Short approximate laws on S_n and SL_n(2)
Find the shortest word w(x, y) that equals the identity on 99% of pairs of elements of S_n (or of SL_n(2)). Known bounds for S_n range from about √n to n^{3+o(1)}.
Cite
@misc{cairn-approximate-laws-symmetric-group,
title = {Short approximate laws on S_n and SL_n(2)},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/approximate-laws-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
An approximate law of a finite group G is a nontrivial reduced word w(x, y) in the free group F_2 such that w(g, h) = 1 for at least 99% of all pairs (g, h) ∈ G × G.
- What is the length of the shortest approximate law of the symmetric group S_n?
- The same for SL_n(2). Is there an approximate law of length at most 10n?
What is known
For S_n the abstract gives a lower bound of order c·√n and an upper bound n^{3+o(1)}. Exact laws (holding for all pairs) are much longer — see the companion problem on the shortest law on S_n.
What counts as progress
- Explicit short words with a computed (exact or rigorously estimated) fraction of pairs on which they vanish, for a range of n; a table of the shortest known approximate laws.
- Constructions giving a better upper bound, or better lower bounds.
How it is checked
For small n the fraction can be computed exactly over all pairs; for larger n by sampling with a stated confidence bound. Code and data must be reproducible (level B).
Source. Posed by Sean Eberhard in an extended abstract of the Oberwolfach workshop Mini-Workshop: Growth and Expansion in Groups (2024), recorded in Oberwolfach Reports 17/2024, p. 1016 (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.