The shortest law on the symmetric group
How long must a nontrivial two-letter word w(x, y) be if it equals the identity for every pair of permutations in S_n? The answer lies somewhere between linear and quasi-polynomial in n.
Cite
@misc{cairn-shortest-law-symmetric-group,
title = {The shortest law on the symmetric group},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/shortest-law-symmetric-group}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
} Also: CITATION.cff · Atom feed of results
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The problem
The question
A law of a group G is a nontrivial reduced word w(x, y) in the free group on two letters such that w(g, h) = 1 for all g, h ∈ G. Every finite group has laws (for example x^e, with e the exponent). Let α(n) be the length of the shortest law of the symmetric group S_n. What is the order of growth of α(n)?
What is known
- A law of S_n must have length at least about 2n (lower bound 2n − 1 recorded in the report); a 2026 preprint improves this to (5/2)n − O(1) for two-variable laws.
- Kozma and Thom showed that S_n has laws of length exp(O((log n)^4 log log n)).
- So the truth is somewhere between linear and quasi-polynomial in n — a huge gap.
What counts as progress
- Exact values for small n. A table of α(n) for n = 3, 4, 5, … with, for each n, an explicit shortest law and an exhaustive search showing that nothing shorter works (with symmetry reductions documented).
- Shorter explicit laws for specific n, or a family beating Kozma–Thom.
- Better lower bounds, or a proof that α(n) grows polynomially / superpolynomially.
How it is checked
A law is a certificate: evaluating the word on all pairs of S_n (with symmetry reductions: conjugation, and one argument up to conjugacy class) confirms it is a law. Minimality claims ship the search code and logs (level B).
Source. Posed by Sean Eberhard in the open problem session of the Oberwolfach workshop Mini-Workshop: Growth and Expansion in Groups (2024), recorded in Oberwolfach Reports 17/2024, p. 1036 (EMS Press, DOI 10.4171/OWR/2024/17), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.