Representations of solvable groups

128 129 '130

131 132 133 134 135 . 136 , 137 138 139

140 141 142 143 144

145 146 147

148 149

150 151

152 153 '154

155 156 157 158 159 160 161

162 163 164

165 .166 , 167 '168 \ 169 170 172 173 174 175 176 177 178

179

180 181

182 183 184

i85 186

187 190

Descriptive set tllCOry and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU TIle subgroup structure of tlle finite classical groups, P.B. KLEIDMAN & M.W.LlEBECK Model tllcory and modules, M. PREST ' Algchraic, cxtremal & metric combinatofics, M-M. DEZA, P. FRANKL & I.G. ROSENDI~RG (cds) Whitchead groups of finite groups, ROBERT OLIVER ' Linear algebraic mOlloids, MOllAN S. PlITCIIA Numbcr tllcory and dynamical systems, M. DODSON & J. VICKERS (cds) Operator algebras alld applicatiolls, I, D. EVANS & M. TAKESAKI (cds) Operator algebras and applications, 2, D. EVANS & M. TAKESAKI (cds) Analysis at Urbana, I, E. BERKSON; T. PECK, & J. UliL (cds)' Analysis at Urbana, II, E. BERKSON, T. PECK, & J. U1IL (cds) Advances ill homotopy theory, S. SALAMON, B. STEER & W. SUTHERLAND (cds) Geometric aspects of Banach spaces, E.M. PEINADOR and A. RODES (cds) Surveys in combinatorics 1989, J, SIEMONS (cd) 'nle geometry of jet bundles, DJ. SAUNDERS 'nle ergodic theory of discrete groups, PETER J. NICIIOLLS Introduction to uniform spaces, I.M. JAMES Homological questions in local algehra, JAN R. STROOKER Cohen-Macaulay modules over CohenoMacaulay rings, Y. YOSHINO Ctllltinuous'ancl tliscrete motlules, S.II. MOIIAMED & BJ. MOLLER 1[dices anti vector bundles, A.N. RUDAKOV et al Solitons, nonlincar evolution equations and inverse scattering, MJ. ABLOWITZ & £l.A. CLARKSON Geometry of low-dimensional manifold~ I, S. DONALDSON & C.B. THOMAS (eUs) Geometry of low-dimensional manifolds 2, S. DONALDSON & C.B. THOMAS (cds) Oligolllorphic pemlUtation groups, P. CAMERON L-f ullctions alld ariUlmetic, J. COATES & M.1. TAYLOR (eds) Nwnbcr tllCory and cryptography, J. LOXTON (cd) C1a~si[jcalion theories of polarized varieties, TAKAO FUJITA Twislors in itlathematics and physics, T.N. BAILEY & R.1. BASTON (cds) Analytic pro-p groups, J.D. DIXON, M.P.F. DU SAUTOY, A. MANN & D. SEGAL.: Geometry of I3anach spaces, P.F.x. MULLER & W. SCHACIIERMA YER (eds) Groups St Andrews 1989 volume I, C.M; CAMPrlELL & E.F. ROOERTSON (cds) Groups St Aildrews 19R9 volume 2, C.M. CAMPBELL & E.F. ROBERTSON (cds) Lectures on bl6ck tllcory, BURKHARD KCrLSIIAMMER llarmonic analysis and represenlation theory lor groups acting on homogeneous trees, A. FIClA-T ALAMANCA & c. NEOI3fA Topics ill varieties of group fepreSClllnliollS, S.M. VOVSI Quasi-symmetric designs, M.S. SIIRIKANDE & S.S. SANE Groups, comhinatorics & geometry, M.W. LlEI3ECK & J. SAXL (cds) Surveys ill cOlllhinatorics, 1991, A.D. KEEDWELL (ed) Stochastic analysis, M.T. BARLOW & N.II. BINGHAM (cds) RcpresentaHons of algebras" II. TAClIlKA WA & S. BRENNER (eds) Boolean function complexity,' M.S. PATERSON (cd) . I\.'lanifolds with singularities anti tlle Adams-Novikov spectral sequence, B.OOTVINNIK Algebraic varieties, GEORGE R. KEMPF Discrete groups and geometry, W.J. HARVEY & c. MACLACIlLAN (cds) Lectures Oil mechanics, J.E. MARSDEN Adams lIlemorial symposium on algebraic topology I, N. RAY & G. WALKER (cds) Adams In 2).

Of course, module (and character) induction is a powerful tool in representation theory, particularly when paired with Clifford's Theorem. Consequently, we need to study "quasi-primitive" linear groups, where those tech-

Olaf Manz

T. R. Wolf

niques do not apply. For solvable groups, the condition of quasi-primitivity

Heidelberg and Frankfurt

Athens, Ohio

Germany

USA

imposes strong restrictions on the normal structure of the group.

~We

study

this extensively in Section I, without restriction on the underlying field. An . important class of solvable (quasi-primitive) linear groups over finite fields are th~ "semi-linear" groups. We study these in Section 2 along with conditions that force a linear .group to indeed be a semi-linear group. Section 3 gives bounds for orders and derived le:ngths of solvable linear groups and permutation groups. Much of Chapters II and III (Sections 4 through 11) deals with orbits of solvable linear groups or, as in Section 5, orbits of permutation groups. Of course, for solvable groups, orbit sizes of linear groups and those of permutation groups are closely related. This becomes clear in Section 6, . where we give a new proof of Huppert's classification of doubly transitive solvable permuta.tion groups. Many of the questions about orbit sizes of linear groups are related to the existence (or non-existence) of ".regular" orbits. Our emphasis here again is on finite fields, because otherwise reglilar, orbits always exist. The main feature of Chapter III, which is critical for, Chapters IV and V, is the study of linear groups with "Sylow centralizers". Chapters IV and V deal with ordinary and modular characters and their degrees. In Section 12, we prove Brauer's height-zero conjecture for solvable G, us}ng material from Sections 5, 6; 9 and 10. In Section 15, we give a character-counting argument and use it to prove the Alperin-McKay conjecture for p-solvable groups. In Section 16, we discuss the derived length and the number of character degrees of a solvable group. This partially relies on a theorem of Berger, presented in Section 8, which unlike other orbit theorems gives the existence of small orbits.

;'.1

ACKNOWLEDGEMENTS

The writing of this book began in Autumn 1988 as part of the project Darstellungstheorie endlicher Gruppen und endlich-dimensionaler Algebren,

sponsored by the Deutsche Forschungsgemeinschaft (DFG). We thank the DFG .for its generous support. We also thank the National Science Foundation, the Mathernatical Sciences Research Institute (Berkeley) and the Ohio University Re'search Council for assistance. We also, thank the following universities for their assistance and resources: Johannes Gutenberg Universitat, Mainz (in particular, Fachbereich lvlathematik), Universitiit Heidelberg (IWR), and Ohio University (Department of Mathematics). We tpank editors Roger Astley and David Tranah of Cambridge University Press for their assistance. Mei Lan Jin (Mathematics Typing Studio, Marion, Ohio) has done a splendid job of preparing a camera-ready manuscript using

'J."EX.

Fina.lly, we thank numerous mathematicians whose work

has influenced these pages and/or with whom we ha.d v'aluable discussions.

Chapter 0

PRELIMIN ARIES

For this manuscript, all groups will be assumed finite. If G is a group and F an (arbitrary) field, an F[ Gl-module V will mean that V is a right F[G]-module and that V is finite dimensional over F. Recall that V is

completely red'll.cible if V is the sum of simpl.e F[ G]-mo