Subgroup S-commutativity degrees of finite groups

TL;DR

This paper introduces a generalization of the subgroup commutativity degree in finite groups, providing new lower bounds and insights into the structure of groups relative to subgroup lattice interactions.

Contribution

It extends the concept of subgroup commutativity degree by considering sublattices of the subgroup lattice, offering new bounds and structural insights.

Findings

Derived new lower bounds for generalized subgroup commutativity degrees

Connected subgroup lattice properties to group classification

Enhanced understanding of subgroup interactions in finite groups

Abstract

The so--called subgroup commutativity degree $sd(G)$ of a finite group $G$ is the number of permuting subgroups $(H,K) \in \mathrm{L}(G) \times \mathrm{L}(G)$, where $\mathrm{L}(G)$ is the subgroup lattice of $G$, divided by $|\mathrm{L}(G)|^2$. It allows us to measure how $G$ is far from the celebrated classification of quasihamiltonian groups of K. Iwasawa. Here we generalize $sd(G)$, looking at suitable sublattices of $\mathrm{L}(G)$, and show some new lower bounds.