Proc. NatL Acad. Sci. USA Vol. 78, No. 2, pp. 689-691, February 1981 Mathematics A construction of F1 as automorphisms of a 196,883-dimensional algebra (Fischer-Griess simple group/sporadic simple group/classification of finite simple groups/ commutative nonassociative algebra/Leech lattice) ROBERT L. GRIESS, JR. School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540; and tDepartment of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 Communicated by Deane Montgomery, September 2, 1980 ABSTRACT In this note, I announce the construction of the Let fl = F23 U {oo}, a set of 24 points, 9f = S9(5, 8, 24) the finite simple group F1, whose existence was predicted indepen- usual Steiner system based on fQ. The Mathieu group M24 is the dently in 1973 by Bernd Fischer and by me. The group has order set of permutations of fi that preserve $1. The 759 octads of 246320597611213 17.19.23.29.31.41.47.59.71 = 808,017,424,794, of the 512,875,886,459,904,961,710,757,005,754,368,000,000,000 and is Yf generate a 12-dimensional linear subspace P(Q), power realized as a group of automorphisms of a 196,883-dimensional set, called the '6-sets. commutative nonassociative algebra over the rational numbers, Let {xili E fl} be an orthonormal basis of Q" Q24 and let which has an associative form. Equivalently, it is a group of au- (,) be one-eighth the usual dot product. Using this basis, A is tomorphisms of a cubic form in 196,883 variables. It turns out that generated by all the relevant arguments and calculations may be done by hand. (i) all (28016) (support an octad; even number of minus signs) Furthermore, existence of the group F1 implies the existence of at a number of other sporadic simple groups for which existence (ii) all (3 123) (take -3xi + j,,,i xj, then change signs every proofs formerly depended on work with computers. We are be- '6-set). ginning to look upon this group as a "friendly giant." A monomial group N24 = 2'2,M24 acts on A and is maximal in 0 (called "dot zero"), the "group of units" of A. The group .1 Motivation = 0/{± 1} is the simple group of Conway. A p-group Q is extraspecial if Q' = ¢D(Q) = Z(Q) ZP. The If F, exists, it contains an involution z whose centralizer C has ~-+ E is a an map (xZ(Q),yZ(Q)) [x,y] Z(Q) nonsingular alternating the following properties: (i) Q: = 02(C) 21+24, extraspecial form on the vector space Q/Z(Q) _ z2, n-1.n When p = 2, 2-group of order 2' and plus type; (ii) C/Q 1, the simple we have additional structure: a form x2 E it quadratic xZ(Q) group of order 22139547211. 13.23 discovered by Conway (1); Z(Q). In fact, Aut(Q)/Inn(Q) 0-(2n,2), E = ±, and where E is -0/{± 1}, where 0 is the group of automorphisms of a rank- = + if and only if the Witt index is maximal (equivalently, if 24 even integral unimodular lattice, A, called the Leech lattice; Q contains ZnI 1). Write Q-21+2n. Then Q has exactly one faith- and (iii) by conjugation, C acts on Q and this action makes Q/ and it Z24 ful irreducible complex representation, of degree 2", may (z) isomorphic to the module A/2A (= as abelian groups) the rationals if E = +. Call the module T. for C/Q- 1. be written over have Suppose e = +. Write Q = EF, where E Z-21 and F Several people, including Conway, Norton, and me (2), Zn. Let (z) = Z(Q). Let 'Pj, ... be the 2" distinct linear characters noticed that if X # 1 is an irreducible character of F1, then A(1) in TIE. Choose a unit eigenvector e(l) for 'pj. Set e(x): = e(l)x, 2 196,883 and, furthermore, that 196,883 is likely to be a de- with Norton (3) and others at Cambridge worked on the con- x E F. Then Qe(x) affords spx: = ep. There is A c GL(T) gree. = if n . classes of F1. In the process, Norton computed the values Q<'A, CA(Q) (z), and A/Qf-Q (2n,2) (nonsplit 4). jugacy Now let 2n = 24, q:A -* Q a set map giving an "isometry" of a hypothetical character X of degree 196,883 and found that Q/(z). Take C,. ' A with Q ' CX and 1. Let (S2y, 1) = (S2X, X) = (S3y, 1) = 1, where SnX denotes the char- A/2A Cd/Q-* acter of the nth symmetric tensor power of a module affording C be the covering group of C,.. We have a commutative diagram X This means that a module affording x is self dual and has an C- C,. F1-invariant commutative algebra structure with an associative form (equivalently, an invariant cubic form). To construct F1, one might attempt to take a Q-vector space M of dimension 196,883 and describe a set of linear transfor- and Z(C)Z2 X Z2 Let QC Q- Q, and (2) = Z(M). An mations on M, then prove that they generate a simple group of irreducible module for C' faithful on (2) looks like TO9 M, M an the required description. The number of possibilities to con- irreducible for '0. If M is not faithful, C induces C,. on TO M. sider in this approach seems too great. By requiring that M have If M is faithful, c induces a group, C, on T 09 M; C % C,.. This an invariant algebra structure, we make the mathematics rigid is a construction of the group C. enough to analyze the possibilities. Now to describe the algebra. We define a QC-module B = U V (D W with a C-invariant positive definite bilinear form Outline of the construction (,),where U S2(Q 0 A) has dimension 300, V is a module We must review facts about A, the Leech lattice (1), and ex- z traspecial groups (4) and establish some notation before de- of dimension 98,280 induced from a subgroup of C containing scribing our construction. Q and corresponding to the image of *2 in C/Q 1 (here z acts trivially and C/(z) acts as a monomial group with Q being represented by 98,280 distinct linear characters), and W T The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertise- ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. t Reprints should be mailed to the University of Michigan address. 689 Downloaded by guest on September 24, 2021 690 Mathematics: Griess, Jr. Proc. Natl. Acad. Sci. USA 78 (1981) Table 1. The algebra product u' E U v(A), A A2 e(x) xi, x EF.iE u E U * -94(uuo(A2))v(A) e(x) 0 p(u,xi)t u(A) -9/4uo(A2) [-38v + qA2(UO(Uii), u(A2))]pA(x)e(xxA) 0 xi jen (i), -k(A,,I)v(A8 + u)t -18&6y uo(uv) + e(y)®x; - - E [-3Av +9A32(uo(uV),u(A2))] pA(x)v(A) AEA2 XA XY The inner product: (uiiu.#) = 48V, (uiiujk) = 0 forj # k. (uU,Ukl) = 28{ij1,({kj} for i # j, k # 1. 0 = (UV) = (VW) = (U,W). (v(A), v(y)) = 8k,a, (e(x) 0 xi, e(x) (0 xj) = SxySV. * The product on U: ui = -253 uii + 11 E ui,,ui u, = joi uiiujj = ll(uii + U#) - E Ukk, UiiUjk = l2Ujk koij UyUjk = -72Uik, u = -66(uii + uj) + 6 I Ukk. k~iij U#Ukl = 0 for i,j, k, 1 distinct t p(ujj,xj) = -69xi; p(uy,xi) = -36xj i 0j; p(u#,xd) = 3xi, i # j; p(uY,xk) = 0 i #j k # i. * is normalized so that (3kA,,u) = 0 or ±36 and P(A,kI) = -36 when {A,M,A + p4 is a triangle of type 222 with each leg in A4 (replace A by -A if necessary to make A + ,u E A2). represented by 98,280 distinct linear characters), and W T of a is critical and amounts to describing the signs. This is not 0 A is a faithful module of dimension 98,304 = 212.24. forced, by any means, and requires some guesswork. Of course, z we want oc to preserve the algebra product (not just be an or- Note that dim B = 196,884, one more dimension than in- thogonal transformation). dicated in the Abstract. We keep the extra dimension for con- We use the refinement B = B24 + B276 + B' + + B4' + are = venience. Both V and W C-irreducible and U Qd ED U0, B2 + B2 + Beven + Bdd (described later) in our description of where d is an invariant quadratic form on Q 0 A and U0 is the o. To prove that (xy)a = xoyo for all x, y E B, it suffices to let z orthogonal complement (and is irreducible). x, y range over a set of basis elements. Since B has eight sum- mands, there are evidently (9) = 36 cases. However, certain Because we are interested in an algebra structure on B with pairs of cases are equivalent under the action of cr or associa- F, in G(B): = {g E GL(B)|g preserves (, ) and the algebra struc- tivity of the form, so that fewer than 36 cases require detailed ture i.e., (ab)r = OM, for all a, b E B}, we consider the possible C-invariant algebra structures.