Counting real characters after Gallagher

TL;DR

This paper develops a method to count the number of real irreducible characters of a finite group that extend a given irreducible character of a normal subgroup, enhancing understanding of character theory.

Contribution

It introduces a new counting technique for real irreducible characters over a normal subgroup, providing explicit formulas and extending previous character theory results.

Findings

Derived formulas for counting real characters over normal subgroups

Extended character theory to include real character counts

Provided explicit examples illustrating the counting method

Abstract

Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let $\theta$ be an irreducible character of $N$. We count the real irreducible characters of $G$ lying over $\theta$