Cycle structure of autotopisms of quasigroups and latin squares

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 α is an automorphism of some quasigroup of order n), provided that either α has at most three cycles other than fixed points or that the non-fixed points of α are in cycles of the same length.

Autoparatopisms of Quasigroups and Latin Squares

“…We find, by computation, that these necessary conditions are sufficient when . The analogous task for Atp( n ) was carried out in and our approach follows a similar direction to that paper, at least initially. For an earlier catalog of Atp( n ) for , see and for related work on partial Latin squares, see .…”

“…It follows from the above two results that any autoparatopism is conjugate to an autoparatopism of the form , or . The first of these possibilities has been well studied in , so we will concentrate mostly on the second and third possibilities. Moreover, in these cases the only salient consideration is the cycle structures of β and γ.…”

“…In this paper, we investigate a fundamental question: What symmetries can a Latin square have? See for a survey of earlier results related to this topic, stretching all the way back to Euler's seminal work. In particular, we note that symmetry has played a critical role in enumerations such as and is helpful for creating Latin squares with desirable properties (see, e.g., the survey ).…”

On Computing the Number of Latin Rectangles

“…This question, along with analogous questions for Latin rectangles, has been a hot research topic for Stones [35], and is linked to the autotopisms and automorphisms of Latin rectangles [4,39] (see also [12,38]), and orthomorphisms and partial orthomorphisms of finite cyclic groups [40,41]. Divisors for the number of even/odd Latin squares have been used in proving special cases of the Alon-Tarsi Conjecture [7] (see also [8,16,37,42]).…”

A Latin square autotopism secret sharing scheme

“…Moreover, often an arbitrary isotopism is not an autotopism of any Latin square of order n [26]. This motivates our choice to use the specific cycle structure.…”