A note on the shortest law for the symmetric group

Adrian Beker; Luka Milićević; Rudi Mrazović

A note on the shortest law for the symmetric group

Combinatorics 2026-07-09 v1 Group Theory

Authors: Adrian Beker , Luka Milićević , Rudi Mrazović

Abstract

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 $\alpha(n)\geq \frac52 n-O(1)$ .

Keywords

symmetric group shortest law two-variable law subgroup separability lower bound

Cite

@article{arxiv.2607.08557,
  title  = {A note on the shortest law for the symmetric group},
  author = {Adrian Beker and Luka Milićević and Rudi Mrazović},
  journal= {arXiv preprint arXiv:2607.08557},
  year   = {2026}
}

Comments

9 pages, 1 figure