Minkowski planes admitting automorphism groups of small type

We prove that a Minkowski plane with an automorphism group of type 5¹ is of order 5 and, if it is of type 4 or 7 it is of order 3 or 5. Received 5 January 1999.     

Translation planes of large dimension admitting nonsolvable groups

In this article, the question is considered whether there exist finite translation planes with arbitrarily small kernels admitting nonsolvable collineation groups. For any integerN, it is shown that there exist translation planes of dimension >N and orderq ³ admittingGL(2,q) as a collineation group.     

Flat laguerre planes admitting 4-dimensional groups of automorphisms that fix at least two parallel classes

We take a further step toward the classification of all flat (2-dimensional) Laguerre planes of group dimension 4 by determining, up to isomorphism, all such Laguerre planes that admit 4-dimensional groups of automorphisms that fix at least two parallel classes. It is shown that these planes occur among those flat Laguerre planes of generalised shear type that admit a 3-dimesional kernel.     

Pseudocharacters on free groups, invariant with respect to some types of endomorphisms

We investigate methods for constructing nontrivial pseudocharacters on a free group F _n invariant with respect to certain types of endomorphisms. We find some conditions for endomorphisms of the free group under which there is a nontrivial pseudocharacter that is invariant with respect to these endomorphisms. We consider free products $ R = \tilde{R}*\prod\limits_{{i = k}}^n {\left\langle {{r_i}} \right\rangle } $ , where one factor is F _n , and the other factor is a group on which there is a pseudocharacter. For such products we obtain a similar result about the conditions of existence of nontrivial pseudocharacters invariant with respect to certain endomorphisms.     

Abelian groups admitting fixed-point-free automorphisms of order qn

For a wide class of Abelian groups, necessary and sufficient conditions under which a group admits an automorphism of order qⁿ are found; we also present necessary and sufficient conditions under which a group admits an automorphism ϕ of order qⁿ such that ϕ^qm is a fixed-point-free automorphism for some m < n. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 239–249, February, 1977.     

Planar groups of automorphisms of stable planes

(Semi-) planar groups of stable planes are introduced, and information about their size and their structure is derived. A special case are the stabilizers of quadrangles in compact connected projective planes (i.e. automorphism groups of locally compact connected ternary fields).     

On 4-dimensional Minkowski planes with 7-dimensional automorphism groups

This paper concerns 4-dimensional (topological locally compact connected) Minkowski planes that admit a 7-dimensional automorphism group . It is shown that such a plane is either classical or has a distinguished point that is fixed by the connected component of and that the derived affine plane at this point is a 4-dimensional translation plane with a 7-dimensional collineation group.This research was supported by a Feodor Lynen Fellowship.     

Translation planes admitting a pair of index three homology groups

It is shown that a translation plane of order which admits two homology groups of order must in fact admit symmetric homology groups of this order. It is further shown that a plane admitting such symmetric index 3 homology groups is, with a finite number of exceptions, a generalized André plane. A list of the possibly exceptional orders is determined. Received 20 March 2000.