Element structure of general affine group of degree one over a finite field

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This article gives specific information, namely, element structure, about a family of groups, namely: general affine group of degree one.
View element structure of group families | View other specific information about general affine group of degree one

This article describes the element structure of the general affine group of degree one over a finite field. We denote the field size by the letter q and the characteristic of the field by the letter p. Note that q is a prime power with underlying prime p.

Summary

Item Value
number of conjugacy classes q
equals number of irreducible representations, see also linear representation theory of general affine group of degree one over a finite field
order q(q - 1)
exponent p(q - 1)
conjugacy class size statistics size 1 (1 class), size q - 1 (1 class), size q (q - 2 classes)

Conjugacy class structure

All the elements of this group are of the form:

x \mapsto ax + v, a \in \mathbb{F}_q^\ast, v \in \mathbb{F}_q


Nature of conjugacy class Size of conjugacy class Number of such conjugacy classes Total number of elements
a = 1, v = 0 1 1 1
a = 1, v \ne 0 (conjugacy class is independent of choice of v) q - 1 1 q - 1
a \ne 1 (conjugacy class is determined completely by choice of a and is independent of choice of v; in other words, each conjugacy class is a coset of the subgroup of translations) q q - 2 q(q - 2)
Total (--) -- q q(q - 1)