On the common transversal probability (Q6634498)

Let \(G\) be a finite group, and let \(H\) be a subgroup of \(G.\) It is a well-known application of Hall's Marriage Theorem that there exists a common set of representatives for the left and the right cosets of \(H\) in \(G.\) Even though the existence of such a double transversal is guaranteed, it seems unlikely that a left transversal of a non-normal subgroup will also be a right transversal. One of the main objectives of this paper is to compute the probability \(P_G(H)\) that a randomly chosen left transversal of \(H\) in \(G\) is also a right transversal. For example it is proved that if \(H\) is not normal in \(G\) and \([G:H]=n,\) then \N\[\N\frac{(n-1)!}{(n-1)^{n-1}}\leq P_G(H)\leq \frac{1}{2}\N\]\Nand \(\lim_{n\to \infty}P_G(H)=0.\) Moreover, the authors define, and denote by \(\mathrm{tp}(G),\) the common transversal probability of \(G\) to be the minimum, taken over all subgroups \(H\) of \(G,\) of \(P_G(H)\). They investigate how \(\mathrm{tp}(G)\) influences the structure of the group \(G.\) For example if \(\mathrm{tp}(G) > (1/2)^{40},\) then either \(G\) is soluble or has a section isomorphic to \(A_5.\)