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# This file is part of the GAP package MAPCLASS
# This file contains a system of GAP programs motivated by Staszewski and Voelklein
# to compute the number of generating tuples
#
# A.James, K. Magaard, and S.Shpectorov 2011
#
# NumberGeneratingTuples
# general function computing the number of generating tuples
#
InstallGlobalFunction(NumberGeneratingTuples,function(group,p,t,g)
local tp,i,full,sz,cc,oc,ct,str,StepDown;
tp:=[];
for i in [1..Length(p)] do
Append(tp,List([1..p[i]],c->i));
od;
full:=[];
sz:=Size(group);
cc:=ConjugacyClasses(group);
oc:=List(t,s->Filtered([1..Length(cc)],i->s in cc[i]));
ct:=CharacterTable(group);
str:=TotalNumberOfTuples(ct,g,List(oc,c->c[1]));
Add(full,rec(gp:=group,sz:=sz,cc:=cc,oc:=oc,str:=str,pr:=0,nx:=0));
## RECURSIVE FUNCTION STARTS HERE
StepDown:=function(no)
local gp,cc,oc,ms,m,sz,p,pp,fl,mcc,moc,ct,str,t,i;
gp:=full[no].gp;
cc:=full[no].cc;
oc:=full[no].oc;
ms:=MaximalSubgroupClassReps(gp);
for m in ms do
sz:=Size(m);
p:=no;
pp:=full[no].pr;
fl:=false;
## Find the next subgroup
while full[p].sz>=sz do
if full[p].sz=sz then
if IsConjugate(group,full[p].gp,m) then
fl:=true;
break;
fi;
fi;
pp:=p;
p:=full[p].nx;
if p=0 then
break;
fi;
od;
if not fl then
mcc:=ConjugacyClasses(m);
moc:=List(oc,s-> Union(List(s, n-> Filtered([1..Length(mcc)],i ->
Representative(mcc[i]) in cc[n]))));
if not ([] in moc) then
ct:=CharacterTable(m);
str:=0;
t:=List(tp,c->1);
repeat
str:=str+TotalNumberOfTuples(ct, g,
List([1..Length(t)],i->moc[i][t[i]]));
i:=1;
while i<=Length(tp) do
if t[i]<>Length(moc[i]) then
t[i]:=t[i]+1;
break;
else
t[i]:=1;
i:=i+1;
fi;
od;
until i>Length(tp);
Add(full,rec(gp:=m,sz:=sz,cc:=mcc,oc:=moc,str:=str,pr:=pp,nx:=p));
if pp<>0 then
full[pp].nx:=Length(full);
fi;
if p<>0 then
full[p].pr:=Length(full);
fi;
StepDown(Length(full));
fi;
fi;
od;
str:=full[no].str;
p:=full[no].nx;
while p<>0 do
if (full[no].sz mod full[p].sz)=0 then
## If is a p-group then try something else because Isomorphic subgroups
## sloooow
if IsPGroup(gp) then
ms:=IsomorphicSubgroups(gp,full[p].gp);
for m in ms do
m:=Image(m);
if IsConjugate(group,m,full[p].gp) then
str:=str-full[p].str*Index(gp,Normalizer(gp,m));
fi;
od;
else
ms:=IsomorphicSubgroups(gp,full[p].gp);
for m in ms do
m:=Image(m);
if IsConjugate(group,m,full[p].gp) then
str:=str-full[p].str*Index(gp,Normalizer(gp,m));
fi;
od;
fi;
fi;
p:=full[p].nx;
od;
full[no].str:=str;
end;
## END OF RECURSIVE FUNCTION
StepDown(Length(full));
return full[1].str;
end);
# End
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