A non empty set Q with an operation is a quasigroup if each of the equations ax=b and xa=b has a unique solution x in Q for all a,b in Q. The associative quasigroups are exactly the groups.
A loop is a quasigroup with a neutral element, i.e. there exists an element 1 with 1x=x1=x for all x in the loop.
Two quasigroups A,B are isotopic if there exist three bijections with
for all x,y in A. The following two simple observations are sometimes usefull.
Lemma. If A is a quasigroup and , then there exists an isotopic loop on the same set A with neutral element e.
Lemma. A quasigroup A isotopic to a group G is a group isomorphic with G.