Basis-Transitive Matroids
Anne Delandtsheer
and Huiling Li
DOI: 10.1023/A:1022411901450
Abstract
We consider the problem of classifying all finite basis-transitive matroids and reduce it to the classification of the finite basis-transitive and point-primitive simple matroids (or geometric lattices, or dimensional linear spaces). Our main result shows how a basis- and point-transitive simple matroid is decomposed into a so-called supersum. In particular each block of imprimitivity bears the structure of two closely related simple matroids, and the set of blocks of imprimitivity bears the structure of a point- and basis-transitive matroid.
Pages: 285–290
Keywords: matroid; geometric lattice; dimensional linear space; transitivity; automorphism group
Full Text: PDF
References
1. Birkhoff, G., Lattice Theory, Amer. Math. Soc. Colloq. Publ. (3rd ed.), Providence, 1967.
2. Buekenhout, E, "Diagrams for geometries and groups," J. Combin. Theory Ser. A 27 (1979), 121-151.
3. Delandtsheer, A., "Basis-homogeneous geometric lattices," J. London Math. Soc. (2) 34 (1986), 385-393.
4. Delandtsheer, A., Classifications of finite highly transitive dimensional linear spaces, Proc. Int. Conf. on Combinatorial Geom. 91, Capri, 1991, to appear in Discrete Math.
5. Delandtsheer, A., Dimensional linear spaces, to appear in the "Handbook of Incidence Geometry," F. Buekenhout ed., North Holland.
6. Kantor, W.M., "Homogeneous designs and geometric lattices," J. Combin. Theory Ser. A 38 (1985), 66-74.
7. Li, H., "On basis-transitive geometric lattices," European J. Combin. 10 (1989), 561-573.
8. Lim, M.H., "A product of matroids and its automorphism group," J. Combin. Theory Ser. B 23 (1977), 151-163.
9. Welsh, D.J.A., Matroid Theory, Academic Press, London, 1976.
10. White, N.L., Theory of matroids, Cambridge Univ. Press, Cambridge, 1986.
2. Buekenhout, E, "Diagrams for geometries and groups," J. Combin. Theory Ser. A 27 (1979), 121-151.
3. Delandtsheer, A., "Basis-homogeneous geometric lattices," J. London Math. Soc. (2) 34 (1986), 385-393.
4. Delandtsheer, A., Classifications of finite highly transitive dimensional linear spaces, Proc. Int. Conf. on Combinatorial Geom. 91, Capri, 1991, to appear in Discrete Math.
5. Delandtsheer, A., Dimensional linear spaces, to appear in the "Handbook of Incidence Geometry," F. Buekenhout ed., North Holland.
6. Kantor, W.M., "Homogeneous designs and geometric lattices," J. Combin. Theory Ser. A 38 (1985), 66-74.
7. Li, H., "On basis-transitive geometric lattices," European J. Combin. 10 (1989), 561-573.
8. Lim, M.H., "A product of matroids and its automorphism group," J. Combin. Theory Ser. B 23 (1977), 151-163.
9. Welsh, D.J.A., Matroid Theory, Academic Press, London, 1976.
10. White, N.L., Theory of matroids, Cambridge Univ. Press, Cambridge, 1986.