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

### Seyed Hassan Alavi

Bu-Ali Sina University, Hamedan, Iran### Ashraf Daneshkhah

Bu-Ali Sina University, Hamedan, Iran### Ali Jafari

Bu-Ali Sina University, Hamedan, Iran

## Abstract

Let $G$ be a finite group and $\mathrm {cd}(G)$ 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 a Mathieu group such that $\mathrm {cd}(G) =\mathrm {cd}(H)$, then there exists an abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. In view of Huppert's conjecture (2000), we also provide some examples to show that $G$ is not necessarily a direct product of $A$ and $H$, and hence we cannot extend this conjecture to almost simple groups.

## Cite this article

Seyed Hassan Alavi, Ashraf Daneshkhah, Ali Jafari, On groups with the same character degrees as almost simple groups with socle Mathieu groups. Rend. Sem. Mat. Univ. Padova 138 (2017), pp. 115–127

DOI 10.4171/RSMUP/138-6