transfer probably to better group (later, AgCase)
# construct a presentation
h:=NaturalHomomorphismByNormalSubgroup(G,N);
# AH still: Try to find a more simple presentation if available.
if Source(h)=G then
q:=ImagesSource(h);
else
q:=Image(h,G);
fi;
fpi:=IsomorphismFpGroup(q);
Info(InfoComplement,2,"using a presentation with ",
Length(MappingGeneratorsImages(fpi)[2])," generators");
factorpres:=[FreeGeneratorsOfFpGroup(Range(fpi)),
RelatorsOfFpGroup(Range(fpi)),
List(MappingGeneratorsImages(fpi)[2],
i->PreImagesRepresentative(fpi,i))];
Assert(1,ForAll(factorpres[3],i->Image(h,PreImagesRepresentative(h,i))=i));
# initialize
com:=[G];
comgens:=[List(factorpres[3],i->PreImagesRepresentative(h,i))];
cen:=[s[1]];
ocrels:=false;
# step down
for i in [2..Length(s)] do
Info(InfoComplement,1,"Step ",i-1);
# we know the complements after s[i-1], we want them after s[i].
#fpcgs:=Pcgs(s[i-1]); # the factor pcgs
#fpcgs:=fpcgs mod InducedPcgsByGenerators(fpcgs,GeneratorsOfGroup(s[i]));
fpcgs:=ModuloPcgs(s[i-1],s[i]);
ncom:=[];
ncomgens:=[];
ncen:=[];
# loop over all complements so far
for j in [1..Length(com)] do
nlcom:=[];
nlcomgens:=[];
nlcen:=[];
# compute complements
ocr:=rec(group:=ClosureGroup(com[j],s[i-1]),
generators:=comgens[j],
modulePcgs:=fpcgs,
factorpres:=factorpres
);
if ocrels<>false then
ocr.relators:=ocrels;
Assert(2,ForAll(ocr.relators,
k->Product(List([1..Length(k.generators)],
l->ocr.generators[k.generators[l]]^k.powers[l]))
in s[i-1]));
fi;
OCOneCocycles(ocr,true);
ocrels:=ocr.relators;
if IsBound(ocr.complement) then
# special treatment for trivial case:
if Dimension(ocr.oneCocycles)=Dimension(ocr.oneCoboundaries) then
l:=[rec(stabilizer:=cen[j],
cocycle:=Zero(ocr.oneCocycles),
complement:=ocr.complement)];
else
#l:=BaseSteinitzVectors(BasisVectors(Basis(ocr.oneCocycles)),
# BasisVectors(Basis(ocr.oneCoboundaries)));
#
# v:=Enumerator(VectorSpace(LeftActingDomain(ocr.oneCocycles),
# l.factorspace,Zero(ocr.oneCocycles)));
#
# dimran:=[1..Length(v[1])];
#
# # fuse
# Info(InfoComplement,2,"fuse ",Length(v)," classes; working in dim ",
# Dimension(ocr.oneCocycles),"/",Dimension(ocr.oneCoboundaries));
#
# opfun:=function(z,g)
# Assert(3,z in AsList(v));
# z:=ocr.cocycleToList(z);
# for k in [1..Length(z)] do
# z[k]:=Inverse(ocr.complementGens[k])*(ocr.complementGens[k]*z[k])^g;
# od;
# Assert(2,ForAll(z,k->k in s[i-1]));
# z:=ocr.listToCocycle(z);
# Assert(2,z in ocr.oneCocycles);
# # sift z
# for k in dimran do
# if IsBound(l.heads[k]) and l.heads[k]<0 then
# z:=z-z[k]*l.subspace[-l.heads[k]];
# fi;
# od;
# Assert(1,z in AsList(v));
# return z;
# end;
#
# k:=ExternalOrbitsStabilizers(cen[j],v,opfun);
l:=COCohomologyAction(ocr,cen[j],GeneratorsOfGroup(cen[j]),false).com;
# if Length(l)<>Length(k) then Error("differ!");fi;
fi;
Info(InfoComplement,2,"splits in ",Length(l)," complements");
else
l:=[];
Info(InfoComplement,2,"no complements");
fi;
for k in l do
q:=k.stabilizer;
k:=ocr.cocycleToComplement(k.cocycle);
Assert(3,Length(GeneratorsOfGroup(k))
=Length(MappingGeneratorsImages(fpi)[2]));
# correct stabilizer to obtain centralizer
v:=Normalizer(q,ClosureGroup(s[i],k));
afu:=function(x,g) return CanonicalRightCosetElement(s[i],x^g);end;
for jj in GeneratorsOfGroup(k) do
if ForAny(GeneratorsOfGroup(v),x->not Comm(x,jj) in s[i]) then
# we are likely very close as we centralized in the higher level
# and stabilize the cohomology. Thus a plain stabilizer
# calculation ought to work.
v:=Stabilizer(v,CanonicalRightCosetElement(s[i],jj),afu);
fi;
od;
Add(ncen,v);
Add(nlcom,k);
Add(nlcomgens,GeneratorsOfGroup(k));
od;
ncom:=Concatenation(ncom,nlcom);
ncomgens:=Concatenation(ncomgens,nlcomgens);
ncen:=Concatenation(ncen,nlcen);
od;
com:=ncom;
comgens:=ncomgens;
cen:=ncen;
Info(InfoComplement,1,Length(com)," complements in total");
od;
if K<>fail then
com:=List(com,x->ClosureGroup(K,x));
fi;
return com;
end;
InstallMethod(ComplementClassesRepresentativesSolvableNC,"using cohomology",
IsIdenticalObj,
[IsGroup,IsGroup],1,
ComplementClassesRepresentativesSolvableWBG);
#############################################################################
##
#E grpcompl.gi
##
|