On Thompson's conjecture for alternating group of large degree

TL;DR

This paper proves that for large alternating groups with certain prime sum conditions, any finite group with the same conjugacy class sizes must be isomorphic to the alternating group itself.

Contribution

It establishes a uniqueness result for large alternating groups based on conjugacy class sizes under specific prime sum conditions.

Findings

Finite groups with the same conjugacy class sizes as large alternating groups are isomorphic to those groups.

The result applies to groups with trivial center and large degree n > 1361.

The proof relies on prime decomposition conditions of n or n-1.

Abstract

For a finite group $G$, let $N(G)$ denote the set of conjugacy class sizes of $G$. We show that if every finite group $G$ with trivial center such that $N(G)$ equals to $N(Alt_n)$, where $n>1361$ and at least one of numbers $n$ or $n-1$ are decomposed into a sum of two primes, then $G\simeq Alt_n$.