Regular 4-polytopes from the Livingstone graph of Janko's first group
Michael I. Hartley
, Isabel Hubard
and Dimitri Leemans
DOI: 10.1007/s10801-011-0300-x
Abstract
The Janko group J 1 has, up to duality, exactly two regular rank four polytopes, of respective Schläfli types {5,3,5} and {5,6,5}. The aim of this paper is to give geometric constructions of these two polytopes, starting from the Livingstone graph.
Pages: 193–214
Keywords: keywords regular abstract polytopes; first group of janko; livingstone graph
Full Text: PDF
References
1. Aschbacher, M.: Flag structures on Tits geometries. Geom. Dedic. 14, 21-31 (1983)
2. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)
3. Buekenhout, F.: The geometry of the finite simple groups. In: Rosati, L.A. (ed.) Buildings and the Geometry of Diagrams. Lect. Notes in Math., vol. 1181, pp. 1-78. Springer, Berlin (1986)
4. Buekenhout, F. (ed.): Handbook of Incidence Geometry. Buildings and Foundations. Amsterdam, Elsevier (1995)
5. Buekenhout, F., Cara, P., Vanmeerbeek, K.: Geometries of the group PSL(2, 11). Geom. Dedic. 83, 169-206 (2000)
6. Gottschalk, H., Leemans, D.: The residually weakly primitive geometries of the Janko group J1. In: Groups and Geometries, pp. 65-79. Birkhäuser, Basel (1998)
7. Hartley, M.I., Leemans, D.: Quotients of a universal locally projective polytope of type {5, 3, 5}. Math. Z. 247(4), 663-674 (2004)
8. Hartley, M.I., Leemans, D.: On thin regular geometries of rank four for the Janko group J1. Innov. Incid. Geom. 1, 181-190 (2005)
9. Hartley, M.I., Leemans, D.: A new Petrie-like construction for abstract polytopes. J. Comb. Theory, Ser. A 115(6), 997-1007 (2008)
10. Janko, Z.: A new finite simple group with abelian 2-Sylow subgroups. Proc. Natl. Acad. Sci. USA 53, 657-658 (1965)
11. Janko, Z.: A new finite simple group with abelian Sylow 2-subgroups and its characterization. J. Al- gebra 3, 147-186 (1966)
12. Lefèvre-Percsy, C., Percsy, N., Leemans, D.: New geometries for finite groups and polytopes. Bull.
2. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)
3. Buekenhout, F.: The geometry of the finite simple groups. In: Rosati, L.A. (ed.) Buildings and the Geometry of Diagrams. Lect. Notes in Math., vol. 1181, pp. 1-78. Springer, Berlin (1986)
4. Buekenhout, F. (ed.): Handbook of Incidence Geometry. Buildings and Foundations. Amsterdam, Elsevier (1995)
5. Buekenhout, F., Cara, P., Vanmeerbeek, K.: Geometries of the group PSL(2, 11). Geom. Dedic. 83, 169-206 (2000)
6. Gottschalk, H., Leemans, D.: The residually weakly primitive geometries of the Janko group J1. In: Groups and Geometries, pp. 65-79. Birkhäuser, Basel (1998)
7. Hartley, M.I., Leemans, D.: Quotients of a universal locally projective polytope of type {5, 3, 5}. Math. Z. 247(4), 663-674 (2004)
8. Hartley, M.I., Leemans, D.: On thin regular geometries of rank four for the Janko group J1. Innov. Incid. Geom. 1, 181-190 (2005)
9. Hartley, M.I., Leemans, D.: A new Petrie-like construction for abstract polytopes. J. Comb. Theory, Ser. A 115(6), 997-1007 (2008)
10. Janko, Z.: A new finite simple group with abelian 2-Sylow subgroups. Proc. Natl. Acad. Sci. USA 53, 657-658 (1965)
11. Janko, Z.: A new finite simple group with abelian Sylow 2-subgroups and its characterization. J. Al- gebra 3, 147-186 (1966)
12. Lefèvre-Percsy, C., Percsy, N., Leemans, D.: New geometries for finite groups and polytopes. Bull.
© 1992–2009 Journal of Algebraic Combinatorics
©
2012 FIZ Karlsruhe /
Zentralblatt MATH for the EMIS Electronic Edition