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

[Marc Roeder]

Dear Forum,

Frédéric Vanhove wrote:

> 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:[...]
I am not aware of any command that does this in GAP. But a slight
variation of your program might do the trick:

listofgroupssizep:=function(g,p)
    local   syl,  ccs,  ccs2,  ccg;
 syl := SylowSubgroup(g, p);;
 ccs:=ConjugacyClasses(syl);;
 ccs2 := Filtered(List(ccs,Representative),t->Order(t)=p);;
 ccg:=List(ccs2,i->ConjugacyClass(g,i));
 return Set(ccg,i->Subgroup(g,[Representative(i)]));
end;

This uses the fact that a Sylow subgroup is represented as a subgroup
(and hence embedded in the original group).

All the best,
marc