Large subgroups of simple groups

Abstract Let G be a finite group. A proper subgroup H of G is said to be large if the order of H satisfies the bound | H | 3 ⩾ | G | . In this note we determine all the large maximal subgroups of finite simple groups, and we establish an analogous result for simple algebraic groups (in this context, largeness is defined in terms of dimension). An application to triple factorisations of simple groups (both finite and algebraic) is discussed.

[1]  Cheryl E. Praeger,et al.  Cyclic Matrices Over Finite Fields , 1995 .

[2]  Gunter Malle,et al.  CORRIGENDA: LOW-DIMENSIONAL REPRESENTATIONS OF QUASI-SIMPLE GROUPS , 2002 .

[3]  Robert A. Wilson,et al.  The Maximal Subgroups of E6(2) and Aut(E6(2)) , 1990 .

[4]  C. Praeger,et al.  Triple factorisations of the general linear group and their associated geometries , 2014, 1405.5276.

[5]  Gary M. Seitz,et al.  SUBGROUPS OF MAXIMAL RANK IN FINITE EXCEPTIONAL GROUPS OF LIE TYPE , 1992 .

[6]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[7]  Cheryl E. Praeger,et al.  A classification of the maximal subgroups of the finite alternating and symmetric groups , 1987 .

[8]  M. Liebeck,et al.  Maximal subgroups of exceptional groups of Lie type, finite and algebraic , 1990 .

[9]  Aner Shalev,et al.  The probability of generating a finite simple group , 1995 .

[10]  M. Liebeck,et al.  Maximal Subgroups of Large Rank in Exceptional Groups of Lie Type , 2005 .

[11]  Arjeh M. Cohen,et al.  The Local Maximal Subgroups of Exceptional Groups of Lie Type, Finite and Algebraic , 1992 .

[12]  Robert A. Wilson,et al.  The finite simple groups , 2009 .

[13]  C. Praeger,et al.  On Triple Factorisations of Finite Groups , 2009, 0909.4393.

[14]  Gary M. Seitz,et al.  The maximal subgroups of positive dimension in exceptional algebraic groups , 2004 .

[15]  P. B. Kleidman,et al.  The maximal subgroups of the Steinberg triality groups 3D4(q) and of their automorphism groups , 1988 .

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

[17]  M. Liebeck,et al.  On the subgroup structure of classical groups , 1998 .

[18]  D. Holt,et al.  The Maximal Subgroups of the Low-Dimensional Finite Classical Groups , 2013 .

[19]  P. B. Kleidman.  The maximal subgroups of the Chevalley groups G2(q) with q odd , 1988 .

[20]  Aner Shalev,et al.  Base sizes for simple groups and a conjecture of Cameron , 2009 .

[21]  J. Conway,et al.  ATLAS of Finite Groups , 1985 .

[22]  Simon P. Norton,et al.  The maximal subgroups of F4(2) and its automorphism group , 1989 .

[23]  Christoph Jansen,et al.  An Atlas of Brauer Characters , 1995 .

[24]  Gary M. Seitz,et al.  On finite subgroups of exceptional algebraic groups , 1999 .

[25]  Martin W. Liebeck,et al.  The Subgroup Structure of the Finite Classical Groups , 1990 .

[26]  Gunter Malle,et al.  The maximal subgroups of 2F4(q2) , 1991 .

[27]  Gary M. Seitz,et al.  On the subgroup structure of exceptional groups of Lie type , 1998 .

[28]  W. A. Manning.  On the order of primitive groups , 1909 .

[29]  S. Norton.  On the group Fi24 , 1988 .

[30]  D. G. Higman,et al.  Geometric $ABA$-groups , 1961 .

[31]  Michael Aschbacher,et al.  On the maximal subgroups of the finite classical groups , 1984 .

[32]  Martin W. Liebeck,et al.  The Finite Primitive Permutation Groups of Rank Three , 1986 .

[33]  S. H. Alavi,et al.  On triple factorizations of finite groups , 2010 .

[34]  Ernest E. Shult,et al.  On a class of doubly transitive groups , 1972 .

[35]  Frank Lübeck.  Small Degree Representations of Finite Chevalley Groups in Defining Characteristic , 2001, LMS J. Comput. Math..

[36]  R. Lawther,et al.  Sublattices generated by root differences , 2014 .

[37]  John Ballantyne,et al.  The Maximal Subgroups of E_7(2) , 2013 .

[38]  M. Liebeck,et al.  On the Orders of Maximal Subgroups of the Finite Exceptional Groups of Lie Type , 1987 .

[39]  Gerhard Hiss,et al.  Low-Dimensional Representations of Quasi-Simple Groups , 2001, LMS J. Comput. Math..

[40]  P. B. Kleidman.  The maximal subgroups of the finite 8-dimensional orthogonal groups PΩ8+(q) and of their automorphism groups , 1987 .

[41]  Peter J. Cameron,et al.  On the degress of primitive permutation groups , 1982 .

[42]  B. Cooperstein.  Maximal subgroups of G2(2n) , 1981 .

[43]  Attila Maróti.  On the orders of primitive groups , 2002 .

[44]  Tonny A. Springer.  Linear Algebraic Groups , 1981 .

[45]  Simon P. Norton,et al.  A correction to the 41-structure of the Monster, a construction of a new maximal subgroup L2(41) and a new Moonshine phenomenon , 2013, J. Lond. Math. Soc..

[46]  M. Liebeck.  On the Orders of Maximal Subgroups of the Finite Classical Groups , 1985 .