Moufang loops with a unique nonidentity commutator (associator, square)

Abstract

In [9], Griess considers a class of loops, called code loops, which have applicability to a construction of the monster and its nonassociative algebra (see [7]). These loops turn out to be Moufang loops which have a unique nonidentity commutator, a unique nonidentity associator, and a unique nonidentity square. Moufang loops with one or more of these properties also play an important role in the authors’ work on loops which have alternative loop rings [2-5, 81. For example, a nonassociative loop which has an alternative loop ring over a ring of characteristic different from two must be a Moufang loop with a unique nonidentity commutator and associator (which coincide). In this paper, we study Moufang loops with the properties in question. In Section 3, we concentrate on loops which have unique nonidentity commutators and/or associators and investigate some of their properties. In Section 4, we consider Moufang loops with a unique nonidentity square and show that these are exactly the code loops of Griess.