Groups whose same-order types are arithmetic progressions

TL;DR

This paper investigates finite groups whose same-order types form arithmetic progressions, focusing on classifying and establishing the existence of such groups with sequences of 3 or 4 elements.

Contribution

It provides new results on the classification and existence of finite groups with same-order types as arithmetic progressions of length 3 or 4.

Findings

Finite groups with 3-element arithmetic progression types are characterized.

Existence of groups with 4-element arithmetic progression types is established.

The maximum length of such arithmetic progression types is 4.

Abstract

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.