A new characterization of A12

Purpose: It is well known that the conjugacy class sizes have an important influence on the structure of a group. This work is considering a different set of ‘sizes’, the number of elements of a given order.Methods: By using the set nse(G) and the order of G, We prove that G is isomorphic to A12.Results: Thompson’s conjecture is true for A112.Conclusions: We proved that a finite group G is isomorphic to A12, the alternating group A12 of degree 12 if, and only if, |G| = |A12| and nse(G)=nse(A12).