General affine group of degree one

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Revision as of 19:32, 3 September 2008 by Vipul (talk | contribs) (New page: ==Definition== 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)</ma...)
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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.