K-loops derived from Frobenius groups Original Research Article

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).