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

## TL;DR

This paper investigates the ratio of the number of subgroups to the order of finite groups, exploring its properties, bounds, and density, with a focus on p-groups and abelian groups.

## Contribution

It determines the second minimum of the subgroup-to-order ratio for p-groups and classifies abelian p-groups where this ratio is at most one for all subgroups.

## Key findings

- The set of subgroup-to-order ratios is dense in [0, ∞).
- The second minimum of the ratio for p-groups of order p^n is identified.
- Classification of abelian p-groups with bounded subgroup ratios.

## 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 $\beta(H)\leq 1, \ \forall \ H\in L(G).$