Sets of elements that pairwise generate a linear group

Let G be any of the groups (P)GL(n,q), (P)SL(n,q). Define a (simple) graph @[email protected](G) on the set of elements of G by connecting two vertices by an edge if and only if they generate G. Suppose that n is at least 12. Then the maximum size of a complete subgraph in @C is equal to the chromatic number of @C if [email protected]?2(mod4), or if n=2(mod4), q is odd and G=(P)SL(n,q). This work was motivated by a question of Blackburn.

[1]  Ron Brown.  Minimal covers of Sn by abelian subgroups and maximal subsets of pairwise noncommuting elements, II , 1991, J. Comb. Theory, Ser. A.

[2]  Jean-Pierre Massias,et al.  Majoration explicite de l'ordre maximum d'un élément du groupe symétrique , 1984 .

[3]  Paul Erdös,et al.  On an extremal problem in graph theory , 1970 .

[4]  M. J. Tomkinson.  Groups as the union of proper subgroups. , 1997 .

[5]  M. Hasse Dresden.  H. Boerner, Darstellungen von Gruppen. 2. überarb. Auflage. (Die Grundlehren der mathematischen Wissenschaften, Band 74). XIV + 317 S. m. 14 Abb. Berlin/Heidelberg/New York 1967. Springer‐Verlag. Preis geb. DM 58,– , 1969 .

[6]  Áron Bereczky,et al.  Maximal Overgroups of Singer Elements in Classical Groups , 2000 .

[7]  P. E. Holmes.  Subgroup coverings of some sporadic groups , 2006, J. Comb. Theory, Ser. A.

[8]  A. R. Rao,et al.  An extremal problem in graph theory , 1968, Journal of the Australian Mathematical Society.

[9]  Cheryl E. Praeger,et al.  Linear Groups with Orders Having Certain Large Prime Divisors , 1999 .

[10]  G. Dirac.  Some Theorems on Abstract Graphs , 1952 .

[11]  Penny Haxell,et al.  A Note on Vertex List Colouring , 2001, Combinatorics, Probability and Computing.

[12]  Penny E. Haxell,et al.  A condition for matchability in hypergraphs , 1995, Graphs Comb..

[13]  J. Conway,et al.  Atlas of finite groups : maximal subgroups and ordinary characters for simple groups , 1987 .

[14]  Simon R. Blackburn.  Sets of permutations that generate the symmetric group pairwise , 2006, J. Comb. Theory, Ser. A.

[15]  Ron Brown.  Minimal covers of Sn by abelian subgroups and maximal subsets of pairwise noncommuting elements , 1988, J. Comb. Theory, Ser. A.

[16]  J. H. E. Cohn,et al.  On $n$-Sum Groups. , 1994 .

[17]  William M. Kantor,et al.  Linear groups containing a singer cycle , 1980 .

[18]  K. Zsigmondy,et al.  Zur Theorie der Potenzreste , 1892 .

[19]  B. Huppert.  Endliche Gruppen I , 1967 .

[20]  L. Pyber,et al.  The Number of Pairwise Non-Commuting Elements and the Index of the Centre in a Finite Group , 1987 .

[21]  Jason E. Fulman.  Cycle indices for the finite classical groups , 1997, math/9712239.

[22]  G. E. Wall.  On the conjugacy classes in the unitary, symplectic and orthogonal groups , 1963, Journal of the Australian Mathematical Society.

[23]  R. Lathe.  Phd by thesis , 1988, Nature.

[24]  Maria Silvia Lucido.  Groups St Andrews 2001 in Oxford: On the covers of finite groups , 2003 .

[25]  V. Rich.  Personal communication , 1989, Nature.

[26]  Roy Meshulam,et al.  The Clique Complex and Hypergraph Matching , 2001, Comb..

[27]  Attila Maróti.  Covering the symmetric groups with proper subgroups , 2005, J. Comb. Theory, Ser. A.

[28]  R. A. Bryce,et al.  Subgroup coverings of some linear groups , 1999, Bulletin of the Australian Mathematical Society.

[29]  John R. Britnell.  Cyclic, separable and semisimple transformations in the finite conformal groups , 2006 .

[30]  Aner Shalev,et al.  Simple Groups, Probabilistic Methods, and a Conjecture of Kantor and Lubotzky , 1996 .

[31]  Thomas Breuer,et al.  Probabilistic generation of finite simple groups, II , 2000 .

[32]  Peter Frankl,et al.  The Erdös-Ko-Rado theorem for vector spaces , 1986, J. Comb. Theory, Ser. A.