×

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

Citations:

Zbl 0673.05010
PDF BibTeX XML Cite
Full Text: DOI EuDML Link

References:

[1] A. Betten, G. Cresp, A. C. Niemeyer, C. E. Praeger, Searching for line-transitive linear spaces preserving a grid structure on points. Preprint, 2002. · Zbl 1108.51006
[2] Cameron P. J., Discrete Math. 118 pp 33– (1993)
[3] Camina A. R., Geom. Dedicata 31 (1989) pp 151– (2000)
[4] Camina A. R., Electron. J. Combin. 3 pp 3– (1996)
[5] A. R. Camina, P. M. Neumann, C. E. Praeger, Alternating groups acting on finite linear spaces. Proc. London Math. Soc. (3) 87 (2003), 29-53. · Zbl 1031.05133
[6] Camina A. R., Bull. London Math. Soc. 25 (1993) pp 309– (2000)
[7] Camina A. R., Aequationes Math. 61 (2001) pp 221– (2002)
[8] Colbourn M. J., Utilitas Math. 17 pp 127– (1980)
[9] European J. Combin. 10 pp 161– (1989)
[10] Delandtsheer A., Geom. Dedicata 29 pp 307– (1989)
[11] P. Dembowski, Finite geometries.Springer 1968. MR 38 #1597 Zbl 0159.50001 · Zbl 0159.50001
[12] J. D. Dixon, B. Mortimer, Permutation groups.Springer 1996. MR 98m:20003 Zbl 0951.20001
[13] Higman D. G., . Illinois J. Math. 5 pp 382– (1961)
[14] W., J. Algebra 106 (1987) pp 15– (2000)
[15] H. Li, W. Liu, Line-primitive 2- v; k; 1 designs with k= k; v c 10. J. Combin. Theory Ser. A 93 (2001), 153-167. MR 2002f:05031 Zbl 0972.05007
[16] W., Utilitas Math. 7 pp 73– (1975)
[17] Nickel W., Algebra Engrg. Comm. Comput. 3 pp 47– (1992)
[18] O’Keefe C. M., Discrete Math. 115 pp 231– (1993)
[19] J. Siemons, B. Webb, On a problem of Wielandt and a question by Dembowski. In: Advances in finite geometries and designs (Chelwood Gate, 1990), 353-358, Oxford Univ. Press 1991. MR 93d:20006 Zbl 0731.20004
[20] J. H. Walter, The characterization of finite groups with abelian Sylow 2-subgroups. Ann. of Math. (2) 89 (1969), 405-514. MR 40 #2749 Zbl 0184.04605 · Zbl 0184.04605
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.