Thompson’s conjecture for alternating groups

Abstract

Given a finite group G, denote by N(G) the set of sizes of its conjugacy classes. We show that if G has trivial center and N(G) equals $N(Al{t}_n)$ or $N(Sy{m}_n)$ for $n\ge 23$ then G has a composition factor isomorphic to the alternating group Alt_k for some $k\le n$ such that the half-interval $(k,n]$ contains no primes. As a corollary, we prove Thompson’s conjecture is valid for simple alternating groups.

Keywords:

• Finite group
• alternating group
• conjugacy class
• Thompson’s conjecture

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D06

Additional information

Funding

The work is supported by the Russian Science Foundation (Project no. 15-11-10025).