Latin bitrades derived from quasigroup autoparatopisms

Abstract

In 2008, Cavenagh, Drápal and Hämäläinen described a method of constructing Latin trades using groups. The Latin trades that arise from this construction are entry-transitive (that is, there always exists an autoparatopism of the Latin trade mapping any ordered triple to any other ordered triple). Moreover, useful properties of the Latin trade can be established using properties of the group. However, the construction does not give a direct embedding of the Latin trade into any particular Latin square. In this paper, we propose a similar approach to the above to construct Latin trades embedded in a Latin square L, via the autoparatopism group of the quasigroup with Cayley table L. We apply this theory to identify non-trivial entry-transitive trades in some group operation tables as well as in Latin squares that arise from quadratic orthomorphisms.