In a recent paper, those quasigroup identities involving at most three variables and of “length” six which force the quasigroup to be a loop or group have been enumerated by computer. We separate these identities into subsets according to what classes of loops they define and also provide humanly-comprehensible proofs for most of the computer-generated results.

This is a survey of the results obtained by K. Głazek and his co-workers. We restrict our attention to the problems of axiomatizations of n-ary groups, classes of n-ary groups, properties of skew elements and homomorphisms induced by skew elements, constructions of covering groups, classifications and representations of n-ary groups. Some new results are added too.

If $(G,·)$ is a group, and the operation $(*)$ is defined by $x*y=x·y^{-1}$ then by direct verification $(G,*)$ is a quasigroup which satisfies the identity $(x*y)*(z*y)=x*z$. Conversely, if one starts with a quasigroup satisfying the latter identity the group $(G,·)$ can be constructed, so that in effect $(G,·)$ is determined by its right division operation. Here the analogous situation is examined for a Moufang loop. Subtleties arise which are not present in the group case since there is a choice of defining identities and the identities produced by...

We prove a version of Hrushovski's Socle Lemma for rigid groups in an arbitrary simple theory.