Abstract. Let G be a group, define an equivalence relation ∼ as below: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 A 5 is the only group with the same-order type {1, 15, 20, 24}. In this paper, among other things, we prove that a nonabelian simple group G has same-order type {r, m, n, k} if and only if G ∼ = A 5 .

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Let G be a group and let T