# Number of conjugacy classes in general affine group of fixed degree over a finite field is polynomial function of field size

## Statement

Suppose $n$ is a natural number. Then, there exists a polynomial function $f$ of degree $n$ such that, for any prime power $q$, the number of conjugacy classes in the general affine group $GA(n,q)$ (i.e., the general affine group of degree $n$ over the finite field of size $q$) is $f(q)$.

## General observations

Below, we list some general observations about the polynomial in $q$ giving number of conjugacy classes in $GA(n,q)$.

Item Value
Degree of polynomial $n$
Leading coefficient of polynomial 1, i.e., it is always a monic polynomial
Factors of polynomial no common factors to all polynomials
Coefficients of polynomial The polynomial is an integer-valued polynomial, i.e., it sends integers to integers. Is it also true that all coefficients of the polynomial are integers? Seems so from first few examples

## Particular cases $n$ (degree of general affine group, degree of polynomial) polynomial of $q$ giving number of conjugacy classes in $GA(n,q)$ More information
1 $q$ element structure of general affine group of degree one over a finite field
2 $q^2 + q - 1$ element structure of general affine group of degree two over a finite field
3 $q^3 + q^2 - 1$ element structure of general affine group of degree three over a finite field
4 $q^4 + q^3 + q^2 - q - 1$ element structure of general affine group of degree four over a finite field
5  ? element structure of general affine group of degree five over a finite field
6  ? element structure of general affine group of degree six over a finite field
7  ?