K-loops derived from Frobenius groups

Abstract

We consider a generalization of the representation of the so-called co-Minkowski plane (due to H. and R. Struve) to an abelian group ( V,+) and a commutative subgroup G of Aut( V,+). If P= G× V satisfies suitable conditions then an invariant reflection structure (in the sense of Karzel (Discrete Math. 208/209 (1999) 387–409)) can be introduced in P which carries the algebraic structure of K-loop on P (cf. Theorem 1). We investigate the properties of the K-loop ( P,+) and its connection with the semi-direct product of V and G. If G is a fixed point free automorphism group then it is possible to introduce in ( P,+) an incidence bundle in such a way that the K-loop ( P,+) becomes an incidence fibered loop (in the sense of Zizioli (J. Geom. 30 (1987) 144–151)) (cf. Theorem 3).