Mihai-Silviu Lazorec, Marius Tarnauceanu: A note on the number of cyclic subgroups of a finite group, 403-416

Abstract:

Let $G$ be a finite group, $L_1(G)$ be its poset of cyclic subgroups and consider the quantity $\alpha(G)=\frac{\vert L_1(G)\vert}{\vert G\vert}$. The aim of this paper is to study the class $\cal{C}$ of finite nilpotent groups having $\alpha(G)=\frac{3}{4}$. We show that if $G$ belongs to this class, then it is a 2-group satisfying certain conditions. Also, we study the membership of some classes of finite groups to $\cal{C}$.

Key Words: Poset of cyclic subgroups, nilpotent groups, subgroup lattice.

2010 Mathematics Subject Classification: Primary 20D15; Secondary 20D60, 20D30.