# Conjugacy class size formula in general linear group over a finite field

This article gives formula(s) for the conjugacy class sizes in a general linear group of finite degree over a finite field with elements, which we denote by .

## Case of semisimple elements

### Elements diagonalizable over

Suppose is diagonalizable over , with eigenvalues having multiplicities respectively (the s are all distinct). Note that .

Then, the centralizer of the diagonal representative of this conjugacy class is isomorphic to:

In fact, if the diagonal entries are arranged so that all the s occur first, then the s, and so on, then the centralizer is the set of invertible block diagonal matrices with blocks of sizes .

The size of the conjugacy class is thus:

This is the same as the -analogue of the multinomial coefficient:

In all cases, this simplifies to a polynomial in , and its degree is .

Some particular cases for the partition of as a sum of s, and the corresponding sizes, are given below.

Partition of | Size of | Size of centralizer of diagonal element | Degree of polynomial giving size | Size of conjugacy class | Degree of polynomial giving size | |
---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 0 | ||

2 | 2 | 4 | 1 | 0 | ||

2 | 1 + 1 | 2 | 2 | |||

3 | 3 | 9 | 1 | 0 | ||

3 | 2 + 1 | 5 | 4 | |||

3 | 1 + 1 + 1 | 3 | 6 | |||

1 | 0 | |||||