A result on the number of cyclic subgroups of a finite group

Proceedings of the Japan Academy, Series A. Mathematical Sciences

article

Marius Tărnăuceanu¹ 

[1] Faculty of Mathematics, ‘‘Al.I. Cuza’’ University

关键词: Finite groups;    p-groups;    number of cyclic subgroups.;   

DOI  :  10.3792/pjaa.96.018

学科分类：数学（综合）

来源: Japan Academy

【 摘 要 】

Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the equality occurs. This gives some sufficient conditions for a finite group to be 4-abelian or abelian.

【 授权许可】

Unknown