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#############################################################################
##
#W oprtglat.gi GAP library Alexander Hulpke
##
#H @(#)$Id: oprtglat.gi,v 4.12 2002/04/15 10:05:06 sal Exp $
##
#Y Copyright (C) 1997, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
## This file contains methods for orbits on subgroups
##
Revision.oprtglat_gi:=
"@(#)$Id: oprtglat.gi,v 4.12 2002/04/15 10:05:06 sal Exp $";
#############################################################################
##
#M GroupOnSubgroupsOrbit(G,H) . . . . . . . . . . . . . . orbit of H under G
##
InstallGlobalFunction( GroupOnSubgroupsOrbit, function(G,H)
return Enumerator(ConjugacyClassSubgroups(G,H));
end );
#############################################################################
##
#M MinimumGroupOnSubgroupsOrbit(G,H [,N_G(H)]) minimum of orbit of H under G
##
InstallGlobalFunction( MinimumGroupOnSubgroupsOrbit, function(arg)
local cont,lim,s,i,j,m,Hc,o,og;
# try some orbit calculation first (at most orbit of length 20) to avoid
# normalizer calculations.
cont:=true;
lim:=QuoInt(Size(arg[1]),Size(arg[2]));
if lim>20 then
cont:=lim<200000; # otherwise give up at once
lim:=20;
fi;
if cont then
o:=[arg[2]];
else
o:=[];
fi;
m:=arg[2];
i:=1;
while cont and i<=Length(o) do
for j in GeneratorsOfGroup(arg[1]) do
if not ForAny(o,x->ForAll(GeneratorsOfGroup(o[i]),y->y^j in x)) then
Hc:=o[i]^j;
Add(o,Hc);
if Hc<m then
m:=Hc;
fi;
cont:=Length(o)<lim;
fi;
od;
i:=i+1;
od;
if not cont then
# orbit is longer -- have to work
s:=ConjugacyClassSubgroups(arg[1],arg[2]);
if Length(arg)>2 then
SetStabilizerOfExternalSet(s,arg[3]);
fi;
s:=Enumerator(s);
if Length(s)>2*lim then
o:=[]; # the orbit is not worth keeping -- test would be too expensive
fi;
for i in [1..Length(s)] do
Hc:=s[i];
if not ForAny(o,x->ForAll(GeneratorsOfGroup(Hc),y-> y in x)) then
if Hc<m then
m:=Hc;
fi;
fi;
od;
fi;
return m;
end );
InstallMethod(SubgroupsOrbitsAndNormalizers,"generic on list",true,
[IsGroup,IsList,IsBool],0,
function(G,dom,all)
local n,l,o,b,r,p,cl,i,sel,selz,gens,ti,t,tl;
n:=Length(dom);
l:=n;
o:=[];
b:=BlistList([1..l],[1..n]);
while n>0 do
p:=Position(b,true);
b[p]:=false;
n:=n-1;
r:=rec(representative:=dom[p]);
cl:=ConjugacyClassSubgroups(G,r.representative);
gens:=GeneratorsOfGroup(r.representative);
r.normalizer:=StabilizerOfExternalSet(cl);
t:=RightTransversal(G,r.normalizer);
tl:=Length(t);
sel:=Filtered([1..l],i->b[i]);
selz:=Filtered(sel,i->Size(dom[i])=Size(r.representative));
if Length(selz)>0 then
i:=1;
while Length(sel)>0 and i<=tl do;
ti:=t[i];
p:=PositionProperty(sel,
j->j in selz and ForAll(gens,k->k^ti in dom[j]));
if p<>fail then
p:=sel[p];
b[p]:=false;
n:=n-1;
RemoveSet(sel,p);
fi;
i:=i+1;
od;
fi;
if all then
cl:=Enumerator(cl);
r.elements:=cl;
fi;
Add(o,r);
od;
return o;
end);
InstallMethod(SubgroupsOrbitsAndNormalizers,"perm group on list",true,
[IsPermGroup,IsList,IsBool],0,
function(G,dom,all)
local n,l,o,b,r,p,cl,i,sel,selz,gens,ti,t,tl;
n:=Length(dom);
l:=n;
o:=[];
b:=BlistList([1..l],[1..n]);
while n>0 do
p:=Position(b,true);
b[p]:=false;
n:=n-1;
r:=rec(representative:=dom[p]);
gens:=GeneratorsOfGroup(r.representative);
r.normalizer:=Normalizer(G,r.representative);
tl:=Index(G,r.normalizer);
sel:=Filtered([1..l],i->b[i]);
selz:=Filtered(sel,i->Size(dom[i])=Size(r.representative));
if tl<=50*Length(selz) then
t:=RightTransversal(G,r.normalizer);
if Length(selz)>0 then
i:=1;
while Length(sel)>0 and Length(selz)>0 and i<=tl do;
ti:=t[i];
p:=PositionProperty(sel,
j->j in selz and ForAll(gens,k->k^ti in dom[j]));
if p<>fail then
p:=sel[p];
b[p]:=false;
n:=n-1;
RemoveSet(sel,p);
RemoveSet(selz,p);
fi;
i:=i+1;
od;
fi;
else
for i in selz do
p:=RepresentativeAction(G,dom[i],r.representative,OnPoints);
if p<>fail then
b[i]:=false;
n:=n-1;
RemoveSet(sel,i);
fi;
od;
fi;
if all then
cl:=ConjugacyClassSubgroups(G,r.representative);
SetStabilizerOfExternalSet(cl,r.normalizer);
cl:=Enumerator(cl);
r.elements:=cl;
fi;
Add(o,r);
od;
return o;
end);
InstallMethod(SubgroupsOrbitsAndNormalizers,"pc group on list",true,
[IsPcGroup,IsList,IsBool],0,
function(G,dom,all)
local n,l,o,b,r,p,cl,i,sel,selz,allcano,cano,can2,p1;
allcano:=[];
n:=Length(dom);
l:=n;
o:=[];
b:=BlistList([1..l],[1..n]);
while n>0 do
p:=Position(b,true);
p1:=p;
b[p]:=false;
n:=n-1;
r:=rec(representative:=dom[p]);
sel:=Filtered([1..l],i->b[i]);
selz:=Filtered(sel,i->Size(dom[i])=Size(r.representative));
if Length(selz)>0 then
if IsBound(allcano[p1]) then
cano:=allcano[p1];
else
cano:=CanonicalSubgroupRepresentativePcGroup(G,r.representative);
fi;
r.normalizer:=ConjugateSubgroup(cano[2],cano[3]^-1);
cano:=cano[1];
for i in selz do
if IsBound(allcano[i]) then
can2:=allcano[i];
else
can2:=CanonicalSubgroupRepresentativePcGroup(G,dom[i]);
fi;
if can2[1]=cano then
b[i]:=false;
n:=n-1;
RemoveSet(sel,i);
Unbind(allcano[i]);
else
allcano[i]:=can2;
fi;
od;
else
r.normalizer:=Normalizer(G,r.representative);
fi;
if all then
cl:=ConjugacyClassSubgroups(G,r.representative);
SetStabilizerOfExternalSet(cl,r.normalizer);
r.elements:=Enumerator(cl);
fi;
Add(o,r);
Unbind(allcano[p1]);
od;
return o;
end);
# destructive version
# this method takes the component 'list' from the record and shrinks the
# list to save memory
InstallMethod(SubgroupsOrbitsAndNormalizers,"generic on record with list",true,
[IsGroup,IsRecord,IsBool],0,
function(G,r,all)
local n,o,dom,cl,i,s,j,t,ti,tl,gens;
dom:=r.list;
Unbind(r.list);
n:=Length(dom);
o:=[];
while n>0 do
r:=rec(representative:=dom[1]);
gens:=GeneratorsOfGroup(dom[1]);
s:=Size(dom[1]);
cl:=ConjugacyClassSubgroups(G,r.representative);
r.normalizer:=StabilizerOfExternalSet(cl);
cl:=Enumerator(cl);
t:=RightTransversal(G,r.normalizer);
tl:=Length(t);
i:=1;
while i<=tl and Length(dom)>0 do
ti:=t[i];
j:=2;
while j<=Length(dom) do
if Size(dom[j])=s and ForAll(gens,k->k^ti in dom[j]) then
# hit
dom[j]:=dom[Length(dom)];
Unbind(dom[Length(dom)]);
else
j:=j+1;
fi;
od;
i:=i+1;
od;
if all then
r.elements:=cl;
fi;
Add(o,r);
od;
return o;
end);
#############################################################################
##
#M StabilizerOp( <G>, <D>, <subgroup>, <U>, <V>, <OnPoints> )
##
## subgroup stabilizer
InstallMethod( StabilizerOp, "with domain, use normalizer", true,
[ IsGroup, IsList, IsGroup, IsList, IsList, IsFunction ],
# raise over special methods for pcgs et. al.
200,
function( G, D, sub, U, V, op )
if not U=V or op<>OnPoints then
TryNextMethod();
fi;
return Normalizer(G,sub);
end );
InstallOtherMethod( StabilizerOp, "use normalizer", true,
[ IsGroup, IsGroup, IsList, IsList, IsFunction ],
# raise over special methods for pcgs et. al.
200,
function( G, sub, U, V, op )
if not U=V or op<>OnPoints then
TryNextMethod();
fi;
return Normalizer(G,sub);
end );
#############################################################################
##
#E oprtglat.gi . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
##
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