ABSTRACT
This chapter applies the module theory introduced in Chapter 10 to address various issues in quasigroup theory. Section 11.1 interprets central piques as modules over a singleton quasigroup, and shows how the free central pique on one generator is the nonassociative analogue of the group of integers, providing indices for nonassociative powers. Section 11.2 uses the concepts of Section 10.5 to define the exponent of a quasigroup. Based on this definition, Section 11.4 formulates Burnside’s problem for quasigroups. As discussed in Section 11.3, Steiner triple systems play a critical role here: although they have exponent 3, they have infinite universal multiplication groups in the variety of all Steiner triple systems. Section 11.5 applies module theory to explain the apparently ad hoc details of the Zassenhaus-Bruck construction of the free commutative Moufang loop on three generators. It also transpires that the module concept due to Eilenberg [51], as interpreted by Loginov for Moufang loops [105], is not strong enough to implement the Zassenhaus-Bruck construction. The final section, Section 11.6, gives a brief survey of extension and cohomology theory for each variety of quasigroups, using the equivalence between modules and self-centralizing congruences described in Section 10.2.
