Even degree characters in principal blocks

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 $p\ne 7$ or if $M_{22}$ is not a composition factor of a group $G$, then the condition above is equivalent to $G/O_{p^{\prime }}(G)$ having odd order.