Character formula for conjugacy classes in a coset

TL;DR

This paper explores the relationship between conjugacy classes in a coset of a finite group and the irreducible characters of the group, providing formulas and consequences for the structure of characters and classes.

Contribution

It introduces a formula connecting conjugacy classes in a coset with irreducible characters that are non-zero on that coset, extending understanding of character theory in finite groups.

Findings

Conjugacy classes in a coset relate to non-zero irreducible characters.

When the quotient is cyclic, the number of extending characters equals the number of classes.

Several structural consequences for character extensions and class counts.

Abstract

Let $G$ be a finite group and $N<G$ a normal subgroup with $G/N$ abelian. We show how the conjugacy classes of $G$ in a given coset $qN$ relate to the irreducible characters of $G$ that are not identically $0$ on $qN$. We describe several consequences. In particular, we deduce that when $G/N$ is cyclic generated by $q$, the number of irreducible characters of $N$ that extend to $G$ is the number of conjugacy classes of $G$ in $qN$.