[GAP Forum] Elementary abelian p-subgroups

[Stephen Linton]

I believe that

ConjugacyClassesSubgroups(LatticeByCyclicExtension(g, IsElementaryAbelian, true));

Does what you want.

	Steve

On 6 Sep 2013, at 10:24, "Ellis, Grahamj" <graham.ellis at nuigalway.ie> wrote:

> Dear GAP forum,
> 
> I too would like to know how best to access the elementary abelian p-subgroups of a finite group G. Attached is an implementation of the Quillen Complex which uses ConjugacyClassesSubgroups. As Jared mentions, this is a very inefficient approach.
> 
> ------------------------------------------------------------------------
> gap> G:=SmallGroup(64,134);;
> gap> Q:=QuillenComplex(G,2);
> Simplicial complex of dimension 2.
> 
> gap> Homology(Q,0);
> [ 0 ]
> gap> Homology(Q,1);
> [  ]
> gap> Homology(Q,2);
> [  ]
> gap> Q!.nrSimplices(2); #The number of 2-simplices
> 168
> gap> Q!.simplices(2,168); #The last 2-simplex
> [ Group([ f6, f2, f3*f4*f5 ]), Group([ f2, f3*f4*f5*f6 ]), 
>  Group([ f2*f3*f4*f5*f6 ]) ]
> ------------------------------------------------------------------------
> 
> 
> Graham
> 
> School of Mathematics, Statistics & Applied Mathematics
> National University of Ireland, Galway
> University Road,
> Galway
> Ireland
> 
> http://hamilton.nuigalway.ie
> tel: 091 493011
> ________________________________________
> From: forum-bounces at gap-system.org [forum-bounces at gap-system.org] on behalf of Jared Warner [jaredwarner4 at gmail.com]
> Sent: Friday, September 06, 2013 1:32 AM
> To: forum at gap-system.org
> Subject: [GAP Forum] Elementary abelian p-subgroups
> 
> Dear GAP forum,
> 
> I'm interested in studying the Quillen Complex of a finite group G, which
> is the lattice of elementary abelian p-subgroups of G.  Magma has a command
> ElementaryAbelianSubgroups which does exactly what I want, but I'd like to
> do this with GAP (to avoid paying for Magma).
> 
> Specifically, given a finite group G I'd like to know:
> 
> 1. The number of conjugacy classes of elementary abelian p-subgroups of a
> certain rank
> 
> 2. The number of subgroups in each such conjugacy class
> 
> 3. A set of generators of a representative from each such conjugacy class
> 
> I'm aware of ConjugacyClassesSubgroups, but I'd like to my command return a
> list of representatives of elementary abelian p-subgroups, not all
> subgroups.
> 
> Also, I'd like to avoid working with ConjugacyClassesSubgroups if possible,
> because it seems like a lot of wasted time and memory to compute the whole
> subgroup lattice when I'm only interested in a small portion of it.
> 
> Thanks for any help!
> 
> Jared
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