SEMISYMMETRIZATIONS OF ABELIAN GROUP ISOTOPES

Abstract

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.