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 $\alpha(n)\geq 2n-O(1)$.

In this short note we give an improvement by proving that $\alpha(n)\geq \frac52 n-O(1)$.