Characterization of the group $A_5\times A_5\times A_5$ by the set of conjugacy class sizes

I. B. Gorshkov^ab, V. V. Pan'shin<sup>c

a</sup> Novosibirsk State Technical University
^b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^c Novosibirsk State University

Abstract: For a finite group $G$, we denote by $N(G)$ the set of its conjugacy class sizes. Recently, the following question was posed: given any $n\in\mathbb{N}$ and an arbitrary non-Abelian finite simple group $S$, is it true that $G\simeq S^n$ if $G$ is a group with trivial center and $N(G)=N(S^n)$? The answer to this question is known for all simple groups $S$ with $n=1$, and also for $S\in\{A_5,A_6\}$, where $A_k$ denotes the alternating group of degree $k$, with $n=2$. It is proved that the group $A_5\times A_5\times A_5$ is uniquely defined by the set $N(A_5\times A_5\times A_5)$ in the class of finite groups with trivial center.

Keywords: finite groups, alternating groups, conjugacy classes.

Received: 22.08.2022
Revised: 06.12.2024

DOI: 10.33048/alglog.2024.63.203

 English version: Algebra and Logic, 2024, 63:2, 105–113