On groups with the same character degrees as almost simple groups with socle the Mathieu groups

TL;DR

This paper proves that finite groups with the same set of irreducible character degrees as certain almost simple groups with Mathieu group socles are closely related to those groups, extending Huppert's conjecture.

Contribution

It establishes a structural link between groups sharing character degrees with almost simple Mathieu group-based groups, advancing the understanding of character degree sets.

Findings

If $cd(G) = cd(H)$, then $G$ has an abelian subgroup with $G/A ext{ isomorphic to } H$

Supports extension of Huppert's conjecture to almost simple groups with Mathieu socles

Provides a step towards classifying groups by their character degrees

Abstract

Let $G$ be a finite group and $cd(G)$ denote the set of complex irreducible character degrees of $G$. In this paper, we prove that if $G$ is a finite group and $H$ is an almost simple group whose socle is Mathieu group such that $cd(G) =cd(H)$, then there exists an Abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. This study is heading towards the study of an extension of Huppert's conjecture (2000) for almost simple groups.