|
#############################################################################
##
#W trans.gi GAP transitive groups library Alexander Hulpke
##
##
#Y Copyright (C) 2001,2016, Alexander Hulpke, Colorado State University
##
## This file contains methods that rely on the transitive groups library
## being available.
##
# computes the perfect subgroups of S_n or A_n. Symconj indicates whether
# they are up to conjugacy in S_n.
BindGlobal("PerfectSubgroupsAlternatingGroup",function(g,symconj)
local dom,deg,S,p,ps,perm,startp,sdps,nsdps,i,j,k,l,m,n,part,
sysdps,syj,nk,kno,nl,lno,khom,kim,lhom,lim,iso,au,ind1,ind2,dc,d,grp,
knom,lnon;
dom:=Set(MovedPoints(g));
deg:=Length(dom);
p:=[TrivialSubgroup(g)];
if deg<5 then
return p;
fi;
if not TransitiveGroupsAvailable(deg) or
ValueOption("NoPrecomputedData")=true then
TryNextMethod();
fi;
Info(InfoPerformance,2,"Using Transitive Groups Library");
S:=SymmetricGroup(deg);
# all partitions with (nontrivial) orbits of length
part:=Filtered(Partitions(deg),i->Length(i)<deg and
ForAll(i,j->j=1 or j>4));
# we shall use implicitly, that the partitions are ordered reversly. I.e.
# all sdps constructed don't have any earlier fixpoints &c.
for i in part do
Info(InfoLattice,1,"Partition: ",i);
# for each partition construct all subdirect products.
sdps:=[];
startp:=1; # point we start on
for j in i do
if j>4 then
Info(InfoLattice,3,j,", ",Length(sdps)," products");
perm:=MappingPermListList([1..j],[startp..startp+j-1]);
# get the transitive ones of this degree.
ps:=AllTransitiveGroups(NrMovedPoints,j,IsPerfectGroup,true);
ps:=List(ps,i->i^perm);
if Length(sdps)=0 then
sdps:=ps;
else
nsdps:=[];
# now we must form spds: run through all pairs
sysdps:=SymmetricGroup(MovedPoints(sdps[1]));
syj:=SymmetricGroup(j);
for k in sdps do
nk:=NormalSubgroups(k);
kno:=Normalizer(sysdps,k);
for l in ps do
nl:=NormalSubgroups(l);
lno:=Normalizer(syj,k);
# run through all combinations of normal subgroups
for m in nk do
knom:=Normalizer(kno,m);
for n in nl do
lnon:=Normalizer(lno,n);
if Index(k,m)=Index(l,n) then
# factor groups have the same order.
khom:=NaturalHomomorphismByNormalSubgroupNC(k,m);
kim:=Image(khom);
lhom:=NaturalHomomorphismByNormalSubgroupNC(l,n);
lim:=Image(lhom);
iso:=IsomorphismGroups(kim,lim);
if iso<>fail then
# they are isomorphic. So there are subdirect
# products. Classify them up to conjugacy (Satz (32)
# in my thesis)
au:=AutomorphismGroup(lim);
# those automorphisms induced by the normalizer of k
ind1:=List(GeneratorsOfGroup(knom),
y->GroupHomomorphismByImagesNC(lim,lim,
GeneratorsOfGroup(lim),
List(GeneratorsOfGroup(lim),
z->Image(iso,
Image(khom,PreImagesRepresentative(khom,
PreImagesRepresentative(iso,z) )^y)
))));
Assert(1,ForAll(ind1,IsBijective));
# those automorphisms induced by the normalizer of l
ind2:=List(GeneratorsOfGroup(lnon),
y->GroupHomomorphismByImagesNC(lim,lim,
GeneratorsOfGroup(lim),
List(GeneratorsOfGroup(lim),
z->Image(lhom,PreImagesRepresentative(lhom,z)^y))));
Assert(1,ForAll(ind1,IsBijective));
dc:=DoubleCosetRepsAndSizes(au,SubgroupNC(au,ind1),
SubgroupNC(au,ind2));
dc:=List(dc,i->i[1]); # only reps
for d in dc do
grp:=ClosureGroup(n,
List(GeneratorsOfGroup(k),i->i*
PreImagesRepresentative(lhom,
Image(d,Image(iso,Image(khom,i)))
))
);
Add(nsdps,grp);
od;
fi;
fi;
od;
od;
od;
od;
sdps:=nsdps;
fi;
fi;
startp:=startp+j;
od;
# S_n classes
nsdps:=[];
for j in sdps do
if ForAll(nsdps,k->Size(k)<>Size(j)
or Set(MovedPoints(k))<>Set(MovedPoints(l))
or RepresentativeAction(
# if they are conjugate in S_deg they are conjugate
# in the smaller S_n on their moved points
Stabilizer(S,Difference([1..deg],
MovedPoints(k)),OnTuples),
j,k)=fail) then
Add(nsdps,j);
fi;
od;
Info(InfoLattice,2,j,", ",Length(sdps)," new perfect groups");
if symconj then
Append(p,nsdps);
else
for j in nsdps do
n:=Normalizer(S,j);
Add(p,j);
if SignPermGroup(n)=1 then
Add(p,ConjugateGroup(j,(1,2))); # Normalizer in A_n: 2 orbits
fi;
od;
fi;
od;
if dom<>[1..deg] then
perm:=MappingPermListList([1..deg],dom);
p:=List(p,i->i^perm);
fi;
return p;
end);
#############################################################################
##
#M RepresentativesPerfectSubgroups
##
InstallMethod(RepresentativesPerfectSubgroups,"alternating",true,
[ IsNaturalAlternatingGroup ], 0,
G->PerfectSubgroupsAlternatingGroup(G,false));
InstallMethod(RepresentativesPerfectSubgroups,"symmetric",true,
[ IsNaturalSymmetricGroup ], 0,
G->PerfectSubgroupsAlternatingGroup(G,true));
[ Seitenstruktur0.40Drucken
etwas mehr zur Ethik
]
|
