A connection between the number of subgroups and the order of a finite group

Abstract

For a finite group $G$, we associate the quantity $\beta(G)=\frac{|L(G)|}{|G|}$, where $L(G)$ is the subgroup lattice of $G$. Different properties and problems related to this ratio are studied throughout the paper. We determine the second minimum value of $\beta$ on the class of $p$-groups of order $p^n$, where $n\geq 3$ is an integer. We show that the set containing the quantities $\beta(G)$, where $G$ is a finite (abelian) group, is dense in $[0,\infty).$ Finally, we consider $\beta$ to be a function on $L(G)$ and we mark some of its properties, the main result being the classification of finite abelian $p$-groups $G$ satisfying $\beta(H)\leq 1, \ \forall \ H\in L(G).$