Number of irreducible Brauer characters equals number of regular conjugacy classes

Statement

Suppose $G$ is a finite group and $p$ is a prime number. Then, the number of irreducible Brauer characters of $G$ for the prime $p$ equals the number of $p$-regular conjugacy classes. Here, a $p$-regular conjugacy class is a conjugacy class in which the orders of elements are relatively prime to $p$.

Related facts

• Number of irreducible representations equals number of conjugacy classes

Particular cases

Particular groups

Note that for a group of prime power order with underlying prime $p$, the number of irreducible Brauer characters as well as the number of $p$-regular conjugacy classes are both equal to $1$. In the table below, we consider other cases.

Group	Order	Second part of GAP ID	Prime number $p$	Number of irreducible Brauer characters for $p$ = number of $p$-regular conjugacy classes	Information on modular representation theory
symmetric group:S3	6	1	2	2	modular representation theory of symmetric group:S3 at 2
symmetric group:S3	6	1	3	2	modular representation theory of symmetric group:S3 at 3
alternating group:A4	12	3	2	3	modular representation theory of alternating group:A4 at 2
alternating group:A4	12	3	3	2	modular representation theory of alternating group:A4 at 3
symmetric group:S4	24	12	2	2	modular representation theory of symmetric group:S4 at 2
symmetric group:S4	24	12	3	4	modular representation theory of symmetric group:S4 at 3
special linear group:SL(2,3)	24	3	2	3	modular representation theory of special linear group:SL(2,3) at 2
special linear group:SL(2,3)	24	3	3	3	modular representation theory of special linear group:SL(2,3) at 3