> < ^ Date: Fri, 15 Aug 1997 16:14:11 -0400 (EDT)
^ From: Darci Kracht <darci@mcs.kent.edu >
> ^ Subject: algebra groups

Dear Gap-Forum,

I am just learning to use GAP and I cannot figure out how
to enter the types of groups ("algebra groups") that I'm working
on. (Finite) Algebra groups are p-groups that are constructed from
finite-dimensional nilpotent algebras over finite fields. (Every-
thing that I look at is finite!)

For example, I might have an algebra J over a finite field F (here
F=GF(q) where q is a power of a prime p) generated by two (non-
commuting) symbols x and y with various relations. (Maybe J^4=0
and y^3=0 and yx-xy-y^2=0, for example).

To get my group G, I just define G=1+J (where 1 is a multiplicative
identity elt in some larger algebra containing J). Note that J is
the Jacobson Radical of the algebra F*1+J. Then G is a finite p-group.
What I want to do right now is to determine the normalizers of certain
algebra subgroups of G (say H = 1+Fy+Fy^2, for example).

I have learned to do a few things with GAP, but not exactly what I need.

1) I have managed to get a finitely presented algebra with the appropriate
generators and relations, but it's always Unital.

2) Also, some of the functions (from the VE package) don't work if q is
not prime. This is the case I need since my conjecture is trivial
if q is prime.

3) Finally, I have no idea how to tell GAP to construct the group from
the algebra.

I suppose I could represent my groups as matrices, but it's much easier
for what I am doing to use generators and relations.

Any hints or suggestions would be most welcome!


Darci L. Kracht
Department of Mathematics and Computer Science
Kent State University
Kent, OH USA

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