NSE characterization of some linear groups

For a finite group G, let nse(G) = {mk | k ∈ πe(G)}, where mk is the number of elements of order k in G and πe(G) is the set of element orders of G. In this paper, we prove that G ∼= Lm(2) if and only if p | |G| and nse(G) = nse(Lm(2)), where m ∈ {n, n + 1} and 2^n − 1 = p is a prime number.