On the number of conjugacy classes of subgroups of a finite group

TL;DR

This paper investigates the ratios measuring how close finite groups are to being Dedekind, establishing density results, structural criteria for nilpotency and Iwasawa groups, and properties of p-groups related to these ratios.

Contribution

It introduces new ratios for subgroup conjugacy classes, proves their density in nilpotent groups, and provides structural criteria for nilpotency and Iwasawa groups based on these ratios.

Findings

The set of ratios in nilpotent groups is dense in [0, 1].

If the ratio exceeds 2/3, the group is nilpotent.

If the ratio exceeds 4/5, the group is an Iwasawa group.

Abstract

Let $k'(G)$ and $L(G)$ be the number of conjugacy classes of subgroups and the subgroup lattice of a finite group $G$, respectively. Our objective is to study some aspects related to the ratios $d'(G)=\frac{k'(G)}{|L(G)|}$ and $d^*(G)=\min\{ d'(S) \mid S\text{ is a section of }G\}$ which measure how close is $G$ from being a Dedekind group. We prove that the set containing the values $d'(G)$, as $G$ ranges over the class of nilpotent groups, is dense in $[0, 1]$. A nilpotency criterion is obtained by proving that if $d^*(G)>\frac{2}{3}$, then $G$ is nilpotent and information on its structure is given. We also show that if $d^*(G)>\frac{4}{5}$, then $G$ is an Iwasawa group. Finally, we deduce a result which ensures that a $p$-group of order $p^n$ ($n\geq 3$) is a Dedekind group. This last result can be extended to the class of nilpotent groups and it also highlights the second maximum values of $d'$ and $d^*$ on the class of $p$-groups of order $p^n$.