  The same-order type $\tau_e(G)$ of a finite group $G$ is a set formed of the
sizes of the equivalence classes containing the same order elements of $G$. In
this paper, we study an arithmetical property of this set. More exactly, we
outline some results on the classification and existence of finite groups whose
same-order types are arithmetic progressions formed of 3 or 4 elements, the
latter being the maximum size of such a sequence.
