Characters and generation of Sylow 2-subgroups

We show that the character table of a finite group G determines whether a Sylow 2-subgroup of G is generated by two elements, in terms of the Galois action on characters. Our proof of this result requires the use of the Classification of Finite Simple Groups and provides evidence for the so-far elusive Alperin–McKay–Navarro conjecture.

[1]  George Lusztig,et al.  On the representations of reductive groups with disconnected cen-tre , 1988 .

[2]  Götz Pfeiffer,et al.  CHEVIE — A system for computing and processing generic character tables , 1996, Applicable Algebra in Engineering, Communication and Computing.

[3]  Noelia Rizo,et al.  Galois action on the principal block and cyclic Sylow subgroups , 2019, Algebra & Number Theory.

[4]  Harold N. Ward,et al.  On Ree’s series of simple groups , 1966 .

[5]  M. Cabanes,et al.  Representation Theory of Finite Reductive Groups: Frontmatter , 2004 .

[6]  G. Malle.  Extensions of unipotent characters and the inductive McKay condition , 2008 .

[7]  Gabriel Navarro,et al.  Characters and blocks of finite groups , 1998 .

[8]  D. Gorenstein,et al.  Finite simple groups of low 2-rank and the families $G_2 \left( q \right),\,D_4^2 \left( q \right),\,q$ odd , 1971 .

[9]  Noelia Rizo,et al.  Groups with few $p'$-character degrees , 2019, 1904.06545.

[10]  G. Navarro,et al.  Brauer characters with cyclotomic field of values , 2008 .

[11]  Meinolf Geck,et al.  Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras , 2000 .

[12]  G. Navarro,et al.  Characters and Sylow 2-subgroups of maximal class revisited , 2018, Journal of Pure and Applied Algebra.

[13]  Peter Sin,et al.  The number of finite p-groups with bounded number of generators , 2004 .

[14]  Meinolf Geck,et al.  Finite groups of Lie type , 1985 .

[15]  G. Navarro,et al.  $p$-rational characters and self-normalizing Sylow $p$-subgroups , 2007 .

[16]  G. M.,et al.  Theory of Groups of Finite Order , 1911, Nature.

[17]  G. Navarro,et al.  Sylow subgroups, exponents, and character values , 2019, Transactions of the American Mathematical Society.

[18]  J. B. Olsson.  On the p-blocks of symmetric and alternating groups and their covering groups , 1990 .

[19]  Linear characters of Sylow subgroups , 2003 .

[20]  G. Navarro.  Character Theory and the McKay Conjecture , 2018 .

[21]  E. Dade.  Blocks With Cyclic Defect Groups , 1966 .

[22]  G. Navarro,et al.  Brauer correspondent blocks with one simple module , 2017, Transactions of the American Mathematical Society.

[23]  R. Carter,et al.  The sylow 2-subgroups of the finite classical groups , 1964 .

[24]  Roger W. Carter,et al.  Finite groups of Lie type: Conjugacy classes and complex characters , 1985 .

[25]  A. S. Fry.  Galois automorphisms on Harish-Chandra series and Navarro’s self-normalizing Sylow $2$-subgroup conjecture , 2017, Transactions of the American Mathematical Society.

[26]  Rationality properties of unipotent representations , 2001, math/0103232.

[27]  G. Navarro,et al.  Brauer's height zero conjecture for the 2-blocks of maximal defect , 2012 .

[28]  G. Malle,et al.  Characters of odd degree , 2015, 1506.07690.

[29]  G. Lehrer,et al.  On Gel'fand-Graev characters of reductive groups with disconnected centre. , 1997 .

[30]  Dan Segal,et al.  Finite Group Theory , 2003 .

[31]  P. Schmid.  Extending the Steinberg representation , 1992 .

[32]  N. Iwahori,et al.  The conjugacy classes of Chevalley groups of type $(F_4)$ over finite fields of characteristic $p\ne 2$ , 1974 .

[33]  Gabriel Navarro,et al.  The McKay conjecture and Galois automorphisms , 2004 .

[34]  Jay Taylor,et al.  On Self-Normalising Sylow $2$-Subgroups in Type A , 2017, 1701.00272.

[35]  G. Malle.  Height 0 characters of finite groups of Lie type , 2007 .

[36]  E. Dade.  Remarks on isomorphic blocks , 1977 .

[37]  Jeremy Sylvestre,et al.  Representations of Finite Groups , 2015, Arch. Formal Proofs.

[38]  I. Isaacs.  Character Theory of Finite Groups , 1976 .

[39]  Abhinav Shrestha.  CLASSIFICATION OF FINITE SIMPLE GROUPS , 2010 .

[40]  G. Navarro,et al.  Characters of relative p ' -degree over normal subgroups , 2013 .

[41]  David A. Craven.  Representation Theory of Finite Groups: a Guidebook , 2019, Universitext.