## Finite line-transitive linear spaces: Parameters and normal point-partitions.(English)Zbl 1030.05022

This paper is a contribution to the ongoing classification of finite linear spaces with line-transitive automorphism group. Let $$G$$ be a group which acts line-transitively on a linear space $$\mathcal L$$ and so that $$G$$ acts imprimitively on the points of $$\mathcal L$$. There are very few known examples, the only infinite family is constructed by considering projective planes.
In 1989 A. Delandtsheer and J. Doyen [Geom. Dedicata 29, 307-310 (1989; Zbl 0673.05010)] proved a simple but very powerful result. Assume $$G$$ is imprimitive with $$d$$ sets of imprimitivity of size $$c$$. Then there exist two natural numbers $$x$$ and $$y$$ so that $c=\frac{\binom{k}{2}-x}{y},\quad d=\frac{\binom{k}{2}-y}{x},$ where $$k$$ is the size of a line.
In this paper the authors consider a number of interesting parameters which can be found using these ideas, especially useful when the imprimitivity comes from the orbits of a normal subgroup. They give very specific information for small values of the parameters and leave the reader with a number of interesting questions.

### MSC:

 05B25 Combinatorial aspects of finite geometries 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures 51E26 Other finite linear geometries

### Keywords:

finite linear spaces; imprimitivity; automorphisms

Zbl 0673.05010
Full Text:

### References:

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