A new characterization of alternating groups

Abstract

Let G be a finite group and let πe(G) be the set of element orders of G. Let k ∈ πe(G) and let mk be the number of elements of order k in G. Set nse(G):={mk|k ∈ πe(G)}. In this paper, we show that if n= r, r + 1, r + 2, r + 3 r + 4, or r + 5 where r ≥ 5 is the greatest prime not exceeding n, then An characterizable by nse and order.