On Thompson’s conjecture for alternating and symmetric groups of degree greater than 1361

Resumo

Let G be a finite group G, and let N(G) be the set of sizes of its conjugacy classes. It is shown that, if N(G) equals N(Alt_n) or N(Sym_n), where n > 1361, then G has a composition factor isomorphic to an alternating group Altm with m ≤ n and the interval (m, n] contains no primes.