Groups whose same-order types are arithmetic progressions
type: Item
instance of: scholarly article
title: Groups whose same-order types are arithmetic progressions
author: Rulin Shen
author: Yutao Jin
published in: International Journal of Group Theory
publication date: 2024-11-28T00:00:00Z
review text: For a 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 was called the \textsl{same-order type} of \(G\) by \textit{R. Shen} [Commun. Algebra 40, No. 6, 2140--2150 (2012; Zbl 1259.20030)]. \textit{M.-S. Lazorec} and \textit{M. Tărnăuceanu} [Quaest. Math. 45, No. 8, 1309--1316 (2022; Zbl 1514.20086)] proved that if \(G\) is a finite group such that its same-order type of \(G\) is an arithmetic progression, then this type is \(\leq 4\). They posed the following problem: Is there any finite non-nilpotent group whose same-order type is an arithmetic progression of length \(4\)?\N\NIn this short note it is proved that there is no finite group whose same-order type is an arithmetic progression of length \(4\).
reviewed by: Grigore Călugăreanu
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 7950847
zbMATH Keywords: element orders
zbMATH Keywords: same-order type
zbMATH Keywords: arithmetic progression
MaRDI profile type: Publication
DOI: 10.22108/IJGT.2024.141416.1903
description: scientific article; zbMATH DE number 7950847