Proportion of Nilpotent Subgroups in Finite Groups and Their Properties
João Victor M. de Andrade, Leonardo Santos da Cruz
TL;DR
This work introduces the function $J(G) = \frac{Nil(G)}{L(G)}$, a multiplicative invariant valued in $(0,1]$ that measures the proportion of nilpotent subgroups among all subgroups of a finite group $G$. It establishes the key structural property that $J(G)=1$ if and only if $G$ is nilpotent, and notes that cyclic, abelian, and finite $p$-groups all satisfy $J(G)=1$, guiding attention to non-nilpotent families such as dihedral groups. The authors derive explicit expressions for $Nil(D_{2n})$ and $L(D_{2n})$ using divisor functions, show that the $J$-values are dense in $(0,1]$ via products of dihedral groups, and connect $J$ to the cyclicity degree via $cdeg(G) \le J(G)$ with equality in certain semidirect-product cases. A probabilistic thread analyzes $J$-values under random dihedral-sample schemes, conjecturing that the sample mean converges to a standard normal distribution as sample size grows, a claim buttressed by extensive simulations and Kolmogorov–Smirnov tests. Overall, the paper expands the toolkit for studying finite-group structure by integrating multiplicative function perspectives with density and probabilistic analyses, offering new insights into subgroup distributions and their asymptotic behavior.
Abstract
This work introduces and investigates the function $J(G) = \frac{\text{Nil}(G)}{L(G)}$, where $\text{Nil}(G)$ denotes the number of nilpotent subgroups and $L(G)$ the total number of subgroups of a finite group $G$. The function $J(G)$, defined over the interval $(0,1]$, serves as a tool to analyze structural patterns in finite groups, particularly within non-nilpotent families such as supersolvable and dihedral groups. Analytical results demonstrate the product density of $J(G)$ values in $(0,1]$, highlighting its distribution across products of dihedral groups. Additionally, a probabilistic analysis was conducted, and based on extensive computational simulations, it was conjectured that the sample mean of $J(G)$ values converges in distribution to the standard normal distribution, in accordance with the Central Limit Theorem, as the sample size increases. These findings expand the understanding of multiplicative functions in group theory, offering novel insights into the structural and probabilistic behavior of finite groups.