Given a group G, we denote by e(G) the set of orders of elements in G. For a natural number m, we denote by sm(G) the number of elements of order m in G, and put nse(G) = {sm(G) ∣ m ∈ e(G)}. The group M is said to be characterizable by NSE if, for every group G, the equality nse(G) = nse(M) implies that G ∼= M. In the present article, we prove that the Mathieu group M12 is characterizable by NSE. This finishes the proof of the characterizability by NSE of all Mathieu simple groups. Thus, some recent results given by C.G. Shao and Q.H. Jiang in 2014 are strengthened in this paper.
