Groups whose same-order types are arithmetic progressions

Author(s):

Rulin Shen * , Yutao Jin

Message:
Article Type:
Research/Original Article (دارای رتبه معتبر)
Abstract:
For any 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 is called the same-order type of $G$. In this short note we prove that there is no finite group whose same-order type is an arithmetic progression of length $4$. This answered an open problem posed by Lazorec and  Tˇarnˇauceanu
Keywords:

Element Orders , Same-Order Type , Arithmetic Progression

Language:
English
Published:
International Journal of Group Theory, Volume:14 Issue: 3, Sep 2025
Pages:
165 to 170
https://www.magiran.com/p2789479