Automorphisms of Latin Squares

Abstract

A Latin Square L of order n is an n×n array containing n symbols from [n] = {1, 2, . . . ,n} such that each element of [n] appears once in each row and each column of L. Rows and columns of L are indexed by elements of [n]. An automorphism α of a Latin square is a permutation such that the triple (α, α, α) maps the Latin square L to itself by permuting its rows, columns and symbols by α. Let Aut(n) be the set of all automorphisms of Latin squares of order n. Whether a permutation α belongs to Aut(n) depends only on the cycle structure of α. Stones et al. [1] characterized α ∈ Aut(n) for which α has at most three non-trivial cycles (that is, cycles other than fixed points). A notable feature of this characterisation is that the length of the longest cycle of α is always divisible by the length of every other cycle of α. In this research we prove a related result for automorphisms with four non-trivial cycles.