Even degree characters in principal blocks

TL;DR

This paper characterizes finite groups where all irreducible characters in the principal p-block have odd degree, revealing specific exceptions and linking the property to the group's structure and composition factors.

Contribution

It provides a complete characterization of such groups, identifying the unique exception involving the group M22 and relating the property to the group's quotient structure.

Findings

Non-abelian simple groups of order divisible by p do not satisfy the condition unless p=7 and the group is M22.

The condition is equivalent to the quotient G/O_{p'}(G) having odd order unless p=7 and M22 is a composition factor.

The paper identifies the specific exception involving the group M22 for the property to hold.

Abstract

We characterise finite groups such that for an odd prime $p$ all the irreducible characters in its principal $p$-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by $p$ unless $p=7$ and the group is $M_{22}$. As a consequence we deduce that if $p\neq 7$ or if $M_{22}$ is not a composition factor of a group $G$, then the condition above is equivalent to $G/O_{p'}(G)$ having odd order.