# Huppert’s conjecture and almost simple groups

### Ashraf Daneshkhah

Bu-Ali Sina University, Hamedan, Iran

## 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 $H_{0}=PSL(2,q)$ with $q=2_{f}$ ($f$ prime) such that $cd(G)=cd(H)$, then there exists an abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. In view of Huppert's conjecture (2000), the main result of this paper gives rise to some examples that $G$ is not necessarily a direct product of $A$ and $H$, and consequently, we cannot extend this conjecture to almost simple groups.

## Cite this article

Ashraf Daneshkhah, Huppert’s conjecture and almost simple groups. Rend. Sem. Mat. Univ. Padova 148 (2022), pp. 173–184

DOI 10.4171/RSMUP/103