Groups whose same-order types are arithmetic progressions

For any group G, define g ∼ h if g, h ∈ G have the same order. The set of sizes of the equivalent classes with respect to this relation is called the same-order type of G. In this short note we prove that there is no finite group whose same-order type is an arithmetic progression of length 4. This answered an open problem posed by Lazorec and Tˇarnˇauceanu.