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 n is a natural number. Then, there exists a polynomial function f of degree n - 1 such that, for any prime power q, the number of conjugacy classes in the projective general linear group PGL(n,q) (i.e., the special linear group of degree n over the finite field of size q) is f(q). Moreover, we only need to consider congruence classes modulo n 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

n (degree of special linear group, also modulus to which we need to consider congruence classes) n - 1 (degree of polynomial) polynomial of q giving number of conjugacy classes in PGL(n,q) More information
1 0 1 the group is a trivial group
2 1 q + 1 for even q, q + 2 for odd q See element structure of projective general linear group of degree two over a finite field
3 2 q^2 + q if q is not 1 mod 3, q^2 + q + 2 if q is 1 mod 3 See element structure of projective general linear group of degree three over a finite field