Smooth Stable Planes and the Moduli Spaces of Locally Compact Translation Planes

 Smooth stable planes have been introduced in [3]. At every point p of a smooth stable plane the tangent spaces of the lines through p form a compact spread (see the definition in Section 2) on the tangent space thus defining a locally compact topological affine translation plane . We introduce the moduli space of isomorphism classes of compact spreads, . We show that for the topology of is not by constructing a sequence of non-classical spreads in that converges to the classical spread in , where . Moreover, we prove that the isomorphism type of varies continuously with the point p. Finally, we give examples of smooth affine planes which have both classical and non-classical tangent translation planes. (Received 15 April 1999; in revised form 22 October 1999)     

Unitals in Topological Affine Translation Planes Need Not Be Strictly Convex

We study the relationship of two incidence geometric convexity notions, namely, ovoids in real affine spaces and compact unitals of codimension 1 in topological affine translation planes. In [3] we showed that every ovoid in a translation plane is a unital, and we asked if the converse is true. Here we introduce the notion of a shell, which is distinctly weaker than that of an ovoid and still implies the unital property if the translation plane is properly chosen (and the shell is not too degenerate). We give an explicit example of a shell that is not an ovoid. The question remains whether or not conversely, every compact unital of codimension 1 in a translation plane is a shell.This paper was written while the third author was supported by a grant from DFG and TÜBITAK.Received March 12, 2002 Published online November 18, 2002     

Polarities of Compact Eight-Dimensional Planes

The polarities of eight-dimensional compact planes with at least 17-dimensional group of automorphisms are determined. We show that, among these planes, the quaternion plane is the only one admitting more than two conjugacy classes of polarities.     

4-Dimensional Elation Laguerre Planes Admitting Non-Solvable Automorphism Groups

 This paper concerns 4-dimensional (topological locally compact connected) elation Laguerre planes that admit non-solvable automorphism groups. It is shown that such a plane is either semi-classical or a single plane admitting the group SL(2, ). Various characterizations of this single Laguerre plane are obtained. Received October 17 2000; in revised form April 23 2001 Published online August 5, 2002     

The Similarity Between Finitary and Artinian-Finitary Groups

Let M be a module over the commutative ring R. The finitary automorphism group of M over R is and the Artinian-finitary automorphism group of M over R is We investigate further the surprisingly close relationship between these two types of automorphism groups. Their group theoretic properties seem practically identical.     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

The Automorphism Group of a Compact Generalized Polygon has Finite Dimension

 For every compact generalized polygon whose point space has finite dimension, we show that the automorphism group has finite dimension as well. This opens the field for the application of Lie theoretic methods. Received 12 November 1997; in revised form 10 March 1998     

Characterization of Finite Groups With Some <Emphasis Type="Italic">S</Emphasis>-quasinormal Subgroups

A subgroup of a finite group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. In this paper we give a characterization of a finite group G under the assumption that every subgroup of the generalized Fitting subgroup of prime order is S-quasinormal in G.     

Totally Permutable Products of Finite Groups Satisfying SC or PST

Finite groups G=AB factorized by two subgroups A and B such that every subgroup of A permutes with every subgroup of B are studied in this paper. The behaviour of such products with respect to the class of finite groups in which Sylow-permutability is transitive is analyzed.     

Precomplete Clones on Infinite Sets which are Closed under Conjugation

We show that on an infinite set, there exist no other precomplete clones closed under conjugation except those which contain all permutations. Since on base sets of some infinite cardinalities, in particular on countably infinite ones, the precomplete clones containing the permutations have been determined, this yields a complete list of the precomplete conjugation-closed clones in those cases. In addition, we show that there exist no precomplete submonoids of the full transformation monoid which are closed under conjugation except those which contain the permutations; the monoids of the latter kind are known.     

4-Transitivity and 6-figures in some Moufang-Klingenberg planes

In this paper, the Moufang-Klingenberg plane over a local alternative ring R of dual numbers is studied. It is shown that its collineation group is transitive on quadrangles. It is therefore shown that the coordinatization of these Moufang-Klingenberg planes is independent of the choice of the coordinatization quadrangle. Also, the concept of 6-figures is extended to these Moufang-Klingenberg planes and it is shown that any 6-figure corresponds to only one inversible m ∈ R.     

Some Examples in the Theory of Subgroup Growth

By estimating the subgroup numbers associated with various classes of large groups, we exhibit a number of new phenomena in the theory of subgroup growth.     

Regularity in a Singular Biharmonic Dirichlet Problem

We consider the singular biharmonic equation with Dirichlet boundary conditions u = f₀ and ∂_nu = f₁ on . In our setup the boundary values f_j (j = 0,1) are elements in two homogeneous Banach spaces B_j (j = 0,1) on . We give a sufficient condition on the spaces B_j (j = 0,1) to ensure that the solution u of this Dirichlet problem has the appropriate boundary values f_j (j = 0,1) in the sense of convergence in spaces B_j (j = 0,1). Our results also apply in the unweighted case.     

Common Extensions for Locally Compact Markov Shifts

 We extend a result of W. Parry by showing that two locally compact transitive Markov shifts have the same Gurevic entropy iff they have a common entropy preserving extension by a locally compact transitive Markov shift. Additionally, the factor maps can be made countable-to-1 biclosing. (Received 12 September 2000; in revised form 16 January 2001)     

Involutory Automorphisms of Groups of Odd Order

The following theorem is proved. Let G be a finite group of odd order admitting an involutory automorphism φ. Suppose that G has derived length d and that C_G(φ) is nilpotent of class c. Assume that C_G(φ) is a m-generator. Then [G,φ]^′ is nilpotent of {c,d,m}-bounded class.     

Characterizations of Some Product Manifolds by Their Spectrum

Let Sⁿ(c) denote the n-dimensional Euclidean sphere of constant sectional curvature c and denote by CPⁿ(c) the complex projective space of complex dimension n and of holomorphic sectional curvature c. In this paper, we obtain some characterizations of the manifolds S²(c) × S²(c′), S⁴(c) × S⁴(c′), CP²(c) × CP²(c′) by their spectrum.     

Some results on supersolvability of finite groups

We get new characterizations of finite supersolvable groups based on the structure of the generalized Fitting subgroup. Then we extend our results to the saturated formations containing the class of all supersolvable groups. Authors’ addresses: M. Asaad, Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt; Piroska Cs?rg?, Department of Algebra and Number Theory, E?tv?s University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary     

Embedding of Shifts of Finite Type into the Dyck Shift

Periodic points data of the Dyck shift will be precisely investigated. A necessary and sufficient condition for an irreducible shift of finite type to be embedded into the Dyck shift will be shown.     

On Closed Weingarten Surfaces

We investigate closed surfaces in Euclidean 3-space satisfying certain functional relations κ = F(λ) between the principal curvatures κ, λ. In particular we find analytic closed surfaces of genus zero where F is a quadratic polynomial or F(λ) = cλ²ⁿ⁺¹. This generalizes results by H. Hopf on the case where F is linear and the case of ellipsoids of revolution where F(λ) = cλ³.