[GAP Forum] Suzuki 2-groups

[Joey Iverson]

Dear Timm, Dear Forum,

I have some code on GitHub
<https://github.com/jwiverson/central-group-frames> that can help construct
what Higman calls A(n,theta) in this paper
<https://projecteuclid.org/download/pdf_1/euclid.ijm/1255637483>. This is a
special example of what we call "B-products" in Section 4 of this paper
<https://arxiv.org/abs/1609.09836>.

To make A(n,theta), download the file "b-prods.g" and put it where GAP can
find it. Then type something like the following:

gap> Read("b-prods.g");
gap> n:=3;
3
gap> theta:=FrobeniusAutomorphism(GF(2^n));
FrobeniusAutomorphism( GF(2^3) )
gap> B:=function(x,y)
> return x*y^theta;
> end;
function( x, y ) ... end
gap> G:=bGroup(n,B);
<pc group of size 64 with 6 generators>

You can replace n and theta with anything want. The result is a pc group.
The function "bEmbed" can help you map a pair of field elements (x,y) into
G, as in Higman's paper:

gap> emb:=bEmbed(B,G,n);
function( x, y ) ... end
gap> x:=Random(GF(2^n));
Z(2)^0
gap> y:=Random(GF(2^n));
Z(2^3)
gap> g:=emb(x,y);
f1*f5

I hope this helps!

All the best,
Joey Iverson

Research Associate
Norbert Wiener Center & Department of Mathematics
University of Maryland, College Park

On Thu, Sep 21, 2017 at 11:26 AM, Timm von Puttkamer <tvonp at gmx.net> wrote:

> Dear all,
>
> I would like to know whether GAP has a method to construct Suzuki
> 2-groups. Recall that a Suzuki 2-group is a non-abelian 2-group containing
> more than one involution which admits a cyclic group of automorphims that
> acts transitively on the set of involutions.
>
> Kind regards,
> Timm
>
>
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