##
**Representation theory of infinite groups and finite quasigroups.**
*(English)*
Zbl 0609.20042

Séminaire de Mathématiques Supérieures. Séminaire Scientifique OTAN (NATO Advanced Study Institute), 101. Université de Montréal, Département de Mathématiques et de Statistique. Montréal (Québec), Canada: Les Presses de l’Université de Montréal. 132 p.; $ 18.00 (1986).

The book under review initiates a study of the representation theory of (non-associative) finite quasigroups; one of the main theses of the work is that the representation theory of a finite quasigroup is equivalent to the representation theory of certain not-necessarily finite groups, namely the universal multiplication group of the quasigroup.

In the first chapter the prerequisites for the subsequent material are introduced: definition of a quasigroup as an algebra satisfying certain identities, concepts from universal algebra, and the basic ideas of centrality. In chapter 2, the notion of combinatorial multiplication group of a quasigroup (Q,.) is first introduced; however, this structure has some defects for further use, and so a universal multiplication group (u.m.g.) is constructed for a quasigroup in a variety of quasigroups; this concept of u.m.g. will be very important in the representation theory. Chapter 3 investigates the representation theory of a finite non- empty quasigroup Q. The normal view of group representation as a module on which the groups acts as a group of automorphisms is not well suited for generalization to a quasigroup. Hence, a module for a group is looked up in another manner, and this gives rise to a satisfactory notion of a module for a quasigroup Q (as an abelian group in the category of quasigroups over Q). Chapter 4 illustrates the techniques by applying them to certain examples of varieties of quasigroups: the variety of groups, of loops, and of commutative Moufang loops of exponent 3. In chapter 5, a (combinatorial) theory of characters for finite non-empty quasigroups is introduced. In the group case there is an intimate connection between the ordinary representation theory of the group and the combinatorial character theory. For general quasigroups this last theory is not powerful enough for a good classification; hence in chapter 6 an analytical character theory is developed, with notions such as periodic and almost periodic functions.

The book offers a good initiation into the study of representations of general quasigroups; the difficulties arising from a direct translation of the notions from groups to quasigroups are explained, and the new notions are introduced in a clear manner. Throughout the work techniques of a wide variety of different subjects are involved: category theory, universal algebra, group theory, functional analysis. Some exercises and open problems are stated.

In the first chapter the prerequisites for the subsequent material are introduced: definition of a quasigroup as an algebra satisfying certain identities, concepts from universal algebra, and the basic ideas of centrality. In chapter 2, the notion of combinatorial multiplication group of a quasigroup (Q,.) is first introduced; however, this structure has some defects for further use, and so a universal multiplication group (u.m.g.) is constructed for a quasigroup in a variety of quasigroups; this concept of u.m.g. will be very important in the representation theory. Chapter 3 investigates the representation theory of a finite non- empty quasigroup Q. The normal view of group representation as a module on which the groups acts as a group of automorphisms is not well suited for generalization to a quasigroup. Hence, a module for a group is looked up in another manner, and this gives rise to a satisfactory notion of a module for a quasigroup Q (as an abelian group in the category of quasigroups over Q). Chapter 4 illustrates the techniques by applying them to certain examples of varieties of quasigroups: the variety of groups, of loops, and of commutative Moufang loops of exponent 3. In chapter 5, a (combinatorial) theory of characters for finite non-empty quasigroups is introduced. In the group case there is an intimate connection between the ordinary representation theory of the group and the combinatorial character theory. For general quasigroups this last theory is not powerful enough for a good classification; hence in chapter 6 an analytical character theory is developed, with notions such as periodic and almost periodic functions.

The book offers a good initiation into the study of representations of general quasigroups; the difficulties arising from a direct translation of the notions from groups to quasigroups are explained, and the new notions are introduced in a clear manner. Throughout the work techniques of a wide variety of different subjects are involved: category theory, universal algebra, group theory, functional analysis. Some exercises and open problems are stated.

Reviewer: G.Crombez

### MSC:

20N05 | Loops, quasigroups |

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

20C15 | Ordinary representations and characters |