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.