# 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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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. $n$ (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
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