Groups whose same-order types are arithmetic progressions
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 \(4\).
scientific article; zbMATH DE number 7950847