On Thompson's conjecture for alternating and symmetric groups

TL;DR

This paper proves that any finite group with conjugacy class sizes matching those of large alternating or symmetric groups must be non-solvable, advancing understanding of group structure based on class sizes.

Contribution

It establishes a new characterization of non-solvable groups through conjugacy class sizes matching large alternating and symmetric groups.

Findings

Groups with class sizes equal to Alt_n (n>4) are non-solvable.

Groups with class sizes equal to Sym_n (n>22) are non-solvable.

Provides a criterion linking conjugacy class sizes to group solvability.

Abstract

For a finite group $G$ denote by $N(G)$ the set of conjugesy class sizes of $G$. We show that every finite group $G$ with the property $N(G)=N(Alt_n), n>4$ or $N(G)=N(Sym_n), n>22$ is non-solvable.