# Number of conjugacy classes in projective general linear group of fixed degree over a finite field is PORC function of field size

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## 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 projective general linear group (i.e., the special linear group of degree over the finite field of size ) is . Moreover, we only need to consider congruence classes modulo to define the PORC function.

A PORC function is a *polynomial on residue classes* -- it looks like different polynomial functions on different congruence classes modulo a particular number.

## Particular cases

(degree of special linear group, also modulus to which we need to consider congruence classes) | (degree of polynomial) | polynomial of giving number of conjugacy classes in | More information |
---|---|---|---|

1 | 0 | 1 | the group is a trivial group |

2 | 1 | for even , for odd | See element structure of projective general linear group of degree two over a finite field |

3 | 2 | if is not 1 mod 3, if is 1 mod 3 | See element structure of projective general linear group of degree three over a finite field |