ABSTRACT
This chapter introduces some of the permutation groups on the underlying set of a quasigroup that result from the quasigroup structure. These groups are key tools of quasigroup theory. The most accessible are the combinatorial multiplication groups of Section 2.1, the faithful permutation groups generated by the right and left multiplications. As discussed in Section 2.2, the combinatorial multiplication group construction yields a functorial assignment only to surjective quasigroup homomorphisms. The diagonal action of the combinatorial multiplication group on the direct square of a quasigroup yields the quasigroup congruences as the invariant equivalence relations (Section 2.3). (This diagonal action is the cornerstone of the combinatorial character theory in Chapters 6 and 7.) Section 2.4 considers point stabilizers in the combinatorial multiplication group, and the extent to which they generalize the inner automorphism groups of groups. Section 2.5 examines transversals to the point stabilizers. The concept of a loop transversal, essentially going back to Baer, shows how loops arise as a generalization of quotient groups when one relaxes the requirement of normality on a subgroup of a group. Section 2.6 discusses an application of the loop transversal concept to algebraic coding theory. It provides a nice illustration of the way that quasigroup-theoretical concepts may yield new insights even within the context of abelian groups. In Section 2.7, the universal multiplication group U(Q;V) of a quasigroup Q in a given variety V of quasigroups, possibly in the variety Q of all quasigroups, is introduced as a completely functorial multiplication group construction. The universal multiplication group of Q acts on Q via its quotient, the usual or combinatorial multiplication group of Q that is defined in Section 2.1. The corresponding stabilizers are examined in Section 2.8, ready for their application to the module theory of Chapter 10.
