A note on the probability of generating alternating or symmetric groups

TL;DR

This paper refines probability estimates for generating alternating and symmetric groups, providing sharp bounds and implications for the minimal number of generators for their direct products.

Contribution

It offers the sharp lower bound for the probability of generating these groups, improving previous estimates and deriving bounds on the number of factors in their direct products.

Findings

Established the quadratic lower bound in terms of n^{-1}

Provided improved bounds on the number of factors in direct products

Enhanced understanding of generation probabilities for symmetric groups

Abstract

We improve on recent estimates for the probability of generating the alternating and symmetric groups $\mathrm{Alt}(n)$ and $\mathrm{Sym}(n)$. 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 $h(\mathrm{Alt}(n))$ such that a direct product of $h(\mathrm{Alt}(n))$ copies of $\mathrm{Alt}(n)$ can be generated by two elements.