Abstract
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.
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