A new characterization of A₇ and A₈

Let G be a finite group and π_e(G) be the set of element orders of G. Let k ∈ π_e (G)and m_k be the number of elements of order k in G. Set nse(G):={m_k|k ∈ π_e (G)}. It is proved that A_n are uniquely determined by nse(A_n), where n ∈ {4,5,6}. In this paper, we prove that if G is a group such that nse(G)=nse(A_n) where n ∈ {7,8}, then G ≅ A_n.