Proportion of Nilpotent Subgroups in Finite Groups and Their Properties

TL;DR

This paper introduces a new function to analyze the proportion of nilpotent subgroups in finite groups, revealing distribution patterns and probabilistic behaviors through theoretical and computational methods.

Contribution

It defines the function J(G) for finite groups, investigates its distribution, and conjectures its normal convergence based on extensive simulations.

Findings

J(G) values are densely distributed in (0,1]

Product density of J(G) in dihedral groups

Conjecture of normal distribution of sample mean of J(G)

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.