On recognition of $A_6\times A_6$ by the set of conjugacy class sizes

TL;DR

This paper proves that the direct product of two alternating groups of degree six, $A_6 imes A_6$, is uniquely identified by its conjugacy class sizes among all finite groups with trivial center, addressing a question about recognizing simple groups.

Contribution

It demonstrates that $A_6 imes A_6$ can be uniquely characterized by its conjugacy class sizes, contributing to the understanding of how group structure relates to class size sets.

Findings

$A_6 imes A_6$ is uniquely determined by $N(A_6 imes A_6)$ among groups with trivial center.

Provides evidence supporting the conjecture that class size sets can identify certain nonabelian simple groups.

Advances the classification of groups based on conjugacy class size data.

Abstract

For a finite group $G$ denote by $N(G)$ the set of conjugacy class sizes of $G$. Recently the following question has been asked: Is it true that for each nonabelian finite simple group $S$ and each $n\in\mathbb{N}$, if the set of class sizes of a finite group $G$ with trivial center is the same as the set of class sizes of the direct power $S^n$, then $G\simeq S^n$? In this paper we approach an answer to this question by proving that $A_6\times A_6$ is uniquely determined by $N(A_6\times A_6)$ among finite groups with trivial center.