A characterization of some alternating group by its order and special conjugacy class sizes

Abstract

Let G be a group and N(G) be the set of the sizes of conjugacy class of G.  Let m_p(G) be the number from N(G) which is not divisible by p and let A_n be the alternating group of degree n.  The alternating groups A₅, A₆, A₇, A₈, and A₉ are characterized by their orders and special conjugacy class sizes.  So in generality, are the alternating groups characterized by their orders and some special conjugacy class size(s)?  In this paper, we show that G is a finite group such that m_p(G) = m_p(A_n) and m₂(G) = m₂(A_n) where n ∈ {p, p + 1, p + 2}, then G is isomorphic to A_n.