A note on the shortest law for the symmetric group

Abstract

Let $α(n)$ denote the length of the shortest non-trivial two-variable law for the symmetric group $S_n$. Buskin's quantitative subgroup-separability argument gives the classical lower bound $α(n)\geq 2n-O(1)$. In this short note we give an improvement by proving that $α(n)\geq \frac52 n-O(1)$.