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Graduate Texts in Mathematics 42· Editorial Board S. Axler F.W. Gehring K.A. Ribet Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo

Graduate Texts in Mathematics T AKEUTIIZARING. Introduction to Axiomatic Set Theory. 2nd ed. 2 OXTOBY. Measure and Category. 2nd ed. 3 SCHAEFER. Topological Vector Spaces. 2nd ed. 4 HIL TON/ST AMMBACH. A Course in Homological Algebra. 2nd cd. 5 MAC LANE. Categories for the Working Mathematician. 2nd ed. SERRE. A Course in Arithmetic. 6 HUGHEsiPIPER. Projective Planes. 7 8 T AKEUTIIZARING. Akiomatic Set Theory. 9 HUMPHREYS. Introduction to Lie Algebras and Representation Theory. 10 COHEN. A Course in Simple Homotopy Theory. 11 CONWAY. Functions of One Complex Variable I. 2nd ed. > 12 BEALS. Advanced Mathematical Analysis. 13 ANDERSON/FULLER. Rings and Categories 35 ALEXANDERIWERMER. Several Complex Variables and Banach Algebras. 3rd ed. 36 KELLEy/NAMIOKA et al. Linear Topological Spaces. 37 MONK. Mathematical Logic. 38 GRAUERTIFRITZSCHE. Several Complex Variables. 39 ARVESON. An Invi~tion to C·-Algebras. 40 KEMENY ISNELIlKNAPP. Denumerable Markov Chains. 2nd ed. 41 APOSTOL. Modular Functions and Dirichlet Series in Number Theory. 2nd ed. 42 SERRE. Linear Representations of Finite Groups. 43 GILLMANlJERlSON. Rings of Continuous Functions. 44 KENDIG. Elementary Algebraic Geometry. 45 LOEVE. Probability Theory I. 4th ed. 46 LOEVE. Probability Theory II. 4th ed. 47 MOISE. Geometric Topology in Dimensions 2 and 3. 48 SACHSlWu. General Relativity for Mathematicians. 49 GRUENBERGlWEIR. Linear Geometry. 2nd ed. 50 Eow ARDS. Fermat's Last Theorem. 51 KLINGENBERG. A Course in Differential Geometry. 52 HARTSHORNE. Algebraic Geometry. 53 MANIN. A Course in Mathematical Logic. 54 GRA VERlW ATKINS. Combinatorics with Emphasis on the Theory of Graphs. 55 BROWNIPEARCY. Introduction to Operator Theory I: Elements of Functional Analysis. 56 MASSEY. Algebraic Topology: An Introduction. 57 CRowELLIF'OX. Introduction to Knot Theory. 58 KOBLITZ. p-adic Numbers, p-adic Analysis, and Zeta-Functions. 2nd ed. 59 LANG. Cyclotomic Fields. 60 ARNOLD. Mathematical Methods in Classical Mechanics. 2nd ed. 61 WHITEHEAD. Elements of Homotopy Theory. 62 KARGAPOLovIMERLZJAKOV. Fundamentals of the Theory of Groups. 63 BOLLOBAS. Graph Theory. 64 EDWARDS. Fourier Series. Vol. 1. 2nd ed. 65 WELLS. Differential Analysis on Complex Manifolds. 2nd ed. (continued after index) of Modules. 2nd ed. 14 GOLUBITSKV/GUILLEMIN. Stable Mappings and Their Singularities. IS BERBERIAN. Lectures in Functional Analysis and Operator Theory. 16 WINTER. The Structure of Fields. t7 ROSENBLA'IT. Random Processes. 2nd ed. 18 HALMOS. MeasUre Theory. 19 HALMOS. A Hilbert Space Problem Book. 2nd ed. 20 HUSEMOLLER. Fibre Bundles. Jrd ed. 21 HUMPHREYS. Linear Algebraic Groups. 22 BARNEslMACK. An Algebraic Introduction to Mathematical Logic. 23 GREUB. Linear Algebra. 4th ed. 24 HOLMES. Geometric Functional Analysis and Its Applications~ 25 HEWITT/STROMBERG. Real and Abstract Analysis. 26 MANES. Algebraic Theories. 27 KELLEY. General Topology. 28 ZARlSKllSAMUEL. Commutative Algebra. Vol.l. 29 ZARlSKIISAMUEL. Commutative Algebra. 30 31 32 Vol.II. JACOBSON. Lectures in Abstract Algebra I. Basic Concepts. JACOBSON. Lectures in Abstract Algebra II. Linear Algebra. JACOBSON. Lectures in Abstract Algebra III. Theory of FieJds and Galois Theory. 33 HIRSCH. Differential Topology. 34 SPITZER. Principles of Random Walk. 2nd ed.

Jean-Pierre Serre Linear· Representations of Finite Groups Translated from the French by Leonard L . Scott Springer

Jean-Pierre Serre College de France 75231 Paris Cedex 05 France Editorial Boord S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA Leonard L. Scott University of Virginia Department of Mathematics Charlottesville, Virginia 22?03 USA .. F.W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA K.A. Ribet Mathematics Department University of California at Berke1ey Berkeley, CA 94720-3840 USA Mathematics Subject Classification: 20Cxx Library of Congress Cataloging in Publication Data Serre, Jean .. Pierre. Linear representations of finite groups. (GradUJte texts in mathematics ; 42) Translation of Representations lineaires des groupes finis, 2. ed. Includes bibliographies and indexes. J. Representations of groups. 2. Finite groups. QA17J.SS313 I. Title. II. Series. 512'.2 76·11585 Translation of the French edition Repr~senlal;ons linla;res des groupes finis, Paris: Hennann 1971 © 1977 by Springer .. Veriag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the writ .. ten pennission of the publisher (Springer .. VerJag New York, Inc., 175 Fifth Avenue, New York, NY 10010; USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in con .. nection with any form of information .storage and retrieval, electronic adaptation, computer soft ware, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even. if the former are Dot especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. The reprint has been authorized by Springer-Verlag (BerlinlHeidelberglNew York) for sale in the People's Republic of China only and not for export therefrom ISBN 0-387-90 190-6 ISBN 3-540 .. 90 190-6 SPIN 10834786 Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH

Preface This book consists of three parts, rather different in level and purpose: The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and charac ters. This is a fundamental result, of constant use in mathematics as well as in quantum chemistry or physics. I have tried to give proofs as elementary as possible, using only the definition of a group and the rudiments of linear algebra. The examples (Chapter 5) have been chosen from those useful to chemists. The second part is a course given in 1966 to second-year students of I'Ecole Nonnale. It completes the first on the following points: (a) degrees of representations and integrality properties of characters (Chapter 6); (b) induced representations, theorems of Artin and Brauer, and applications (Chapters 7-1 J); (c) rationality questions (Chapters] 2 and ) 3). The methods used are those of linear algebra (in a wider sense than in the first part): group algebras, modules, noncommutative tensor products, semisimple algebras. The third part is an introduction to Brauer theory: passage from characteristic 0 to characteristic p (and conversely). I have freely used the language of abelian categories (projective modules, Grothendieck groups), which is well suited to this sort of question. The principal results are: (a) The fact that the decomposition homomorphism is surjective: all irreducible representations in characteristic p can be lifted "virtually" (i.e., in a suitable Grothendieck group) to characteristic o. (b) The Fong-Swan theorem, which allows suppression of the word' 'virtually" in the preceding statement, provided that the group under consideration is p-solvable. I have also given several applications to the Artin representations. • v

Preface I take pleasure in thanking: Gaston Berthier and Josiane Serre, who have authorized me to reproduce Part I, written for them and their students in Quantum Chemistry; Yves Balasko, who drafted a first version of Part II from some lecture notes; Alexandre Grothendieck, who has authorized me to reproduce Part III, which first appeared in his Sc!minaire de Gc!omc!trie Algc!brique, I.H.E.S., 1965/66.

Contents 1 3 3 4 5 7 7 9 10 10 13 15 17 18 21 23 25 25 26 28 32 32 32 33 vii ~, Part I Representations and Characters ] Generalities on linear representations 1.1 Definitions 1.2 Basic examples 1.3 Subrepresentations 1.4 I.S Tensor product of two representations 1.6 Symmetric square and alternating square Irreducible representations 2 Character theory 2.1 The character of a representation 2.2 Schur's lemma; basic applications 2.3 Orthogonality relations for characters 2.4 Decomposition of the regular representation 2.5 Number of irreducible representations 2.6 Canonical decomposition of a representation 2.7 Explicit decomposition of a representation 3 Subgroups, products, induced representations 3.] Abelian subgroups 3.2 Product of two groups 3.3 Induced representations 4 Compact groups 4. ) Compact groups 4.2 4.3 Linear representations of compact groups Invariant measure on a compact group

Contents 5 Examples 5.1 The cyclic Group c,. 5.2 The group Coo 5.3 The dihedral group On 5.4 T~e group D.h 5.5 The group 0 00 5 .6 The group Dooll 5.7 The alternating group 214 5.8 The symmetric group 6 4 5.9 The group of the cube Bibliography: Part I Part II Representations in Characteristic Zero 6 The group algebra 6.1 Representations and modules 6.2 Decomposition, of e[G ] 6.3 The center of C [G ] 6.4 Basic properties of integers 6.5 Induced representations; Mackey's criterion 7.1 7.2 The character of an induced representation; Induction Integrality properties of characters. Applications 7 the rec iprocity formula 7.3 Restriction to subgroups 7.4 Mackey's irreducibility criterion 8 Examples of induced representations 8.1 Normal subgroups; applications to the degrees of the irreducible representations 8.2 Semidirect products by an abelian group 8.3 A review of some classes of finite groups 8.4 Sylow's theorem 8.5 Linear representations of supersolvable groups 9 Artin 's theorem 9.1 The ring R(G) 9.2 Statement of Artin's theorem 9.3 First proof 9.4 Second proof of (i) ~ (ii) 10 A theorem of Brauer 10.1 p-regular elements; p-elementary subgroups 10.2 Induced characters arising from p-elementary subgroups 10.3 Construction of characters ) 0.4 Proof of theorems ) 8 and ) 8' 10.5 Brauer's theorem viii 35 35 36 36 38 39 40 41 42 43 44 45 47 47 48 50 50 52 54 54 55 58 59 61 61 62 63 65 66 68 68 70 70 , 72 74 74 75 76 78 78

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