Tactical decompositions and some configurationsv 4

The study of configurations or — more generally — finite incidence geometries is best accomplished by taking into account also their automorphism groups. These groups act on the geometry and in particular on points, blocks, flags and even anti-flags. The orbits of these groups lead to tactical decompositions of the incidence matrices of the geometries or of related geometries. We describe the general procedure and use these decompositions to study symmetric configurationsv ₄ for smallv. Tactical decompositions have also been used to construct all linear spaces on 12 points [2] and all proper linear spaces on 17 points [3].     

Regular orbits for projective orthogonal groups over finite fields of odd characteristic

In this paper we prove that the projective orthogonal groups over finite fields of odd characteristic acting on the set of points of the corresponding quadrics, have regular orbits apart from a finite number of explicitly listed exceptions occurring in dimension 2 and 3.Lavoro eseguito nell'ambito dei gruppi nazionali del C.N.R. (G.N.S.A.G.A.) e del finanziamento del M.P.I.     

Tactical decompositions in uniform normal designs

Results on the block structure of tactical decompositions are given for uniform normal multiplicative designs. Emphasis is placed on the case with three replications. Reducible uniform normal designs are shown to be degenerate.     

Two-transitive orbits in finite projective planes

The problem of classifying finite projective planes of order n with an automorphism group G and a point orbit on which G acts two-transitively is investigated in considerable detail, under the assumption that has length at last n. Combining old and new results a rather satisfying classification is obtained, even though some cases for orbit lengths n and n + 1 remain unsolved.     

Asymmetric decompositions of abelian groups

A subsetA of an Abelian groupG is said to be asymmetric ifg+S⊄A for any elementg∈G and any infinite symmetric subsetS⊂G (S=−S). The minimal cardinality of a decomposition of the groupG into asymmetric sets is denoted by ν(G). for any Abelian groupG, the cardinal number ν(G is expressed via the following cardinal invariants: the free rank, the 2-rank, and the cardinality of the group. In particular, . Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 10–19, July, 1999.     

Local decompositions of pseudo-orthogonal groups

We find all the local decompositions of the Lie algebra 50(1,n+1) in the sum of two Lie subal-gebras. We also find all the connected transitive subgroups in the conformai group of Euclidean space, the transitive subgroup is either contained in the group of motions or contains the group of displacements.     

Critical orbits of holomorphic maps on projective spaces

We study the dynamics of iterated holomorphic maps of a complex projective space onto itself. Relations between the Fatou set and the orbits of critical points are investigated. In particular, results concerning critically finite maps on the Riemann sphere are generalized to higher dimensional case.     

Schauder decompositions and the Fremlin projective tensor product of Banach lattices

In this paper, we first introduce a lattice decomposition and finite-dimensional lattice decomposition (FDLD) for Banach lattices. Then we show that for a Banach lattice with FDLD, the following are equivalent: (i) it has the Radon-Nikodym property; (ii) it is a KB-space; (iii) it is a Levi space; and (iv) it is a σ-Levi space. We then give a sequential representation of the Fremlin projective tensor product of an atomic Banach lattice with a Banach lattice. Using this sequential representation, we show that if one of the Banach lattices X and Y is atomic, then the Fremlin projective tensor product has the Radon-Nikodym property (or, respectively, is a KB-space) if and only if both X and Y have the Radon-Nikodym property (or, respectively, are KB-spaces).     

Product decompositions of solvable Lie groups

Summary The article investigate the structure of real solvable connected Lie groups. It is described how one can decompose a solvable Lie group in direct and semidirect products of closed connected subgroups. In particular, the commutator group, Cartan subgroups, the center, maximal compactly embedded subgroups and tori are considered. Furthermore, one can find special solvable Lie groups and their product decompositions, namely compactly generated solvable Lie groups and those Lie groups which are generated by maximal compactly embedded subgroups. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Direct decompositions of mixed abelian groups

We consider a sufficiently large subcategory of the category of mixed abelian groups of finite torsion-free rank and its quotient catego ry obtained by annihilating those homomorphisms which factor through the torsion. We prove that the second category provides a good approximation to the first category, but is much simpler: the groups of morphisms in the second category are finite rank torsion-free groups. This renders it possible to exam ine direct decompositions of mixed groups by the same methods as in the case of finite rank torsion-free groups.     

Relatively projective groups as absolute Galois groups

By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields. Dedicated to Yuri Ershov on the occasion of his 60-th birthday Heisenberg-Stipendiat der Deutschen Forschungsgemeinschaft (KO 1962/1-1).     

Complex projective surfaces and infinite groups

((Without abstract)) Submitted: March 1997, Final version: September 1997