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Quasigroup representation theory. (English) Zbl 0772.20023
Universal algebra and quasigroup theory, Lect. Conf., Jadwisin/Pol. 1989, Res. Expo. Math. 19, 195-207 (1992).
[For the entire collection see Zbl 0745.00017.]
The author presents a study of quasigroup representation theory. The paper is set out as follows. Section 1, introduction. One of the first steps in quasigroup representation theory is thus to find a multiplication group construction that does behave functorially. This construction, the universal multiplication group, is given by the author in the second section. In the third section the author describes the most general possible representations of a quasigroup $$Q$$. The topic of section 4 are representations in varieties. The Burnside problem for quasigroups is introduced in section 5. There, the author generalizes exponents of groups and power-associative loops. Using this exponent, the Burnside problem may be posed for quasigroup varieties. In section 6 it is shown how the representation of $$Z^ 2$$ in the variety of commutative Moufang loops gives a natural account of the baroque details of Bruck’s description of the free commutative Moufang loop on 3 generators that in turn rested on Zassenhaus’ construction of the first non-associative Moufang loop. In the final section the author summarizes some other aspects and problems of the subject.

##### MSC:
 20N05 Loops, quasigroups 05B15 Orthogonal arrays, Latin squares, Room squares 20-02 Research exposition (monographs, survey articles) pertaining to group theory