# Element structure of special affine group of degree two over a finite field

## Contents

This article gives specific information, namely, element structure, about a family of groups, namely: special affine group of degree two.
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This article describes the element structure of $SA(2,q)$, the special affine group of degree two over a finite field of size $q$, where $q$ is a prime power. We denote by $p$ the characteristic of the field, so $p$ is a prime number. Thus, $q = p^r$ for some positive integer $r$.

## Summary

Item Value
order of the group $q^2(q^3 - q) = q^3(q^2 - 1) = q^5 - q^3$
number of conjugacy classes Case $q$ even (e.g., $q = 2,4,8,\dots$): ?
Case $q$ odd (e.g., $q = 3,5,7,9,11,13,\dots$): $2q + 4$