On the Characterization of Alternating Groups by Codegrees

TL;DR

This paper proves that the set of codegrees of irreducible characters uniquely identifies alternating groups of degree at least 5, providing a new characterization based on character theory.

Contribution

It establishes that alternating groups are uniquely determined by their codegree sets, a novel characterization in the representation theory of finite groups.

Findings

Alternating groups are characterized by their codegree sets.

Codegree sets uniquely identify $ ext{A}_n$ for $n geq 5$.

The result advances understanding of group characterization via character invariants.

Abstract

Let $G$ be a finite group and $\mathrm{Irr}(G)$ the set of all irreducible complex characters of $G$. Define the codegree of $\chi \in \mathrm{Irr}(G)$ as $\mathrm{cod}(\chi):=\frac{|G:\mathrm{ker}(\chi) |}{\chi(1)}$ and denote by $\mathrm{cod}(G):=\{\mathrm{cod}(\chi) \mid \chi\in \mathrm{Irr}(G)\}$ the codegree set of $G$. Let $\mathrm{A}_n$ be an alternating group of degree $n \ge 5$. In this paper, we show that $\mathrm{A}_n$ is determined up to isomorphism by $\mathrm{cod}(\mathrm{A}_n)$.