# [GAP Forum] how to compute all subgroups of order p up to conjugacy

Laurent Bartholdi laurent.bartholdi at gmail.com
Tue May 27 11:04:21 BST 2008

```Hi Frederic!
Subgroups of order p are generated by a single element -- so you might
look for all such subgroups, and then eliminate all redundancies by
keeping 1 subgroup for each conjugacy class and choice of generator.

I won't be more specific since I see you're a student... Feel free to
ask for more details if this isn't for an exam.

Best, Laurent

On Tue, May 27, 2008 at 9:56 AM, Frédéric Vanhove
<fvanhove at cage.ugent.be> wrote:
> 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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--
Laurent Bartholdi \ laurent.bartholdi<at>gmail<dot>com
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