Let G be a group. The same-order type of G may be defined to be the set of sizes of equivalence classes for the equivalence relation ∼ on G defined by ∀ g, h ∈G g ∼ h ⇐⇒ |g| = |h|. Shen et al. (Monatsh Math 160:337–341, 2010), showed that A5 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 with just four members if and only if G ∼= A5
