Pacific Journal of Mathematics IN THIS ISSUE— Walter Feit and John Griggs Thompson, Chapter I, from Solvability of groups of odd order, Pacific J. Math, vol. 13, no. 3 (1963 .................................................... 775 Walter Feit and John Griggs Thompson, Chapter II, from Solvability of groups of odd order, Pacific J. Math., vol. 13, no. 3 (1963 .................................................... 789 Walter Feit and John Griggs Thompson, ChapterIII, from Solvability of groups of odd order, Pacific J. Math., vol. 13, no. 3 (1963 .................................................. 803 Walter Feit and John Griggs Thompson, Chapter IV, from Solvability of groups of odd order, Pacific J. Math., vol. 13, no. 3 (1963 .................................................. 845 Walter Feit and John Griggs Thompson, Chapter V, from Solvability of groups of odd order, Pacific J. Math., vol. 13, no. 3 (1963 .................................................... 943 Walter Feit and John Griggs Thompson, Chapter VI, from Solvability of groups of odd order, Pacific J. Math., vol. 13, no. 3 (1963 .................................................. 1011 Walter Feit and John Griggs Thompson, Bibliography, from Solvability of groups of odd order, Pacific J. Math., vol. 13, no. 3 (1963 .................................................. 1029 Vol. 13, No. 3 May, 1963 PACIFIC JOURNAL OF MATHEMATICS EDITORS RALPH S. PHILLIPS J. DUGUNDJI Staruord University University ot Southern California Staniord, Calilornia Los Angeles 7, California M. G. ARSOVE LOWELL J. PAIGE University of Washington University of California Seattle 5, Washington Los Angeles 24, California ASSOCIATE EDITORS E. F. BECKENBACH D. DERRY H. L. ROYDEN E. G. STRAUS T. M. CHERRY M. OHTSUKA E. SPANIER F. WOLF SUPPORTING INSTITUTIONS UNIVERSITY OF BRITISH COLUMBIA STANFORD UNIVERSITY CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF TOKYO UNIVERSITY OF CALIFORNIA UNIVERSITY OF UTAH MONTANA STATE UNIVERSITY WASHINGTON STA1L UNIVERSITY UNIVERSITY OF NEVADA UNIVERSITY OF WASHINGTON NEW MEXICO STATE UNIVERSITY * * * OREGON STATE UNIVERSITY AMERICAN MATHEMATICAL SOCIETY UNIVERSITY OF OREGON CALIFORNIA RESEARCH CORPORATION OSAKA UNIVERSITY SPACE TECHNOLOGY LABORATORIES UNIVERSITY OF SOUTHERN CALIFORNIA NAVAL ORDNANCE TEST S1AT1OIM Mathematical papers intended for publication in the Pacific Journal of Mathematics should be typewritten ^double spaced), and the author should keep a complete copy. Manuscripts may be sent to any one of the four editors. All other communications to the editors should be addressed to the managing editor, L. J. Paige at the University of California, Los Angeles 24, California. 50 reprints per author of each article are furnished free of charge; additional copies may be obtained at cost in multiples of 50. The Pacific Journal of Mathematics is published quarterly, in March, June, September, and December. Effective with Volume 13 the price per volume (4 numbers) is $18.00; single issues, $5.00. Special price for current issues to individual faculty members of supporting institutions and to individual members of the American Mathematical Society: $8.00 per volume; single issues $2.50. Back numbers are available. Subscriptions, orders for back numbers, and changes of address should be sent to Pacific Journal of Mathematics, 103 Highland Boulevard, Berkeley 8, California. Printed at Kokusai Sunken hibatsusha (international Academic Printing Co., Ltd.), No. 6, 2 chome, Fujimi-cho, Chiyoda-ku, Tokyo, Japan. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS, A NON-PROFIT CORPORATION The Supporting Institutions listed above contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its content or policies SOLVABILITY OF GROUPS OF ODD ORDER WALTER FEIT AND JOHN G. THOMPSON CHAPTER I 1. Introduction The purpose of this paper is to prove the following result: THEOREM. All finite groups of odd order are solvable. Some consequences of this theorem and a discussion of the proof may be found in [11]. The paper contains six chapters, the first three being of a general nature. The first section in each of Chapters IV and V summarizes the results proved in that chapter. These results provide the starting point of the succeeding chapter. Other than this, there is no cross reference between Chapters IV, V and VI. The methods used in Chapter IV are purely group theoretical. The work in Chapter V relies heavily on the theory of group characters. Chapter VI consists primarily of a study of generators and relations of a special sort. 2. Notation and Definitions Most of the following lengthy notation is familiar. Some comes from a less familiar set of notes of P. Hall [20], while some has arisen from the present paper. In general, groups and subsets of groups are denoted by German capitals, while group elements are denoted by ordinary capitals. Other sets of various kinds are denoted by English script capitals. All groups considered in this paper are finite, except when explicitly stated otherwise. Ordinary lower case letters denote numbers or sometimes elements of sets other than subsets of the group under consideration. Greek letters usually denote complex valued functions on groups. However, Received November 20, 1962. While working on this paper the first author was at various times supported by the U. S. Army Research Office (Durham) contract number DA-30-115-ORD-976 and National Science Foundation grant G-9504; the second author by the Esso Education Foundation, the Sloan Foundation and the Institute for Defense Analyses. Part of this work was done at the 1960 Summer Conference on Group Theory in Pasadena. The authors wish to thank Professor A. A. Albert of the University of Chicago for making it possible for them to spend the year 1960-61 there. The authors are grateful to Professor E. C. Dade whose careful study of a portion of this paper disclosed several blunders. Special thanks go to Professor L. J. Paige who has expedited the publication of this paper. 775 776 SOLVABILITY OF GROUPS OF ODD ORDER a and r are reserved for field automorphisms, permutations or other mappings, and e is used with or without subscripts to denote a root of unity. Bold faced letters are used to denote operators on subsets of groups. The rational numbers are denoted by &, while <ff* denotes the field of nth roots of unity over &. Set theoretic union is denoted by U. If 21 and S3 are sets, 21 — 93 denotes the elements of 21 which are not in S3. 21 c S3 means that 21 is a proper subset of S3. <(• • • | • • •> the group generated by • • • such that <1> will be identified with 1. {•••!•••} the set of • • • such that the group defined by the generators • • • with the relations 13EI the number of elements in the set X. V the set of non identity elements in the set X. TZ a set of primes. If n = {p}, we customarily identify n with p. n' the complementary set of primes. 7r-number a non zero integer all of whose prime factors are in nm the largest 7r-number dividing the non zero integer n. rc-group a group X with | X | = | X I*. 7r-element a group element X such that <X> is a 7r-group. S^-subgroup of X a subgroup @ of X with | @ | = | X |^. S-subgroup of X a S^-subgroup of X for suitable TU. Hall subgroup of X a S-subgroup of X. 5?<X S is a normal subgroup of X. S? char X $ is a characteristic subgroup of X. /(X mod 31) the inverse image in X of /(X/5JZ). Here 31 < X, and / is a function from groups to subgroups. the maximal normal 7r-subgroup of X. O,n(XmodO,1,....,n_1(X)). 7r-closed group we say that X is 7r-closed if and only if X has a normal S^-subgroup. the Fitting subgroup of X, the maximal normal nilpotent subgroup of X. W) the Frattini subgroup of X, the intersection of all maximal subgroups of X. the nth term in the ascending central series of X, defined inductively by: Z0(X) = 1, ZX(H) = 2. NOTATION AND DEFINITIONS 777 Z(X) = center of X, ZW+1(X) = Z(X mod Zn(X)). O*(X) the smallest normal subgroup ?) of X such that X/$ is a 7r-group. [X, F] X^Y^XY = X~lXT. [Xu • • •, Xn] [[Xu . • •, Xn^l Xn], n^S. [21, SB] <[A, JB] | A e 2i, B e S3>, 21 and 35 being subsets of a group. >. If IS?),3£S is called the normal closure of X in 2). I' [3E, X], the commutator subgroup of X. Cn(X) the nth term of the descending central series of X, defined inductively by: Cx(X) = X, CW+1(X) = [Cn(X), I]. £n(X) the subgroup of the p-group 3£ generated by the elements of order at most pn. ffn(3E) the subgroup of the p-group X generated by the pnth powers of elements of X. m(X) the minimal number of generators of X. mp(X) mity), ?$ being a Sp-subgroup of X. cl (X) the class of nilpotency of the nilpotent group X, that is, the smallest integer n such that X = Zn(X). C$Q(2I) the largest subset of S3 commuting element- wise with 21, 21 and S3 being subsets of a group X. In case there is no danger of confusion, we set C(2I) = CS(2I). the largest subset of S3 which normalizes 21,21 and S3 being subsets of a group X. In case there is no danger of confusion, we set iV(2I) = ker (X —• 59) the kernel of the homomorphism a of the group X into the group 2). a will often be suppressed. ccla(«) {2F | Xe X}, 21 being a subset of X. x F(ccls(2I);33) <2l | X e X, 21* s S3>, the weak closure of cclx(2I) in S3 with respect to the group X. Here 21 and 33 are subgroups of X. If 21 = F(ccl3E(2l); S3), we say that 21 is weakly closed in S3 with respect to X.