Number of conjugacy classes in general affine group of fixed degree over a finite field is polynomial function of field size
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 .
Below, we list some general observations about the polynomial in giving number of conjugacy classes in .
|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|
|(degree of general affine group, degree of polynomial)||polynomial of giving number of conjugacy classes in||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|