General affine group of degree one: Difference between revisions

Revision as of 02:33, 2 November 2010

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 degree $1$ 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.

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 Let <math>p</math> be a [[prime number]] and <math>q = p^r</math> be a power of <math>p</math>. The '''general affine group''' or '''collineation group''' <math>GA(1,q)</math> is defined as follows. Let <math>\mathbb{F}_q</math> denote the field with <math>q</math> elements. Then <math>GA(1,q)</math> is the semidirect product of the additive group of <math>\mathbb{F}_q</math> with its multiplicative group.
-Equivalently it is the [[general affine group]] of order <math>1</math> over the field of <math>q</math> elements.
+Equivalently it is the [[general affine group]] of degree <math>1</math> over the field of <math>q</math> elements.
 ==Group properties==