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Cycle structure of autotopisms of quasigroups and latin squares

✍ Scribed by Douglas S. Stones; Petr Vojtěchovský; Ian M. Wanless

Publisher
John Wiley and Sons
Year
2012
Tongue
English
Weight
542 KB
Volume
20
Category
Article
ISSN
1063-8539

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✦ Synopsis

Abstract

An autotopism of a Latin square is a triple (α, β, γ) of permutations such that the Latin square is mapped to itself by permuting its rows by α, columns by β, and symbols by γ. Let Atp(n) be the set of all autotopisms of Latin squares of order n. Whether a triple (α, β, γ) of permutations belongs to Atp(n) depends only on the cycle structures of α, β, and γ. We establish a number of necessary conditions for (α, β, γ) to be in Atp(n), and use them to determine Atp(n) for n⩽17. For general n, we determine if (α, α, α)∈Atp(n) (that is, if αis an automorphism of some quasigroup of order n), provided that either αhas at most three cycles other than fixed points or that the non‐fixed points of α are in cycles of the same length. © 2011 Wiley Periodicals, Inc. J Combin Designs

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