# Number of irreducible Brauer characters equals number of regular conjugacy classes

From Groupprops

## Statement

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

## Related facts

## Particular cases

### Particular groups

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

Group | Order | Second part of GAP ID | Prime number | Number of irreducible Brauer characters for = number of -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 |