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The Friendly Giant Invent. math. 69, 1-102 (1982) II~ventiongs mathematicae Springer-Verlag 1982 The Friendly Giant Robert L. Griess, ,lr. Department of Mathematics, University of Michigan, Ann Arbor, Mi 48109, USA Table of Contents 1. Introduction .................................. 1 2. Preliminary Results ............................... 4 3. Faithful Modules for Extraspecial Groups ..................... 27 4. The Groups C, C~. and C and the Vector Space B .................. 28 5. Tensor Products of Irreducibles of C ........................ 33 6. The Algebra Product .............................. 39 7. The Groups F and a ............................... 40 8. The Action of Elements of P on the v().) and the e(x)| x, . ............. 48 9. The Belas ................................... 50 10. The Definition of~r ............................... 53 1 l. A Proof that o- is an Algebra Automorphism .................... 59 12. The Identification of G =(C, a) .......................... 81 13. Consequences .................................. 86 14. The Happy Family and the Pariahs ........................ 91 15. Concluding Remarks .............................. 96 List of Notations and Definitions ........................... 97 last of Tables ................................... 99 References ..................................... 100 1. Introduction In this paper, we demonstrate the existence of the Friemtly Giant, a finite simple group of order 2463205976 112133 . 17.19.23.29.31.41.47.59.71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. Evidence for the existence of this group was produced independently in November, 1973, by Bernd Fischer in Bielefeld and by this author in Ann Dedicated to the memory of Richard D. Brauer, February 10, 1901-April 17, 1977 Research supported in part by NSF grants MCS-78-02463 and MCS-80-03027 0020-9910/82/0069/0001/$20.40 2 R.L. Griess, Jr. Arbor. Serious work on this group - mainly a study of subgroups and con- jugacy classes - began the first weekend of that month in both locations. Additional details of this early work are discussed in Sect. 15. For now, we add only that such a simple group appeared likely to have a complex irreducible character of degree 196883; in 1974, this number was established as a lower bound for the degree of a nonprincipal irreducible character [13, 37]. While this evidence for the existence was very persuasive, it did not constitute a proof. Our existence proof was announced on January 14, 1980 and more formally in [-29]. Our method is to take a 196884-dimensional module B for a particular group C of shape (21+24)( 1), define on B the structure of a commutative nonassociative algebra with a symmetric nondegenerate associative bilinear form, then define an automorphism ~ of this algebra. The group G-~C, c~) is the simple group of the title (the usual symbol for this group is F1). The extra rigidity required by expecting our linear group to preserve an algebra structure enables us to make precise definitions of the relevant linear transformations and verify their required properties. The reason we thought of this approach is the following. Simon Norton had computed the values of a hypothetical character Z of degree 196883 and computed that (S2x, 1)=1, ($3)~, 1)=1, (S3Z, Z)= 1 and Z is rational-valued. It follows that if M is a module affording Z, M has the structure of a commutative (but not necessarily associative) algebra with a nondegenerate associative symmetric bilinear form. This finding of Norton was the inspiration for this paper. See Sect. 15 for additional comments on algebras associated to finite simple groups. We comment on some over-all aspects of the construction. In some sense, the algebra B is described using only basic linear algebra. The group theory used is descriptive in nature. Thus, one could say that the construction of G =(C, or) is elementary. That is, starting from scratch, one may construct M24, then 0 and finally G, with each stage depending on the previous one. See two paragraphs ahead and look at Table 1.1. However, the identification of G as a finite simple group with the right properties requires deep results from the classification of finite groups. It is possible that this dependence can be elim- inated, for instance, by counting configuration of vectors in B permuted by G. An enumeration of any such configurations may be long and difficult, however. Section 2 contains various preliminary results, mainly about group repre- sentations, the Leech lattice, Conway groups and the classification of finite simple groups. Sections 3 and 4 set up basic notation. In Sect. 5, we compute the C-invariant algebra structures on the module B, and in Sect. 6 we select the one we work with in the rest of the paper (modulo a choice of F made in Sect 7). Sections 7, 8 and 9 discuss various technicalities needed both in the definition of a (Sect. 10) and in the proof of the "main result," Proposition 11.2, that cr is an algebra automorphism. Section 7 is concerned with a choice of complement F which will cause the function // to behave well, while Sects. 8 and 9 develop techniques for analyzing the action of certain elements of C on basis elements, mainly for the purpose of being able to analyze/3. Nearly all of Sect. 11 is concerned with a proof of the main result, which in turn amounts to verifying a list of identities involving configurations of vectors in the Leech The Friendly Giant 3 lattice; this is where the correctness of the plus and minus signs in the definition of cr is so critical. In Sects. 12, 13, and 14, the mathematics departs from that of preceding sections in that we require results from the classification theory, and, in Sect. 14, we refer to work of others on the group Fa, only some of which has appeared. In Sect. 12, we identify G=(C, c~) as a finite simple group of order 2r176 It is not obvious that G is finite, and if G is finite, it is not obvious that the containment C< CG(z), (z)=Z(C), is equality, a necessary step in the identification of G. This prob- lem is handled by a "reduction modulo p" procedure. In Sect. 13, we derive existence of a number of sporadic groups (besides G). These other groups had been constructed earlier; in some of these cases, existence proofs required com- puter work. All we need to do is name appropriate subquotients of G (although we use results from the classification theory to identify these subquotients), using little more than notation already established earlier in the paper. Also in Sect. 13, we derive existence of a number of nonsplit group extensions; hence we get nonvanishing of certain degree 2 cohomology groups. In Sect. 14, we determine that the simple groups LyS, J3, "]4, O'S and Ru are not involved in the Friendly Giant. The sporadic groups which are in- volved in the Friendly Giant constitute the Happy Family and those which are not are called the Pariahs. The membership of every sporadic group in one of those two categories is settled, except for J~. The twenty sporadics M~I, M~z , M22, M23, M24, J2, Held, HiS, McL, Suz, .1, .2, .3, F22, F23, F24, F~, F 2, Fa, Fs are involved in the Friendly Giant in a "visible" manner. A glance at the group orders shows that LyS and J~ must be Pariahs, but it is certainly not obvious for J3, Ru and O'S. The group J1 has order "only" 175,560 and one might easily imagine a copy of J~ floating as a tiny speck within F~. We point which that J~ is a subgroup of O'S (the fixed points of an outer automor- phism), which is not involved in F 1. In any case, suitable information is available (using outside sources) to carry out the arguments of Sect. 14. In Sect. 15, we conclude with some comments on background and the proof. A list of notations and definitions and a list of tables to assist the reader has been placed before the references. We make it clear that our construction of G (Sect. 2 through 11) is direct, explicit and is carried out entirely by hand. The identification of G, however, requires hard theorems from the classification of finite simple groups. A few of our arguments in Sect. 14 require computer calculations, but this is the only place in the paper where we make explicit reference to computer work. Some work in the theory of finite simple groups does involve computing machines and a few of the references we use do have some ultimate dependence on such work (e.g., in determining conjugacy classes and character tables). With these exceptions, the results of this paper are free of machine calculations. This work was carried out mainly at the Institute for Advanced Study during the academic year 197%80. We thank the Institute for Advanced Study for the privileges of membership during this time (and during Winter term, 1981), the National Sciences Foundation for partial financial support and the University of Michigan for partial financial support during that sabbatical year. 4 R.L. Griess, .Jr. Special thanks go to my wife, Pamela Schwarzmann, for her support and patience with me during the year at the Institute for Advanced Study and to Enrico Bombieri for many words of encouragement. We acknowledge helpful remarks from Allan Adler, George Glauberman, Melvin Hochster, Michael O'Nan, Steven Smith and Ronald Solomon which led to clarifications and we acknowledge the computer work by Charles Sims and Steven Smith which settled a few points in Sect. 14. We thank the referee for investing an enormous amount of work in reading this paper and providing thoughtful and detailed commentary. In particular the referee caught some mistakes in the preprint version.
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