A CHARACTERIZATION OF PROJECTIVE SPECIAL LINEAR GROUP L\(_3\)(5) BY nse

Abstract

Let G be a group and ω(G) be the set of element orders of G.  Let k ∈ ω(G) and s_k be the number of elements of order k in G.  Let nse(G) = {s_k | k ∈ ω(G)}.  In Khatami et al. and Liu’s works, the authors proved that the groups L₃(2) and L₃(4) are unique determined by nse.  In this paper, we prove that if G is a group such that nse(G) = nse(L₃(5)), then G \(\cong\) L₃(5).