A characterization of A₅ by its Same-order type

Let G be a group, define an equivalence relation s as below: 8 g; h 2 G g s h () jgj = jhj the set of sizes of equivalence classes with respect to this relation is called the same-order type of G. Shen et al. (Monatsh. Math. 160 (2010), 337-341.), showed that A5 is the only group with the same-order type f1; 15; 20; 24g. In this paper, among other things, we prove that a nonabelian simple group G has same-order type fr; m; n; kg if and only if G ?= A5.