Frobenius Quasigroups and Regular Polygons

In terms of regular n-gons a left distributive quasigroup operation is defined on the complex plane. This operation can be expressed by means of a semidirect product G of the translation group (which is sharply transitive on the points of the plane and hence may be identified with the plane) by a finite cyclic group of rotations of order n. That observation makes possible a wide generalization of this geometric quasigroup construction. The connection in general between algebraic properties of the quasigroup and various properties of the group G is discussed, in particular it is studied what the consequences for the quasigroup Q are if G is interpreted as a topological group or an algebraic group. © 2004, Birkhäuser Verlag, Basel.