A note on the probability of generating alternating or symmetric groups

摘要

We improve on recent estimates for the probability of generating the alternating and symmetric groups A (n) and S (n) . In particular, we find the sharp lower bound if the probability is given by a quadratic in n (-1). This leads to improved bounds on the largest number h(A (n) ) such that a direct product of h(A (n) ) copies of A (n) can be generated by two elements.