Fixed point ratios in actions of finite classical groups, III

Abstract This is the first in a series of four papers on fixed point ratios in actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G -set then either fpr ( x ) ≲ | x G | − 1 2 for all elements x ∈ G of prime order, or ( G , Ω ) is one of a small number of known exceptions. Here fpr ( x ) denotes the proportion of points in Ω which are fixed by x . In this introductory note we present our results and describe an application to the study of minimal bases for primitive permutation groups. A further application concerning monodromy groups of covers of Riemann surfaces is also outlined.

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

[2]  Robert M. Guralnick,et al.  Generation of finite almost simple groups by conjugates , 2003 .

[3]  Jonathan I. Hall,et al.  Generators for Finite Simple Groups, with Applications to Linear Groups , 1992 .

[4]  Daniel Frohardt,et al.  Composition factors of monodromy groups , 2001 .

[5]  H. Enomoto.  The conjugacy classes of Chevalley groups of type ($G_2$) over finite fields of characteristic 2 or 3 , 1970 .

[6]  Gary M. Seitz,et al.  Fixed point spaces in actions of exceptional algebraic groups , 2002 .

[7]  Timothy C. Burness.  On base sizes for actions of finite classical groups , 2007 .

[8]  Martin W. Liebeck,et al.  Minimal Degrees of Primitive Permutation Groups, with an Application to Monodromy Groups of Covers of Riemann Surfaces , 1991 .

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

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

[11]  N. Spaltenstein,et al.  Caractères unipotents de $${}^3D_4 (\mathbb{F}_q )$$ , 1982 .

[12]  D. Gluck,et al.  Character and Fixed Point Ratios in Finite Classical Groups , 1995 .

[13]  Timothy C. Burness.  Fixed point ratios in actions of finite classical groups, I , 2006 .

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

[15]  Robert Steinberg,et al.  Endomorphisms of linear algebraic groups , 1968 .

[16]  M. Liebeck,et al.  Character Degrees and Random Walks in Finite Groups of Lie Type , 2005 .

[17]  Ross Lawther,et al.  Correction to 'Jordan block sizes of unipotent elements in exceptional algebraic groups' , 1998, Communications in Algebra.

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

[19]  Peter J. Cameron.  Groups, Combinatorics & Geometry: Some open problems on permutation groups , 1992 .

[20]  Peter J. Cameron,et al.  Random Permutations: Some Group-Theoretic Aspects , 1993, Comb. Probab. Comput..

[21]  Nicolas Bourbaki,et al.  Groupes et algèbres de Lie , 1971 .

[22]  Gary M. Seitz,et al.  Fixed point ratios in actions of finite exceptional groups of lie type. , 2002 .

[23]  M. Graber.  Group Theory: 7 , 1905 .

[24]  Timothy C. Burness.  Fixed point spaces in actions of classical algebraic groups , 2004 .

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

[26]  Michael Aschbacher,et al.  Corrections to “Involutions in Chevalley groups over fields of even order” , 1976, Nagoya Mathematical Journal.

[27]  M. Liebeck,et al.  Reductive subgroups of exceptional algebraic groups , 1996 .

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

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

[30]  R. Guralnick,et al.  Finite groups of genus zero , 1990 .

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

[32]  R. Guralnick,et al.  Probabilistic Generation of Finite Simple Groups , 2000 .

[33]  William M. Kantor,et al.  Subgroups of classical groups generated by long root elements , 1979 .

[34]  Aner Shalev,et al.  Simple groups, permutation groups, and probability , 1999 .

[35]  J. Humphreys.  Conjugacy classes in semisimple algebraic groups , 1995 .

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

[37]  Grassmannian Fixed Point Ratios , 2000 .

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

[39]  R. W. Carter,et al.  ‘GROUPES ET ALGEBRES DE LIE’ CHAPTERS 2, 3 , 1974 .

[40]  Cheryl E. Praeger,et al.  Minimal degree of primitive permutation groups , 1976 .

[41]  I. G. MacDonald,et al.  Lectures on Lie Groups and Lie Algebras: Simple groups of Lie type , 1995 .

[42]  D. Gorenstein,et al.  The Classification of the Finite Simple Groups, Number 2 , 1995 .

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