##
**An introduction to quasigroups and their representations.**
*(English)*
Zbl 1122.20035

Studies in Advanced Mathematics. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-537-8/hbk). xii, 340 p. (2007).

As we know the theory of quasigroups is one of the oldest branches of algebra and combinatorics [see L. Euler, Recherches sur une nouvelle espèce de quarrés magiques. Mém. de la Societé de Vlissungue 9, 85-239 (1782)]. The aim of this book is to show how the representations for groups are fully capable of extension to general quasigroups, and to illustrate the added depth and richness that result from such an extension.

The author divides the book into three parts. 1) The first three chapters cover elements of the theory of quasigroups and loops, including certain key examples and construction techniques that are needed for a full appreciation of the representation theory. 2) The bulk of this book is devoted to the three main branches of the representation theory itself: permutation representations, characters, and modules. 3) Finally, three brief appendices summarize some essential topics from category theory, universal algebra, and coalgebras.

About the chapters of the book. Chapter 1 provides a quick and necessary introduction to quasigroups and loops, as well as some of the most important special classes such as semi-symmetric quasigroups, Steiner triple systems, and Moufang loops. Chapter 2 discusses the group actions on the underlying set of a quasigroup that result from the quasigroup structure. These actions are the key tools of quasigroup theory. Chapter 3 looks at the quasigroup analogues of Abelian groups, namely central quasigroups and piques. Chapters 4 and 5 are devoted to the theory of permutation representations of quasigroups.

Chapters 6 through 9 treat the oldest branch of quasigroup representation theory, the combinatorial character theory. This theory extends the ordinary character theory of finite groups. For example in Chapter 6, the combinatorial characters of a finite quasigroup are obtained from the action of the combinatorial multiplication group on the quasigroup. Chapter 7 develops those parts of quasigroup character theory that form natural generalizations of group character theory. In Chapter 8 the topics do not have direct counterparts in group theory. Chapter 9 serves a twofold purpose. On the one hand, it uses properties of the permutation action of the multiplication group of a quasigroup to describe some of the algebra structure associated with a homogeneous space for that quasigroup. On the other hand, it also introduces the characters of a quasigroup that are associated with permutation actions of the quasigroup.

In Chapter 10 the fundamental theorems show that categories of modules over quasigroups are equivalent to categories of modules over certain rings, quotients of group algebras of universal stabilizers. The topics discussed in Chapter 11 include the indexing of nonassociative powers, the exponent of a quasigroup, Burnside’s problem for quasigroups, construction of free commutative Moufang loops, and a quick synopsis of cohomology and extension theory for quasigroups. Chapter 12 introduces analytical characters of a finite quasigroup, as certain almost-periodic functions.

Appendix A covers the main constructions of category theory used at various points throughout the book. Appendix B provides a quick introduction to universal algebraic concepts such as congruences, free algebras, and identities. Appendix C summarizes the basic facts about coalgebras that are needed for the treatment of permutation representations.

The author divides the book into three parts. 1) The first three chapters cover elements of the theory of quasigroups and loops, including certain key examples and construction techniques that are needed for a full appreciation of the representation theory. 2) The bulk of this book is devoted to the three main branches of the representation theory itself: permutation representations, characters, and modules. 3) Finally, three brief appendices summarize some essential topics from category theory, universal algebra, and coalgebras.

About the chapters of the book. Chapter 1 provides a quick and necessary introduction to quasigroups and loops, as well as some of the most important special classes such as semi-symmetric quasigroups, Steiner triple systems, and Moufang loops. Chapter 2 discusses the group actions on the underlying set of a quasigroup that result from the quasigroup structure. These actions are the key tools of quasigroup theory. Chapter 3 looks at the quasigroup analogues of Abelian groups, namely central quasigroups and piques. Chapters 4 and 5 are devoted to the theory of permutation representations of quasigroups.

Chapters 6 through 9 treat the oldest branch of quasigroup representation theory, the combinatorial character theory. This theory extends the ordinary character theory of finite groups. For example in Chapter 6, the combinatorial characters of a finite quasigroup are obtained from the action of the combinatorial multiplication group on the quasigroup. Chapter 7 develops those parts of quasigroup character theory that form natural generalizations of group character theory. In Chapter 8 the topics do not have direct counterparts in group theory. Chapter 9 serves a twofold purpose. On the one hand, it uses properties of the permutation action of the multiplication group of a quasigroup to describe some of the algebra structure associated with a homogeneous space for that quasigroup. On the other hand, it also introduces the characters of a quasigroup that are associated with permutation actions of the quasigroup.

In Chapter 10 the fundamental theorems show that categories of modules over quasigroups are equivalent to categories of modules over certain rings, quotients of group algebras of universal stabilizers. The topics discussed in Chapter 11 include the indexing of nonassociative powers, the exponent of a quasigroup, Burnside’s problem for quasigroups, construction of free commutative Moufang loops, and a quick synopsis of cohomology and extension theory for quasigroups. Chapter 12 introduces analytical characters of a finite quasigroup, as certain almost-periodic functions.

Appendix A covers the main constructions of category theory used at various points throughout the book. Appendix B provides a quick introduction to universal algebraic concepts such as congruences, free algebras, and identities. Appendix C summarizes the basic facts about coalgebras that are needed for the treatment of permutation representations.

Reviewer: C. Pereira da Silva (Curitiba)

### MSC:

20N05 | Loops, quasigroups |

20C15 | Ordinary representations and characters |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |