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arXiv · 2607.08557

A note on the shortest law for the symmetric group

Adrian Beker·Luka Milićević·Rudi Mrazović
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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 $α(n)\geq \frac52 n-O(1)$.

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Adrian Beker, Luka Milićević, Rudi Mrazović. 2026-07-09. A note on the shortest law for the symmetric group. https://arxiv.org/abs/2607.08557

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