Permutability degrees of finite groups

TL;DR

This paper introduces the permutability degree of finite groups, a new measure based on subgroup permutability, and explores its relationships with existing probabilities of subgroup and element commutativity, revealing structural insights.

Contribution

It defines the permutability degree for finite groups and establishes inequalities relating it to subgroup and element commuting probabilities, advancing structural understanding.

Findings

Established inequalities between permutability degree, subgroup commuting probability, and element commuting probability.

Connected permutability degree with structural restrictions of finite groups.

Provided new bounds and relations among key probabilistic measures in group theory.

Abstract

Given a finite group $G$, we introduce the \textit{permutability degree} of $G$, as $$pd(G)=\frac{1}{|G| \ |\mathcal{L}(G)|} {\underset{X \in \mathcal{L}(G)}\sum}|P_G(X)|,$$ where $\mathcal{L}(G)$ is the subgroup lattice of $G$ and $P_G(X)$ the permutizer of the subgroup $X$ in $G$, that is, the subgroup generated by all cyclic subgroups of $G$ that permute with $X\in \mathcal{L}(G)$. The number $pd(G)$ allows us to find some structural restrictions on $G$. Successively, we investigate the relations between $pd(G)$, the probability of commuting subgroups $sd(G)$ of $G$ and the probability of commuting elements $d(G)$ of $G$. Proving some inequalities between $pd(G)$, $sd(G)$ and $d(G)$, we correlate these notions.