The blocks with four irreducible characters

TL;DR

This paper investigates the structure of Brauer p-blocks with exactly four irreducible characters, establishing that under the Alperin–McKay conjecture, the defect group must have order 4 or 5.

Contribution

It proves a classification result for Brauer blocks with four irreducible characters assuming the Alperin–McKay conjecture.

Findings

If a Brauer p-block has exactly 4 irreducible characters, then its defect group has order 4 or 5.

The result is conditional on the validity of the Alperin–McKay conjecture.

Provides insight into the structure of blocks with small numbers of irreducible characters.

Abstract

Suppose that $B$ is a Brauer $p$-block with defect group $D$. If $B$ exactly contains 4 irreducible characters, then we show that $D$ has order 4 or 5, assuming the Alperin--McKay conjecture.