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**The Moufang loops of order less than \(64\).**
*(English)*
Zbl 0964.20043

Commack, NY: Nova Science Publishers. xvii, 287 p. (1999).

This is a monograph listing all 158 nonassociative loops of order less than 64. In 1974, O. Chein [Trans. Am. Math. Soc. 188, 31-51 (1974; Zbl 0286.20088)] determined all nonassociative Moufang loops of order less than 32, and later [Mem. Am. Math. Soc. 197 (1978; Zbl 0378.20053)] he extended his work to all those of order less than 64. For each such loop, Chein gave a “normal form” for its elements in terms of generators and then described the normal form of the product of two such elements. In the monograph under review, full multiplication tables for each loop of order less than 32 are exhibited. For all nonassociative loops of order between 32 and 64, such tables would have become too cumbersome, and only basic information about them is provided. In an appendix of the monograph, the loops of order less than 64, whose loop rings are alternative but not associative, are being listed.

Reviewer: R.Artzy (Haifa)