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**Theory of K-loops.**
*(English)*
Zbl 0997.20059

Lecture Notes in Mathematics. 1778. Berlin: Springer. x, 186 p. (2002).

K-loops can be regarded as a non-associative generalization of Abelian groups. They appeared in the framework of loop and quasigroup theory quite early, but their study gained momentum especially during the last ten years. This is due to the fact that K-loops are naturally connected to special relativity (as shown by Ungar), to the more studied Bol-loops and they are also intimately related to the structure theory of strictly 2-transitive groups.

The present book provides the first systematic and state-of-the-art account of the theory and it is appropriately divided into three parts. In the first six chapters, the basic theory is developed in a coherent and self-contained way, with complete proofs, and the relationship among K-loops, Bol-loops, Kikkawa-loops is thoroughly elucidated. This part could perfectly serve as a textbook. Chapters 7 and 8 are more specialized and gather recent research results about Frobenius groups with many involutions and loops with fibration. The final chapters 9-12 describe examples of K-loops stemming from various sources: in §9, they are obtained via classical groups over ordered fields, in §10 via the relativistic velocity addition and in §11 from general linear groups over rings. In §12, a method (called derivation) is presented for obtaining loops by modifying the operation in a group. It generalizes the well-known procedure which manufactures near-fields and quasi-fields from ordinary fields and gives many examples and counterexamples.

This book can be considered as an extended survey of the existing literature on the subject as well, in that specific contributions from various authors are carefully pointed out, leading to a comprehensive bibliography. The historical appendix casts further light on the whole subject.

The present book provides the first systematic and state-of-the-art account of the theory and it is appropriately divided into three parts. In the first six chapters, the basic theory is developed in a coherent and self-contained way, with complete proofs, and the relationship among K-loops, Bol-loops, Kikkawa-loops is thoroughly elucidated. This part could perfectly serve as a textbook. Chapters 7 and 8 are more specialized and gather recent research results about Frobenius groups with many involutions and loops with fibration. The final chapters 9-12 describe examples of K-loops stemming from various sources: in §9, they are obtained via classical groups over ordered fields, in §10 via the relativistic velocity addition and in §11 from general linear groups over rings. In §12, a method (called derivation) is presented for obtaining loops by modifying the operation in a group. It generalizes the well-known procedure which manufactures near-fields and quasi-fields from ordinary fields and gives many examples and counterexamples.

This book can be considered as an extended survey of the existing literature on the subject as well, in that specific contributions from various authors are carefully pointed out, leading to a comprehensive bibliography. The historical appendix casts further light on the whole subject.

Reviewer: Elena Zizioli (Brescia)

### MSC:

20N05 | Loops, quasigroups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |