Bounds for the probability of generating the symmetric and alternating groups.
The authors give explicit, asympotically sharp bounds for the probability \(p(S_n)\) that a pair of random permutations of degree \(n\) generates either \(S_n\) or \(A_n\) and also for the probability \(p(A_n)\) that a pair of random even permutations of degree \(n\) generates \(A_n\). Precisely they prove that if \(n\geq 4\) and \(X\in\{A_n,S_n\},\) then \(1-1/n-13/{n^2}<p(X)\leq 1-1/n+2/{3n^2}\). As an application they answer a question of Wiegold in the case of alternating groups, proving that \(A_n^t\) is 2-generated if \(t\leq\sqrt{|A_n|}\) and \(n\geq 5\).
scientific article; zbMATH DE number 5872602
scientific article