Gallagher's Theorem for Real-valued Brauer Characters

Abstract

Let $G$ be a finite group, let $N$ be a normal subgroup of $G$, and let $θ$ be an irreducible $p$-Brauer character of $N$, for a prime $p$. It is well known that the number of irreducible Brauer characters of $G$ lying over $θ$ equals the number of $p$-regular $θ$-good conjugacy classes of $G_θ/N$. In this note we express the number of real-valued irreducible Brauer characters of $G$ lying over $θ$ in terms of the $p$-regular $θ$-good conjugacy classes of $G_θ/N$. This is a modular analogue of one of the main results in a recent paper (arXiv:2301.01177) by John Murray.