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 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