Alexander Kreuzer

On K-Loops Of Finite Order

In this note we undertake an axiomatic investigation of K-loops (or gyrogroups, as A.A.Ungar used to name them) and provide new construction methods for finite K-loops. It is shown how, more or less, the axioms are independent from each other. Especially (K6) is independent as A.A. Ungar already had conjectured. We begin with right loops (L,⊕) and add step by step further properties. So the connection between K-loops, Bol-loops, Bruck-loops and the homogeneous loops of Kik-kawa became clear. The smallest examples of proper K-loops possess 8 elements; there are exactly 3 non-isomorphic of these.At last it is shown that one gets quite naturally a Frobenius-group as a quasidirect product of a K-loop (L,⊕) and a group D of automorphisms of (L,⊕) if D is fixed point free except from 0.

INNER MAPPINGS OF BRUCK LOOPS

K -loops have their origin in the theory of sharply 2-transitive groups. In this paper a proof is given that K -loops and Bruck loops are the same. For the proof it is necessary to show that in a (left) Bruck loop the left inner mappings L ( b ) L ( a ) L ( ab ) −1 are automorphisms. This paper generalizes results of Glauberman [ 3 ], Kist [ 8 ] and Kreuzer [ 9 ].

Beispiele endlicher und unendlicher K-Loops

In this note examples are given for non trivial K-loops. There are commutative examples as well as non commuative, finite examples as well as infinite. Furthermore it will be shown that under an additional condition K-loops and Brück loops coincide.

Zur Einbettung von Inzidenzräumen und angeordneten Räumen

We characterize subsets of incidence spaces, which are locally complete resp. bundletrue (bündeltreu). Then we specify necessary and sufficient conditions to extend the embedding of a subset of a space into another space to an embedding of the complete space. Using these results we show that every ordered space (not necesserily fulfilling (O3), cf. §4) can be embedded into a projective space. The last result is a generalization of former results using the additional assumption (O3) (cf. [2,9,10]).

Locally projective spaces which satisfy the Bundle Theorem

We give a complete and short proof of KAHNs Theorem that every locally projective space (M,M) with dim M≥3 satisfying the Bundle Theorem is embeddable in a projective space. The central tool of KAHNs proof is the fact that (M,M) is locally projective, while we use mainly the Bundle Theorem.

Forcing linearity numbers for injective modules over PID's

Abstract. In this paper the forcing linearity numbers for injective modules over principal ideal domains are determined. As a consequence one obtains the forcing linearity numbers for divisible abelian groups.

On the definition of isomorphisms of linear spaces

AbstractLet (P,

Klassifizierung von Halbordnungen

\mathfrak{L}