Moufang loops with a unique nonidentity commutator (associator, square) (Q909803)

A class of loops, called code loops, turns out to be Moufang loops which have some properties like a unique nonidentity associator, a unique nonidentity square, and a unique nonidentity commutator. Moufang loops with one or more of these properties play an important role for loops which have alternative loop rings. In this article, the authors study Moufang loops with the properties listed above. So, in Section 3, we have loops which have unique nonidentity commutators and/or associators, and some properties are investigated. In Section 4, the authors consider Moufang loops with a unique nonidentity square, and show that these are exactly the code loops studied by \textit{R. L. Griess} jun. [J. Algebra 100, 224-234 (1986; Zbl 0589.20051)].