Number of conjugacy classes in projective general linear group of fixed degree over a finite field is PORC 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 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.
|(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|