Number of irreducible Brauer characters equals number of regular conjugacy classes

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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

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