SURVEYS OF MODERN MATHEMATICS

Transcription

1 SURVEYS OF MODERN MATHEMATICS SMM 1: SMM 2: SMM 3: SMM 4: SMM 6: SMM 7: SMM 8: SMM 9: Analytic Methods in Algebraic Geometry by Jean-Pierre D ly Lie Theory and Representation Theory Naihong Hu, Bin Shu and Jianpan Wang, eds. Application of Elementary Differential Geometry to Influence Analysis Yat-Sun Poon and Wai-Yin Poon, eds. An Introduction to Rota-Baxter Algebra by Li Guo Open Problems and Surveys of Contemporary Mathematics Lizhen Ji, Yat-Sun Poon and Shing-Tung Yau, eds. Introductory Lectures on Manifold Topology: Signposts by Thomas Farrell and Yang Su Lie-Bäcklund-Darboux Transformations by Y. Charles Li and Artyom Yurov Compressible Flow and Euler s Equations by Demetrios Christodoulou and Shuang Miao SMM 10: Finite Groups: An Introduction by Jean-Pierre Serre

2 SURVEYS OF MODERN MATHEMATICS Series Editors Shing-Tung Yau Department of Mathematics Harvard University Cambridge, Massachusetts U.S.A. Lizhen Ji Department of Mathematics University of Michigan, Ann Arbor U.S.A. Yat-Sun Poon Department of Mathematics University of California at Riverside U.S.A. Jean-Pierre D ly Laboratoire de Mathématiques Institut Fourier Saint-Martin d Hères, France Eduard J. N. Looijenga Mathematisch Instituut Universiteit Utrecht The Netherlands Neil Trudinger Mathematical Sciences Institute Australian National University Canberra, Australia Jie Xiao Department of Mathematics Tsinghua University Beijing, China

3 Surveys of Modern Mathematics Volume 10 Finite Groups: An Introduction Jean-Pierre Serre Honorary professor at the Collège de France, Paris With assistance in translation provided by: Garving K. Luli (University of California at Davis) Pin Yu (Tsinghua University, Beijing) International Press HIGHER EDUCATION PRESS

4 Surveys of Modern Mathematics, Volume 10 Finite Groups: An Introduction by Jean-Pierre Serre (honorary professor at the Collège de France, Paris) Copyright 2016 by International Press, Somerville, Massachusetts, U.S.A., and by Higher Education Press, Beijing, China. This work is also published and sold in China, exclusively by Higher Education Press. All rights reserved. Individual readers of this publication, and non-profit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgement of the source is given. Republication, systematic copying, or mass reproduction of any material in this publication is permitted only under license from International Press. ISBN: Printed in the United States of America

5 Contents Preface Conventions and Notation ix x 1 Preliminaries Group actions Normal subgroups, automorphisms, characteristic subgroups, simple groups Filtrations and Jordan-Hölder theorem Subgroups of products: Goursat s lemma and Ribet s lemma Exercises Sylow theorems Definitions Existence of p-sylowsubgroups Properties of the p-sylowsubgroups Fusion in the normalizer of a p-sylow subgroup Local conjugation and Alperin s theorem Other Sylow-like theories Exercises Solvable groups and nilpotent groups Commutators and abelianization Solvable groups Descending central series and nilpotent groups Nilpotent groups and Lie algebras Kolchin s theorem Finite nilpotent groups v

6 Contents vi 3.7 Applications of 2-groupstofieldtheory Abelian groups The Frattini subgroup Characterizations using subgroups generated by two elements Exercises Group extensions Cohomology groups A vanishing criterion for the cohomology of finite groups Extensions, sections and semidirect products Extensions with abelian kernel Extensions with arbitrary kernel Extensions of groups of relatively prime orders Liftings of homomorphisms Application to p-adicliftings Exercises Hall subgroups π-subgroups Preliminaries: permutable subgroups Permutable families of Sylow subgroups Proof of theorem Sylow-like properties of the π-subgroups A solvability criterion Proof of theorem Exercises Frobenius groups Union of conjugates of a subgroup An improvement of Jordan s theorem Frobenius groups: definition Frobenius kernels Frobenius complements Exercises

7 Contents vii 7 Transfer Definition of Ver : G ab H ab Computation of the transfer A two-century-old example of transfer: Gauss lemma An application of transfer to infinite groups Transfer applied to Sylow subgroups Application: groups of odd order < Application: simple groups of order The use of transfer outside group theory Exercises Characters Linear representations and characters Characters, hermitian forms and irreducible representations Schur s lemma Orthogonality relations Structure of the group algebra and of its center Integrality properties Galois properties of characters The ring R(G) Realizing representations over a subfield of C, for instance the field R Application of character theory: proof of Frobenius s theorem Application of character theory: proof of Burnside s theorem The character table of A Exercises Finite subgroups of GL n Minkowski s theorem on the finite subgroups of GL n (Q) Jordan s theorem on the finite subgroups of GL n (C) Exercises

8 Contents viii 10 Small Groups Small groups and their isomorphisms Embeddings of A 4, S 4 and A 5 in PGL 2 (F q ) Exercises Bibliography 165 Index 171 Index of names 177

9 Preface This book is based on a course given at École Normale Supérieure de Jeunes Filles, Paris, in Its aim is to give an introduction to the main elementary theorems of finite group theory. Handwritten notes were taken by Martine Buhler and Catherine Goldstein (Montrouge, 1979); they were later type-set by Nicolas Billerey, Olivier Dodane and Emmanuel Rey (Strasbourg-Paris, 2004), and made freely available through arxiv:math/ In 2013, they were translated into English by Garving K. Luli and Pin Yu. In , I revised and expanded them (by a factor 2) for the present publication: I gave many references to old and recent results, I added two chapters on finite subgroups of GL n, and on small groups, and I also added about 160 exercises. I thank heartily all the people mentioned above, without whom this book would not have been published. Jean-Pierre Serre, Paris, Spring 2016 ix

10 Conventions and Notation The symbols Z, Q, F p, F q, R, C have their usual meaning. Set theory If X Y, the complement of Y in X is written X Y. ThenumberofelementsofafinitesetX is denoted by X. Rings Rings have a unit element, written 1. If A is a ring, A is the group of invertible elements of A. The word field means commutative field. Group theory We use standard notation such as (G : H), G/H, H\G when H is a subgroup of a group G. AgroupG is abelian (= commutative) if xy = yx for every x, y G. If A is a subset of G, the centralizer of A in G is written C G (A); it is the set of all g G such that ga = ag for every a A. The normalizer of A is written N G (A); it is the set of all g G such that gag 1 = A. If A, B aresubsetsofg, the set of all products ab with a A and b B is written either A.B or AB; the subgroup of G generated by A and B is written A, B. The formula G =1means that G =1;whenGis abelian, and written additively, we write G =0instead. Symmetric groups The symmetric and alternating groups of permutations of {1,...,n} are written S n and A n. The group of permutations of a set X is written S X. Linear groups If A is a commutative ring, and n is an integer 0, then: x

11 Contents xi M n (A) =A-algebra of n n matrices with coefficients in A, GL n (A) =M n (A) = group of invertible n n matrices with coefficients in A, SL n (A) = Ker(det : GL n (A) A ). We use End(V ), GL(V ) and SL(V ) for the similar notions relative to a vector space of finite dimension. Let k be a field. If n 1, there is a natural isomorphism of k onto the center of GL n (k); thequotientgl n (k)/k is the n-th projective linear group PGL n (k). The image of SL n (k) into PGL n (k) is denoted by PSL n (k).

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13 About the Author Jean-Pierre Serre was awarded the Fields Medal in 1954 for his work in algebraic topology. In 2003 he was awarded the first Abel Prize by the Norwegian Academy of Science and Letters. Serre attended the École Normale Supérieure ( ) and the Sorbonne (Ph.D.; 1951), both now part of the Universities of Paris. Between 1948 and 1954 he was at the National Centre for Scientific Research in Paris, and after two years at the University of Nancy he returned to Paris for a position at the Collège de France. He retired in Between 1983 and 1986 Serre served as vice president of the International Mathematical Union. Serre was awarded the Fields Medal at the International Congress of Mathematicians in Amsterdam in Serre s mathematical contributions leading up to the Fields Medal were largely in the field of algebraic topology, but his later work ranged widely in algebraic geometry, group theory, and especially number theory. By seeing unifying ideas, he helped to unite disparate branches of mathematics. One of the more recent phenomena to which he was a principal contributor was the application of algebraic geometry to number theory applications now falling into a separate subclass called arithmetic geometry. He was one of the second generation of members of Nicolas Bourbaki (publishing pseudonym for a group of mathematicians) and a source of inspiration for fellow medalists Alexandre Grothendieck and Pierre Deligne. An elegant writer of mathematics, Serre published Groupes algébriques et corps de classes (1959, Algebraic Groups and Class Fields); Corps locaux (1962, Local Fields); Lie Algebras and Lie Groups (1965); Abelian l-adic Representations and Elliptic Curves (1968); Cours d arithmétique (1970, A Course in Arithmetic); Cohomologie Galoisienne (1964, Galois Cohomology); Représentations linéaires des groupes finis (1967, Linear Representations of Finite Groups); Algèbre locale, multiplicités (1965, Local Algebra: Multiplicities); Arbres, amalgames, SL 2 (1977, Trees); and, with Uwe Jannsen and Steven L. Kleiman, Motives (1994). His collected works were published in A Leroy P. Steele Prize in 1995 was awarded to Serre on the basis of A Course in Arithmetic.