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Amalgams Determined By Locally Projective Actions

Published online by Cambridge University Press:  22 January 2016

A. A. Ivanov  and
S. V. Shpectorov
A. A. Ivanov
Affiliation:
Department of Mathematics, Imperial College, 180 Queen’s Gate, London, SW7 2BZ, UK
S. V. Shpectorov
Affiliation:
Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, USA
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Abstract

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A locally projective amalgam is formed by the stabilizer G(x) of a vertex x and the global stabilizer G{x, y} of an edge (containing x) in a group G, acting faithfully and locally finitely on a connected graph Γ of valency 2n - 1 so that (i) the action is 2-arc-transitive; (ii) the subconstituent G(x)Γ(x) is the linear group SLn(2) = Ln(2) in its natural doubly transitive action and (iii) [t, G{x, y}] < O2(G(x) n G{x, y}) for some t G G{x, y} \ G(x). D. Z. Djokovic and G. L. Miller [DM80], used the classical Tutte’s theorem [Tu47], to show that there are seven locally projective amalgams for n = 2. Here we use the most difficult and interesting case of Trofimov’s theorem [Tr01] to extend the classification to the case n > 3. We show that besides two infinite series of locally projective amalgams (embedded into the groups AGLn(2) and O2n+(2)) there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups M22, M23, Co2, J4 and BM. For each of the exceptional amalgam n = 3, 4 or 5.

Keywords

20B2520E0820D08
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 176 , 2004 , pp. 19 - 98
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

References

[B78] Bell, G., On the cohomology of finite special linear groups I and II, J. Algebra, 54 (1978), 216–238 and 239–259.CrossRefGoogle Scholar
[B84] Borcherds, R., The Leech lattice and other lattices, Ph.D. Thesis, Cambridge (1984).Google Scholar
[Cp82] Cameron, P. J. and Praeger, C. E., Graphs and permutation groups with projective subconstituents, J. London Math. Soc. (2), 25 (1982), 62–74.CrossRefGoogle Scholar
[Ch99] Ching, K., Graphs with projective linear stabilizers, Europ. J. Combin., 20 (1999), 29–44.CrossRefGoogle Scholar
[ATLAS] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of Finite Groups, Clarendon Press, Oxford, 1985.Google Scholar
[DS85] Delgado, A. and Stellmacher, B., Weak (B,N)-pairs of rank 2, Groups and Graphs: New Results and Methods, Birkhäuser, Basel (1985), pp. 58–244.Google Scholar
[DM80] Djoković, D. Z. and Miller, G. L., Regular groups of automorphisms of cubic graphs, J. Combin. Theory (B), 29 (1980), 195–230.Google Scholar
[GAP] The GAP Group, , GAP – Groups, Algorithms, and Programming, Version 4.2, Aachen, St. Andrews, 1999. (http://www-gap.dcs.st-and.ac.uk/~gap)Google Scholar
[Iv87] Ivanov, A. A., On 2-transitive graphs of girth 5, Europ. J. Combin., 8 (1987), 393–420.CrossRefGoogle Scholar
[Iv90] Ivanov, A. A., The distance-transitive graphs admitting elations, Math. USSR Izvestiya Math., 35 (1990), 307–335.CrossRefGoogle Scholar
[Iv92] Ivanov, A. A., A presentation for J4 , Proc. London Math. Soc. (3), 64 (1992), 369–396.CrossRefGoogle Scholar
[Iv93] Ivanov, A. A., Graphs with projective subconstituents which contain short cycles, Surveys in Combinatorics (Walker, K., ed.), Cambridge Univ. Press, Cambridge (1993), pp. 173–190.Google Scholar
[Iv99] Ivanov, A. A., Geometry of Sporadic Groups I. Petersen and Tilde Geometries, Cambridge Univ. Press, Cambridge, 1999.Google Scholar
[Iv04] Ivanov, A. A., The Fourth Janko Group, Clarendon Press, Oxford, 2004.Google Scholar
[IM99] Ivanov, A. A. and Meierfrankenfeld, U., A computer-free construction of J4 , J. Algebra, 219 (1999), 113–172.Google Scholar
[IP04] Ivanov, A. A. and Pasechnik, D. V., Minimal representation of locally projective amalgams, J. London Math. Soc., 70 (2004), 142–164.Google Scholar
[IP98] Ivanov, A. A. and Praeger, C. E., On locally projective graphs of girth 5, J. Algebraic Comb., 7 (1998), 259–283.Google Scholar
[IS02] Ivanov, A. A. and Shpectorov, S. V., Geometry of Sporadic Groups II. Representations and Amalgams, Cambridge Univ. Press, Cambridge, 2002.Google Scholar
[Ja73] James, G., The modular characters of the Mathieu groups, J. Algebra, 27 (1973), 57–111.CrossRefGoogle Scholar
[JP76] Jones, W. and Parshall, B., On the 1-cohomology of finite groups of Lie type, Proc. Conf. on Finite Groups (Scott, W. R. and Gross, F., eds.), Acad. Press, San Diego (1976), pp. 313–327.Google Scholar
[K87] Karpilovsky, G., The Schur Multipliers, Oxford Univ. Press, Oxford, 1987.Google Scholar
[K60] Kurosh, A. G., The Theory of Groups. II, Chelsea, New York, 1960.Google Scholar
[MS93] Meierfrankenfeld, U. and Stellmacher, B., Pushing up weak BN-pairs of rank two, Comm. Algebra, 21 (1993), 825–934.Google Scholar
[P94] Pasini, A., Diagram Geometries, Clarendon Press, Oxford, 1994.Google Scholar
[SW88] Stroth, G. and Weiss, R., Modified Steinberg relations for the group J4 , Geom. Dedic., 25 (1988), 513–525.CrossRefGoogle Scholar
[Tim84] Timmesfeld, F. G., Amalgams with rank 2 groups of Lie type in characteristic 2, preprint, Math. Inst. Univ. Giessen (1984).Google Scholar
[Tr91a] Trofimov, V. I., Stabilizers of the vertices of graphs with projective suborbits, Soviet Math. Dokl., 42 (1991), 825–828.Google Scholar
[Tr91b] Trofimov, V. I., More on vertex stabilizers of the symmetric graphs with projective subconstituents, Int. Conf. Algebraic Combin., Vladimir, USSR (1991), pp. 36–37, (Russian).Google Scholar
[Tr92] Trofimov, V. I., Graphs with projective suborbits, Russian Acad. Sci. Izv. Math., 39 (1992), 869–894.Google Scholar
[Tr95a] Trofimov, V. I., Graphs with projective suborbits. Cases of small characteristics. I, Russian Acad. Sci. Izv. Math., 45 (1995), 353–398.Google Scholar
[Tr95b] Trofimov, V. I., Graphs with projective suborbits. Cases of small characteristics. II, Russian Acad. Sci. Izv. Math., 45 (1995), 559–576.Google Scholar
[Tr98] Trofimov, V. I., Graphs with projective suborbits. Exceptional cases of characteristic 2. I, Izv. Math., 62 (1998), 1221–1279.CrossRefGoogle Scholar
[Tr00] Trofimov, V. I., Graphs with projective suborbits. Exceptional cases of characteristic 2. II, Izv. Math., 64 (2000), 173–192.CrossRefGoogle Scholar
[Tr01] Trofimov, V. I., Graphs with projective suborbits. Exceptional cases of characteristic 2. III, Izv. Math., 65 (2001), 787–822.Google Scholar
[Tr03a] Trofimov, V. I., Vertex stabilizers of locally projective groups of automorphisms of graphs. A summary, Groups, combinatorics and geometry (Durham, 2001) (Ivanov, A. A., Liebeck, M. W. and Saxl, J., eds.), World Sci. Publishing, River Edge, NJ (2003), pp. 313–326.Google Scholar
[Tr03b] Trofimov, V. I., Graphs with projective suborbits. Exceptional cases of characteristic 2. IV, Izvestiya Akad. Nauk, Mat., 67 (2003), 193–222, (Russian).Google Scholar
[Tu47] Tutte, W., A family of cubical graphs, Proc. Camb. Phil Soc., 43 (1947), 459–474.CrossRefGoogle Scholar
[W77] Weiss, R., Über symmetrische Graphen und die projektiven Gruppen, Arch. Math., 28 (1977), 110–112.CrossRefGoogle Scholar
[W78] Weiss, R., Symmetric graphs with projective subconstituents, Proc. Amer. Math. Soc., 72 (1978), 213–217.Google Scholar
[W79] Weiss, R., Groups with a (B, N)-pair and locally transitive graphs, Nagoya Math. J., 74 (1979), 1–21.CrossRefGoogle Scholar
[W81] Weiss, R., s-Transitive graphs, Algebraic Methods in Graph Theory, North Holland, Amsterdam (1981), pp. 827–847.Google Scholar
[W82] Weiss, R., Graphs with subconstituents containing L3(p), Proc. Amer. Math. Soc., 85 (1982), 666–672.Google Scholar