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. To cut down the possibilities, analysis. After writing (xy) and xoyv in terms of basis elements, we find that equality of corresponding coefficients is equiva- we require that dim B2 = 196,883. By studying C-invariant lin- lent, in each case, to some identity involving configurations of ear maps S2B -) B with image in the C-submodule U0 + V + vectors in A; these are all verified. As one example, let 0 be W of codimension 1, we find that the set of C-invariant com- an octad, J a subset of 0, even, v E A, A= mutative products on B for which the form is associative [i.e., IJI =Ii)=j 4xi and E then (ab,c) = (a,bc) for all a, b, c E B] is described by six rational iEa 2xi, X F(2); parameters. fPA+,,,(TX)( 1)12(v,)+1/21J1 8(- 1)1/2J(sXfnc)+JI We choose a set of parameters, then proceed to describe o, AE;A2,supp(/)=o E G(B) - C. These choices are given in Table 1, and a descrip- in this article. tion of of is given in Table 2. The choice of parameters is mo- This notation is explained later we we = This turns out tivated by imagining that F, is really in G(B), then deriving con- Once get a E G(B), define G: (C,o). it is not sequences in the linear algebra from this. In particular, we do to be the simple group of the title. However, obviously the following. Let N be the normalizer in F1 of (z, z)Z2 finite and it is not clear that the inequality CG(z) 2 C is an equality. We solve this problem by reduction modulo p, where X Z2, where z1 = q(8xx.). Then N has shape (22.211.211 X 2"1). p E p, an infinite set of primes. Using the basis for B indicated (M24 X 3), and N n C has shape (22.211.211 X 211)(M24 X Z2). in Table 1, all structure constants lie in Z['/6] and all matrix Also, N" = N"' has shape (22.211.211 X 211).M24, and N is a sem- entries for elements of G lie in we take = idirect product N"(T,o), where Ioj = 1il = 2 and (r,a) - and Z['/2]. So, may p {pl p is a p - 5}. We use the suffix to indicate reduction T= q(A,), A. = -3x. + X x E A. Since N" < C, our es- prime, (p) tablished notation is directly useful for N" but only indirectly so for oc. If H1 and H2 are subgroups of C that satisfy H' = H2, Table 2. Definition of o- then the set of irreducible constituents for H1 on B is trans- B24 i= Uii formed by cr to those for H2 on B. By making useful choices of B276 u?=v=ij + vij {H1,H2} among the subgroups of N", we get more and more B 4,+ (vv ++vV) = vy + VY- detailed information about the linear transformation o. Even- B42fi (-v0 + vof=uu tually, we find two bases, el, e2, ... and ej, e', ... so that el B2 v(Ak)°= 'A 2a(,A)v(),: a(;,A)= +1 B2 4CA2,Supp SUpp A =±+ej, e2 =±+e2,. Even though e, +e may be imprecise, V(Ai (_ )(A-,AiS) e(,rxs) (0 x, (e2) = (e:)2 is precise. In this manner, we obtain five linearly B23 S7= 0 = in 0 independent relations among the six parameters and choose a Beven (e(x) xi)a (1),i Sx e(x) xi, 0 = ( nonzero set of six satisfying the relations. An exact description Botd (e(rxs) xi)' 1)(AOAiS) v(Ai s). Downloaded by guest on September 24, 2021 Mathematics: Griess, Jr. Proc. Natl. Acad. Sci. USA 78 (1981) 691 modulo p. The image G(p) of G is finite since it is a subgroup Uij, basis element for U corresponding to xixj. of GL(196,884,Fp). Since cG(p)(z(p)) is finite, techniques from uo, the orthogonal projection U -* UO. the classification theory of finite simple groups, mainly Gold- A(2k), {A E A|all coordinates of A are in 2kZ}, k 0. schmidt's classification of finite groups with a strongly closed An, {A E A|(A,A) = 2n}, n 2 0. abelian 2-subgroup (5), may be invoked to prove that An, AJ{±1}. cG(P)(z(p)) = C(p). Since C(p) C, work of Smith (6) implies A2 = A2 U A2 U A2, where that either G(p) is a simple group of order 2463205976112 A4 = all vectors of shape (42Q22) 13317.19.23.29.31.41.47.59.71 (using a group order calculation Ai = all vectors of shape (28016) of mine or else G(p) = C(G(p))C(p). Properties of o eliminate A3 = all vectors of shape (3 12). this last alternative. Since the reduction modulo p of G has the E0, E = (z) X EO; we arrange q(A(4)) = E0. same cardinality for all p E p and p is infinite, we get G -G(p) (PA, linear character of Q corresponding to q(A). for all p E p. v(A), basis vector of V corresponding to A E A2(v(A) = v(-A)). Consequences vij, v(4xi + 4x,). For a recent survey article on the classification of finite simple vp,, v(4x, - 4xj). groups, which contains a discussion of the known simple XA, unique element of F with CE(xA) = E n ker (p. groups, see ref. 7. There are elements Xk E C of orders k = 1, Ai = -3xi + Y>,i Xj E A3,A3k = Ai with the signs changed 2, 3, 5 such that Fk: = CC(xk)/(xk) is a "new" simple group: at the (-set S; AiAis E A2 F. G. F(2), a hyperplane of F satisfying E.F(2) = (q(A(2))); there F2: Fischer's {3,4}-transposition group, constructed by Leon is a natural isomorphism F(2) -'/(Q ), Xs ++ S, X and Sims in 1976; order 241313567211. 13.17.19.23.31.47. Sx. F3: Thompson's group (8), constructed by Thompson and P. B2A = span of all uii. Smith; order 215310537213.19.31. B276 = span of all u,-,, i # j. F5: Harada's group (9), constructed by Harada and Norton B2 = span of all v.. + vy'. (3); order 2143 567.11.19. B2 = span of all vij - vy.. More precisely, F2 was discovered in 1973 before F1 was dis- B2 = span of all v(A), A E A2. covered, whereas F3 and F5 were discovered as components in B3 = span of all v(A), A E A . F1 shortly after F1 was discovered. Beven = span of all e(x) O xi, x E F(2), i E Q. One can do this for other elements X7 (order 7) and X3* (order Bodd = span of all e(x) 0 xi, x E F - F(2), i E fl. 3) in C and get simple groups (these were not "new" in 1973): a(;,A) =aM (JA) = (-1)iin S+oj in T+I/2(Ay.) (P~X )AAIW- F7: Held's group, constructed by Higman and McKay in 36, where ku= Ais, v = AjT and A = ±+ +iv; this is 1968; order 21033527317. defined when A, ; E A2, supp (;) = supp (A). F3*: Fischer's 3-transposition group F' (10, 11), constructed by Fischer in 1969; order 221316527311. 13.17.23.29. Given F24 NG((x3*))/(x3*), we also get existence for Fz and This research was supported in part by National Science Foundation The constructions of F2, F3, F5, and Held referred to re- Grant MCS-77-18723 (02). I thank the University of Michigan for partial F22. financial support during my sabbatical year 1979-1980 and the Institute quired computer work. Since X3* may be chosen so that for Advanced Study for extending me the privileges of membership (X7,X3*) is a certain Frobenius group of order 21, we get an during the time this work was carried out. embedding of Held into F' , a fact proved by Norton in 1977. By Lagrange's theorem, the sporadic groups LyS (12) andhJ 1. Conway J. (1971) in Finite Simple Groups, ed. Powell, M. B. & (13) are not involved in F1 (37,67 E ir(LyS) - ir(FI) and 37,43 Higman, G. (Academic, London). E - 2. Griess, R. L., Jr. (1976) in Proceedings ofthe Conference on Finite (J4) wr(F1)). It is clear that 20 of the 26 sporadic simple Groups (Academic, New York). groups are involved in Fl. We have made progress settling the 3. Norton, S. P. (1975) Dissertation (Univ. Cambridge, London). four last cases. 4. Gorenstein, D. (1968) Finite Groups (Harper & Row, New York). In addition, we get a number of results in group extension 5. Goldschmidt D. (1974) Ann. Math. 99, 70-117. theory. The hard part of calculating degree 2 cohomology of a 6. Smith, S. D. (1979) J. Algebra 58, 251-281. finite group is often establishing the lower bound. Hence, find- 7. Gorenstein, D. (1979) Bull. Am. Math. Soc. 1, 43-199. 8. Thompson, J. (1974) in Finite Groups, Sapporo and Kyoto, ed. ing a nonsplit extension "in nature" is of some interest. Among Iwahori, N. (Japan Society for the Promotion of Science, Tokyo, the subgroups of F are nonsplit extensions Japan). 38.fQ(8,3) and 3 .0-(8,3), 9. Harada, K. (1976) in Proceedings of the Conference on Finite 2.F2, Groups (Academic, New York). (2 X 2).2E6(2) (perfect) [shown to exist by Griess (14)], 10. Fischer, B. (1970) University of Warwick preprint. 3.F' (shown to exist by Norton in 1977), 11. Fischer, B. (1971) Invent. Math. 13, 232-246. and others. In particular, the Schur multiplier situation for the 12. Lyons, R. N. (1972) . Algebra 20, 540-569. 13. Janko, Z. (1976) J. Algebra 42, 564-596. known finite simple groups is now completely settled (15). 14. Griess, R. L., Jr. Trans. Am. Math. Soc. 183, 355-421. We require some additional notation before presenting Ta- 15. Griess, R. L., Jr. in The Santa Cruz Conference on Finite Groups bles 1 and 2: (Am. Math. Soc., Providence, RI). Downloaded by guest on September 24, 2021