Number of conjugacy classes in general affine group of fixed degree over a finite field is polynomial function of field size
From Groupprops
Statement
Suppose is a natural number. Then, there exists a polynomial function
of degree
such that, for any prime power
, the number of conjugacy classes in the general affine group
(i.e., the general affine group of degree
over the finite field of size
) is
.
General observations
Below, we list some general observations about the polynomial in giving number of conjugacy classes in
.
Item | Value |
---|---|
Degree of polynomial | ![]() |
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
![]() |
polynomial of ![]() ![]() |
More information |
---|---|---|
1 | ![]() |
element structure of general affine group of degree one over a finite field |
2 | ![]() |
element structure of general affine group of degree two over a finite field |
3 | ![]() |
element structure of general affine group of degree three over a finite field |
4 | ![]() |
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 | ? |