Groups whose same-order types are arithmetic progressions

For any group G, define g\sim h if g,h\in 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 ۴. This answered an open problem posed by Lazorec and  Tˇarnˇauceanu

کلیدواژه ها:

element orders ، Same-order type ، Arithmetic progression