On characterisation of a finite group by the set of conjugacy class sizes

TL;DR

This paper explores conditions under which the set of conjugacy class sizes uniquely characterizes a finite group, extending Thompson's conjecture by relaxing some of its original assumptions.

Contribution

The paper generalizes previous results by proving that conjugacy class sizes can determine a finite group without requiring the group to have a trivial center or the simple nature of the related group.

Findings

Thompson's conjecture is extended to broader cases.

Conjugacy class sizes can uniquely identify certain finite groups.

The paper provides new proofs for generalized conditions.

Abstract

Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes. In the 1980's Thompson conjectured that the equality $N(G)=N(S)$, where $Z(G)=1$ and $S$ is simple, implies the isomorphism $G\simeq S$. In a series of papers of different authors Thompson's conjecture was proved. In this paper, we show that in some cases it is possible to omit the conditions $Z(G)=1$ and $S$ is simple and prove a more general result.