[GAP Forum] Elementary abelian p-subgroups

[Mathieu Dutour]

Well, I think that this can be programmed easily:
1> we simply need to consider one Sylow p-group.

2> the ConjugacyClassesSubgroups is an iterative algorithm
starting from a small group and expanding it. Maybe we can
adapt by starting with conjugacy classes in G of order p
and adding commuting elements.

The real challenge, I would say is whether or not the full
list of elementary p-subgroup is stored in memory. If yes,
then memory will be limiting. If not, then the difficulty will be
in programming with only the knowledge of the conjugacy
classes.

  Mathieu


On Fri, Sep 6, 2013 at 11:24 AM, 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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