  In 2008, Cavenagh and Dr\'{a}pal, et al, 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 generalize 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

Combinatorics