On the common transversal probability in finite groups

Let $G$ be a finite group, and let $H$ be a subgroup of $G$. We compute the probability, denoted by $P_G(H)$, that a left transversal of $H$ in $G$ is also a right transversal, thus a two-sided one. Moreover, we define, and denote by $\mathrm{t}\mathrm{p}(G)$, the common transversal probability of $G$ to be the minimum, taken over all subgroups $H$ of $G$, of $P_G(H)$. We prove a number of results regarding the invariant $\mathrm{t}\mathrm{p}(G)$, like lower and upper bounds, and possible values it can attain. We also show that $\mathrm{t}\mathrm{p}(G)$ determines structural properties of $G$. Finally, several open problems are formulated and discussed.