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

Abstract

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

15 pages