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Dear GAP Forum,

John Murray wrote

Is there a way of getting GAP to regard

a vector space as an abelian group, and

a matrix as a homomorphism of the

underlying vector space abelian group?Also, has anyone got a construction of

Holomorph(G) where G is a finite group?What I want is a natural way of working

with the affine group V . GL(V), where V

is a finite dimensional vector space over a

finite field. By natural, I mean describing

the group as ordered pairs (v,m), where

v is a vector and m is a nonsingular matrix,

and (v,1)^{(1,m)} = (v*m,1), where v*m is

the image of the (row) vector v under m.Clearly this can be done using

affine:=SemidirectProduct(v,hom,gl); where

v := CyclicGroup(q)^n; and

gl := AutomorphismGroup(v); and

hom is the identity homomorphism gl->gl,

q the size of the underlying field and n is

the dimension of V. But this is not natural.More generally I'd like a construction for

V . H, where H is a subgroup of GL(V),

or indeed H has a factor group which is

isomorphic to a subgroup of GL(V).

Since groups (and thus in particular abelian groups) in GAP

are always written multiplicatively,

a vector space itself is not regarded as a group,

and vice versa.

Matrices can be regarded as homomorphisms between vector spaces

in the sense that they describe linear mappings,

but these are not group homomorphisms for GAP.

A quite natural way to deal with affine groups in GAP

would be to use the embedding of (GF(q)^n).GL(n,q) into GL(n+1,q);

then there is an easy translation between a pair (v,m)

and the corresponding element g of this group,

via m = g{ [ 1 .. n ] }{ [ 1 .. n ] } and v = g[ n+1 ]{ [ 1 .. n ] }.

This allows one to avoid the technical overhead of the

general `SemidirectProduct' construction,

and the matrix group acting on the vector space may be any

subgroup of GL(n,q).

For the more general case that a linear action of a given group H

on a vector space is not faithful, it may be still useful to

work with the extended matrices, and to use an explicit

epimorphism from H onto the matrix group in question.

I hope this helps.

Kind regards,

Thomas Breuer

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