# General affine group of degree one

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## Definition

Let $p$ be a prime number and $q = p^r$ be a power of $p$. The general affine group or collineation group $GA(1,q)$ is defined as follows. Let $\mathbb{F}_q$ denote the field with $q$ elements. Then $GA(1,q)$ is the semidirect product of the additive group of $\mathbb{F}_q$ with its multiplicative group.

Equivalently it is the general affine group of order $1$ over the field of $q$ elements.

## Group properties

### Frobenius group

The additive subgroup $\mathbb{F}_q$ is a Frobenius kernel and the multiplicative subgroup is a Frobenius complement. (note: the case $q = 2$ is an exception, where it fails to be a Frobenius group on account of the multiplicative group being trivial).

### Solvable group

The group is solvable of solvable length two. Specifically, its commutator subgroup is precisely the additive group of the field.