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This note begins a study of the structure of quasigroup semisymmetrizations. For the class of quasigroups isotopic to abelian groups, a fairly complete description is available. The multiplication group is the split extension of the cube of the abelian group by a cyclic group whose order is identified as the semisymmetric index of the quasigroup. For a finite abelian group isotope, the dual of the semisymmetrization is isomorphic to the opposite of the semisymmetrization. The character table of the semisymmetrization is readily computed. The simplicity question for semisymmetrizations is raised. It is shown that a simple, non-abelian quasigroup need not have a simple semisymmetrization.

This paper forms part of the general development of the theory of quasigroup permutation representations. Here, the concept of sharp transitivity is extended from group actions to quasigroup actions. Examples of nontrivial sharply transitive sets of quasigroup actions are constructed. A general theorem shows that uniformity of the action is necessary for the existence of a sharply transitive set. The concept of sharp transitivity is related to two pairwise compatibility relations and to maximal cliques within the corresponding compatibility graphs.

The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank among all skew-symmetric matrices over F whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. We apply techniques from the minimum (symmetric) rank problem and from skew-symmetric matrices to obtain results about the minimum skew rank problem.

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