Groups whose same-order types are arithmetic progressions

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The same-order type τe (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.