Permutability degrees of finite groups

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.