[GAP Forum] how to compute all subgroups of order p up to conjugacy
Frédéric Vanhove
fvanhove at cage.ugent.be
Tue May 27 08:56:14 BST 2008
Hello,
many thanks for letting me on this list. This is my very first question.
This is my problem : let G be a finite group, the order of which
divisibly by a prime p. I would like to get a list of all subgroups of
G of order p, up to conjugacy.
Is there a command in GAP for that. Right now, I only know two
alternatives:
listofgroupssizep:=function(g,p)
local hom,image,syl,ccs,ccs2,elem;
hom := NiceMonomorphism(g);;
image:=Image(hom);;
syl := SylowSubgroup(image, p);;
ccs:=ConjugacyClasses(syl);;
ccs2 := Filtered(List(ccs,Representative),t->Order(t)=p);;
elem := List(ccs2,t->PreImage(hom,t));
pgroups:=List(elem,x->Subgroup(g,[x]));;
return pgroups;
end;
This function needs a group g and a prime p as arguments, and it returns
a list of subgroups of order p. All possibilites up to conjugacy will
appear at least once, but unfortunately many of them will appear more
than once...
The other alternative is :
Filtered(ConjugacyClassesSubgroups(g),x->Size(Representative(x))=p);
which works fine as long as g is pretty small, because it's absolutely
not efficient and it just doesn't finish the job when g is a bit bigger.
Does anyone know some advice?
Thank you very much,
Frédéric Vanhove
Ghent University
Belgium
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