# 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 $p$.

## 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