Cycle structure of autotopisms of quasigroups and Latin squares

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 every n we determine if (α,α,α) in 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 all non-fixed points of α are in cycles of the same length.

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