[GAP Forum] Elementary abelian p-subgroups

[Ellis, Grahamj]

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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