A note on the shortest law for the symmetric group

Let $\alpha (n)$ 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 $\alpha (n)\ge 2n-O(1)$. In this short note we give an improvement by proving that $\alpha (n)\ge \frac{5}{2}n-O(1)$.