General affine group

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Definition

In terms of dimension

Let $n$ be a natural number and $K$ be a field. The general affine group or affine general linear group of degree $n$ over $K$, denoted $GA(n,K)$, $GA_{n}(K)$, $AGL(n,K)$, or $AGL_{n}(K)$, is defined as the external semidirect product of the vector space $K^{n}$ by the general linear group $GL(n,K)$, acting by linear transformations.

While $GA(n,K)$ cannot be realized as a subgroup of $GL(n,K)$, it can be realized as a subgroup of $GL(n+1,K)$ in a fairly typical way: the vector from $K^{n}$ is the first $n$ entries of the right column, the matrix from $GL(n,K)$ is the top left $n\times n$ block, there is a $1$ in the bottom right corner, and zeroes elsewhere on the bottom row.

In terms of vector spaces

Let $V$ be a $K$-vector space (which may be finite- or infinite-dimensional). The general affine group of $V$, denoted $GA(V)$, is defined as the external semidirect product of $V$ by $GL(V)$.

Notation for general affine group over a finite field

For $q=p^{n}$ a prime power ($p$ prime), we write $GA(n,q)=GA(n,\mathbb {F} _{q})$ for the general affine group over the finite field with $q$ elements.

Particular cases

Finite general affine groups

Degree one

Further information: General affine group of degree one

$q$ (field size)	$p$ (underlying prime, field characteristic)	$GA(1,q)$	Order	Second part of GAP ID
2	2	cyclic group:Z2	2	1
3	3	symmetric group:S3	6	1
4	2	alternating group:A4	12	3
5	5	general affine group:GA(1,5)	20	3
7	7	general affine group:GA(1,7)	42	1
8	2	general affine group:GA(1,8)	56	11
9	3	general affine group:GA(1,9)	72	39

Infinite general affine groups

• General affine group:GA(1,Q)

Linear representation theory

Further information: Linear representation theory of general affine group

Important subgroups

A particular subgroup of note is the special affine group.