On Thompson's conjecture for alternating and symmetric groups

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 $N(G)=N(Al{t}_n),n>4$ or $N(G)=N(Sy{m}_n),n>22$ is non-solvable.