Permutability Degrees of Finite Groups

D. E. Otera, F. G. Russo

Given a finite group $G$, we introduce the permutability degree of $G$, as \[ pd(G)=\frac1{|G||\mathcal L(G)|}um_{Xı\mathcal L(G)}|P_G(X)| \] 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.