On the common transversal probability

Abstract

Let 𝐺 be a finite group, and let 𝐻 be a subgroup of 𝐺. We compute the probability, denoted by P G ⁢ ( H ) 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 tp ⁡ ( G ) \operatorname{tp}(G) , the common transversal probability of 𝐺 to be the minimum, taken over all subgroups 𝐻 of 𝐺, of P G ⁢ ( H ) P_{G}(H) . We prove a number of results regarding the invariant tp ⁡ ( G ) \operatorname{tp}(G) , like lower and upper bounds, and possible values it can attain. We also show that tp ⁡ ( G ) \operatorname{tp}(G) determines structural properties of 𝐺. Finally, several open problems are formulated and discussed.