Thompson's conjecture for alternating group

TL;DR

This paper proves Thompson's conjecture for simple alternating groups by analyzing the conjugacy class sizes of finite groups with trivial centers, establishing conditions under which such groups contain alternating group composition factors.

Contribution

It demonstrates that if a finite group with trivial center has conjugacy class sizes matching those of an alternating or symmetric group, then it contains an alternating group as a composition factor, confirming Thompson's conjecture for simple alternating groups.

Findings

Finite groups with trivial center and specific conjugacy class sizes contain alternating group factors.

Thompson's conjecture holds for simple alternating groups under the given conditions.

Conditions on prime intervals are crucial for the main result.

Abstract

Let $G$ be a finite group, and let $N(G)$ be the set of sizes of its conjugacy classes. We show that if a finite group $G$ has trivial center and $N(G)$ equals to $N(Alt_n)$ or $N(Sym_n)$ for $n\geq 23$, then $G$ has a composition factor isomorphic to an alternating group $Alt_k$ such that $k\leq n$ and the half-interval $(k, n]$ contains no primes. As a corollary, we prove the Thompson's conjecture for simple alternating groups.