Invent. math. 79, 513-540 (1985) Inventiones mathematicae Springer-Verlag 1985

A simple construction for the Fischer-Griess monster group

J.H. Conway Department of Mathematics, 16 Mill Lane, Cambridge, CB2 1SB/UK

To one who will understand. 8 July 1984

Contents w Introduction ...... 513 w The Golay code c~ and cocode c~, ...... 515 w Parker's Moufang loop P ...... 516 w The standard automorphisms of P ...... 516 w Some notational conventions ...... 517 w Triples and triple maps. Definitions of various groups ...... 517 w The relation with the Leech lattice A ...... 519 w8. Monomial representations of N, and Nxl ...... 521 w The construction of Gxl ...... 523 w10. The representations 196884x, 196884y, 196884~ ...... 524 w11. The dictionary, and the group G O ...... 525 w12. The algebra ...... 526 w13. The finiteness of G o. Identifying G O ...... 527 w14. Axial vectors ...... 528 w15. Transpositions explicitly computed from their axes ...... 529 w16. Norton's inequality ...... 530 w17. Subalgebras and their units ...... 531 Appendix 1. Some remarks about extraspecial groups ...... 533 Appendix 2. The dictionary defines isometries ...... 534 Appendix 3. The dictionary is invariant under Nxy ~ ...... 534 Appendix 4. The complete symmetry of the algebra ...... 535 Appendix 5. The algebra is the same in all three languages ...... 535 Appendix 6. The eigenvalues of v~v t for a particular t ...... 539 Appendix 7. The mean square eigenvalue of various maps ...... 539 Appendix 8. Some information about characters of the Monster ...... 540

Note. The construction is contained in w167 except that various tedious calculations are relegated to the Appendices. We recommend the reader merely to sample these calculations - a nice example is Calculation (1) of Appendix 5.

w 1. Introduction

In his monumental paper, "The Friendly Giant" I-1], R.L. Griess has con- structed the finite simple group variously known as the Fischer-Griess group, 514 J.H. Conway the Friendly Giant, or the Monster, and denoted by the symbols FG, F1, or M. The main aim of this paper is to present a simplified construction. We shall first briefly describe Griess's construction, and then compare it with our own. Griess constructs the group in a particular representation as a group of automorphisms of a certain algebra in Euclidean 196884-space. He defines the group as the compositum of two particular subgroups, which are in fact the centraliser C of a suitable involution, and the normaliser N of a certain fourgroup containing that involution. After a detailed study of the restrictions of the representation to these groups and their intersection (which has index 3 in N), he is able to construct the group C as a fairly explicit group of matrices, and then to specify the action of a particular involution a in N but not in C. There are a number of extremely subtle problems in determining the signs of various matrix entries, and Griess solves these partly by calculation and partly by inspired guesswork. The group so defined might well have been infinite. In order to apply the techniques of finite group theory, Griess therefore considers instead the group obtained by reducing the generating matrices modulo a suitable prime p, and is able to show that for infinitely many p the finite group so obtained actually does have C for its involution centraliser, and so by using a theorem of S. Smith, must be the desired Monster group. This result can then be transfer- red to the original group. We enumerate the main differences between our paper and Griess's. Using a remarkable Moufang loop discovered by R.A. Parker [2], we are able to give an extremely simple and explicit definition for the group we call N, which is a fourfold cover of Griess's N. We therefore work from N to C, rather than C to N. Certain complicated double-cover constructions are no longer needed, and the sign problems no longer require solution, since they have vanished into the definition of Parker's loop. We also obtain simpler structure constants by working with a modified form of the algebra in which the fixed vector receives a more natural treatment. Often a formula in [1] consists of a fairly simple expression defining some vector, multiplied by a more or less complicated sign factor. We can summar- ise the improvements so far by saying that our analogue will usually be obtained by omitting the sign factor and slightly simplifying the remaining expression. The construction from our N has a threefold symmetry, which we exhibit by using three "languages", any one of which may be used to name any given object. The core of our proof is then the verification that the results of various calculations are independent of the language in which we perform them. This is fundamentally similar to Griess's verification that his element a preserves the algebra, but is made easier by the greater simplicity and symmetry of the formulae. It has become easy to "play around" in the algebra. We shall define certain transposition vectors, and describe the subalgebras generated by any two of them. This leads to a simple proof that the group is finite, since we can show that there are only finitely many transposition vectors. Our results suggested the conjecture, since proved by S.P. Norton, that the annihilator of any A simple construction for the Fischer-Griess monster group 515 subalgebra is again a subalgebra. We include Norton's results, with his permis- sion, and derive some interesting consequences. In a certain sense, the algebra is "more than 99 ~o associative", and many theorem that hold for associative algebras can be extended to it. It is by no means obvious a priori that the group N obtained from Parker's loop has the correct structure, but once this is assumed, the remaining steps are forced, and so our paper might easily have been written without reference to Griess's. In fact when any difficulty arose during the working out of our construction, we freely used Griess's paper to see what the answer should be, and it was reassuring to check at every stage that our formulae were essentially translations of Griess's. Griess, of course, had no such recourse. The work [-4] of J.Tits appeared after this paper was written. Tits avoids the need for explicit consideration of the sign problems by a more abstract discussion of the underlying representations. He also has a very elegant proof of finiteness. In a sense, Tits' improvements are orthogonal to ours. He wishes to avoid all calculations in the Monster, while we would like to make it easy for the reader to perform such calculations for himself. w2. The Golay code ~ and cocode eft,

Let ~ be a set of size 24, and construct the vector space IF224 as the cartesian product 1-I IF2. Then the Golay code cs is a 12-dimensional subspace of IF~ 4 iEfk whose smallest non-zero weight is 8. This characterises the code up to per- mutation of the base-set 4, and it is known to have weight-distribution 01 8759 122s76 16759 241"

In group-theoretical papers it is customary to identify subsets of ~ with the vectors that are their characteristic functions, so that cg becomes a subspace of the power-set P(~) under symmetric difference. The elements of this space are then called Cg-sets, and the terms octad and dodecad are used for Cg-sets of cardinal 8 and 12 respectively. Let us call two subsets of ~2 congruent if their symmetric difference is a ~- set. This gives us the Golay cocode ~f* as the quotient space P((2)/~. By a slight abuse of language the elements of c~, are called cg*-sets, and the weight of a cg*-set is the weight of its smallest representative. Then cg, has weight-distribu- tion 01124 2276 3202441771 '

and the only congruences between sets of size at most 4 are that each tetrad (set of size exactly 4) is one of a family of six mutually disjoint and congruent ones (a sextet). The group of all permutations of O that preserve ~f (and therefore also act on off,) is the Mathieu group ME4. We presume some familiarity with M24 and the Golay code (see [3]). 516 J.H. Conway w3. Parker's Moufang loop P

In [2], a remarkable Moufang loop P is defined, and shown to simplify the construction of the Fischer group Fi24. We summarise the main properties of P for the reader who is unable to consult [2]. The operation in P is written multiplicatively, and the loop inverse of a is written a. There is a centre {_ 1} of order 2, modulo which P becomes isomorphic to the Golay code cg. We shall write d-+d for the corresponding projection map: p+cg. Every element squares to + 1, each pair of elements either commutes or anticommutes, and every triple of elements either associates or antiassociates. Using [d, e] for the commutator and [d, e, f] for the associator, we have

d 2 =(-- 1)~lal =(- 1)ran) [d, e] =(- 1)~la~t =(- 1) n(e'e) (1) [d, e, f] = ( - 1)la~il = ( _ 1)N(a,e, y), where N(d), N(d, e), N(d, e,f) are the respective numbers of dodecads in the lists

We can see that these equations provide an abstract definition for P. For let us call the sequence dl ..... d,2 a base for P if the image sequence d~ .... ,d~2 is a base for g. Then every element of P can be represented (in many ways) as an iterated product F(d a .... , d12), and all we need to know is just when two such products F(d 1 .... ,d12 ) and G{dl ..... dj2 ) have the same value in P. For this to hold, it must certainly be true that when the signs are ignored (so that the calculation is performed in g), the resulting Cg-sets must coincide. But if this holds, then we could prove the two products equal if we were allowed to use the associative, commutative, and square laws, since these are just the laws defining elementary abelian 2-groups. By following the steps of such a 'proof' and inserting the appropriate factors from (1), we obtain a proof in P that F(dl, ..., d,2)= +_G(d, ..... d,2 ) with a known sign. w 4. The standard automorphisms of P

Let d 1 .... ,d12 be a base for P, and ~ any element of M24. Then 72 takes d 1 ..... d12 to another base for oK, say el ..... e12. If e 1.... ,e12 are any pre- images for ~1, ..., ~12 in P, then there is a unique automorphism rt of P for which zt: d~--,e i (i= 1.... , t2).

We call these the standard automorphisms of P. There are 212 standard auto- morphisms mapping to each element of M2,. A standard automorphism is called even or odd according as it fixes or interchanges the two preimages (which we call g2 and -O) of the universal Cg-set ~. A simple construction for the Fischer-Griess monster group 517

The standard automorphisms mapping to the identity element of M24 are indexed by the cg*-sets, and are given the same names. Indeed, let 6 be any subset of t). Then the map x~x~=( - 1)lx~lx is such an automorphism, which is the identity map whenever 6 is a Cg-set. The map x~x ~ is therefore defined even if 6 is only regarded as a ~*-set. We call these the diagonal automorphisms of P. w 5. Some notational conventions

From now on we shall not use separate notations for elements of ~ itself, but understand that if F is a function normally defined on sets, then F(d, e,f, ...) will mean F(d, ~,f, ...). Thus

"subset of f2" means "subset of f2" "i~d" means "/ca" "(- 1) means "(- 1) ~la~l''.

However, it will be important to know when a symbol denotes an element of P (or, loosely, of cg), and when an element of cg,. Since it would be cumbersome to indicate this explicitly in the notation in all cases, we merely hint at it by the following conventions: a,b, c,d, e,f,.., denote elements of P, or, loosely, of c4, 6, ~, ~o.... denote elements of e4*, i,j, k .... denote elements of t2, often regarded as particular cg*-sets, ij and dc~e denote the sets {i,j} and dc~, regarded as cg*-sets.

It is convenient to use the term preset for the typical element of P, and the terms preoctad, predodecad for the preimages in P of octads and dodecads. w 6. Triples and triple maps. Definitions of various groups

We shall enlarge P to Po=Pw{O}, by adjoining a formal zero satisfying Od =dO=O0=O. Then a triple is an ordered triple (a,b,c) of elements of Po. A triple map is a function from triples to triples. The following triple maps (specified by the image of (a, b, c)) generate an important group we call N:

x a" (dad, db, cd) Yd: (ad, dbd, dc) z d" (da, bd, dcd--) x~ = y~ = z~ : (a ~, b ~, c ~) (n even) x~ : (~, U, 6 ~) y,: (U, b", if") z~: (b~, ~, U) (re odd),

where d e P and ~z is a standard automorphism of P. It is usually sufficient to compute the effect of an element of N on just two of a, b, c, since the above generators take triples satisfying abc = 1 to other triples of the same type. 518 J.H. Conway

The particular elements

o,~1 = y f~ 7, _ ~ ff{'Z = Z ~ X _ f~ o,~{'3 = x o y _ o are called the kernel elements, and we define

Kx={1,~} K2={1,~2} K3={1,~3} and Ko={1,~,Jgz,~3}.

It turns out that the quotient group N o= N/K o is the normaliser of a certain fourgroup in the Monster, and that its fourfold cover N has a simpler represen- tation theory that considerably simplifies the construction. For various other groups F we shall use the convention that F, denotes F/K,, (n = 0, 1, 2, 3).

Gxl Gx2 Gz3

(fourfoLd covers) (doubLe covers ) I

Ny2 Nz3 kiLL K 2 k~LLK 3

Qx"1/ \ z Qxl Qxyz kiLL kilL kill K 2 , K3 K1,K 3 K1,K 2

(subgroups) ~0 ~ The Monster

Gx ~ Gyo Gz0 ~ 21.2/, . Co 1

22.211 " 2 22 ' S3xM2z'

2 1§ .211 .M2/* Structures '~ xO\ /~.~I- zO 2 2 .211 .2 22. M2t.

22 .211 .222.$3

21.2/*

22" 211 QxyzO Fig. l. Some subgroups of F1, and double and fourfold covers A simple construction for the Fischer-Griess monster group 519

The structure of N can be read from

N= (22 • 22) 211 222 (S 3 • M24 ) .

,~'1, o~z"2,,~3 X- l,y I,Z-I x6=yo~z6 XdlYd,Zd X6,y6,ZO X~=yzr=Z~ even 6 odd rc even By omitting the M24 generators we obtain a group Q; by omitting instead the S 3 generators a group N~:, and by retaining only the (22• 22) 211 generators a group Qx:. We shall usually abbreviate the elements x ~,y_~, z I to x, y, z. The centraliser of x in N is called Nx, and has structure

Nx= 22 . 21+ 24 . 2il . M24.

agh ~2, ~3 Xa,xe ya,za x~

By omitting the 211 and M24 generators we obtain a group Qx, and from Nx, Qx we get groups Ny, Qy and N~, Qz by applying the triality symmetry which is obtained by performing the cyclic permutations (xyz), (123), (abc) on all our notations. In future we shall often make just one of three definitions, and suppose that the reader will obtain the others by triality. Our plan is this (see Fig. l). It will turn out that N o and Nxy~o are respectively the normaliser and centraliser of {1, x, y, z} in the Monster. The groups Nxo, Nyo, N~o are the centralisers in N o of the three involutions x, y, z. Their Monster centralisers are larger groups G~o, Gy o, G~o which have double covers Gxl, Gr 2, G~3 whose representation theory is slightly simpler. We con- struct the Monster as the compositum G o of groups G~o, Gro, Gzo acting on an explicitly defined 196884-dimensional space. w 7. The relation with the Leech lattice A

Theorem 1. The elements x d, x~ satisfy

x2=l=[xi, x~]; 1-[xi=l if 6~cg; iE6 [xa, xo] =(-- 1) ra~ol++jal'l~l x 2 = xt-lal, [xa, Xe] = X l_ n X_d=XXd~ XdXe~-Z ~[d e[.Xde. Xdc~e.

The group Qx they generate contains K 1 and has order at most 226. Proof. It will suffice to verify the last relation, which is the hardest. We have

xaxe: (a, b, c)~(~.dad.~, e.db, cd.e) (1) Xae: (a, b, c)o(ed.a.gd, de.b, c.de). (2)

Now since z: (a, b, c)-,(-a, -b, c) and [d, e] =(- 1)~la~el we have z ~la~eb .Xa~: (a, b, c)~(gd.a, dg, ed.b, c. de) 520 J.H. Conway which needs only some re-association to become (1). But the diagonal auto- morphism d ne actually re-associates any three-term product two of whose terms are d and e. For example, since ldnel is even, we have

(C. de) ane = Cane. de = ( - 1) bc~a~el, c. de = cd. e. Accordingly z ~lanel. xde. x~o e : (a, b, c)~(?, e. db, cd. e), agreeing with (1). [Since triples with abc= 1 are taken to other such triples, we need only compute effects on two elements of a generic triple.] Now ~=yoz~, which using the relation xoy~zo=l we can express as x~z_ 1 =x~z. So indeed of1 s Qx. Modulo x and ~1, the relations indicate that the group becomes abelian, with the x i and xo generating a subgroup S of order at most 213. Then modulo S the last two relations become X d=Xd=Xod and XdXe=.~de, showing that S has at most 211 cosets. []

Theorem2. The group Q,I is extraspecial of type 21+24--+ with a homomorphism x,~). r onto A/2A, where A= A24 is the Leech lattice, such that

,L= I_J_ eg (-- 3oni, l~ls~wh~re)

2 d--l/~_1 (2o.d, 0ol~who~o) [Xr, Xs"] ___x(2r, 2s) X2 = xtype (2r).

Note. The Leech lattice may be defined as the integral lattice spanned by the vectors 2i and 22 of the statement, the coordinates being indexed by the members 1 of f2. From now on we shall omit the factors ~-~, although their presence is always to be understood, so that the inner product (v, w) of the vectors with coordinates (vi) and (w~) is ~Y, viw i. The type of a vector v is half of its inner product with itself, namely ~ S v 2. Proof. It is a trivial matter to check the images of the relations of Theorem 1. We consider only the last relation, and name some useful vectors on the way. Working in A/2A, we find:

).~ = 2~- 2 2 i = (8o~ i, 0~l~wh~r

2f~ "[- 2i j = ( 4on i,j, 0else ) and so for any even subset ~ of f2: 0o,~e). Now if 6z~r then is even and the last displayed vector is in 2A, whence 2~ =0. So for arbitrary 6, 2~ is well defined (as an element of A/2A) even if 6 is only specified as a ~'*-set. A simple construction for the Fischer-Griess monster group 521

Finally 2a+2r 2ona~, 0~1~) and this can be written as

2o + )~ae+ 2a~e in agreement with the last relation of Theorem 1, which in Qxl reads

Xd. Xe = (Xo)~la~el Xa ~ . Xa~e" []

Since we shall use them in many places, we shall give the subscripts r that appear on our x's a life of their own by defining x,.s=x,.x~, etc. If 2, is congruent modulo 2A to a type 2 (minimal) vector of A, we call r and 2, and x~ short. We define x~+ and x i to be xQ., and x_~.~ when these are short, and otherwise xr, and will employ a similar notation in other contexts, for example 5~+, etc. The short elements of A are easily seen to be

21j: (4o, i, -4o, s, 0else ) 2+: (4o, i, ~, 0~1~) 2d+a ' (--2o.~, 2o, a\a, 0~s~), where Id]=8, and 6 is an even subset of d 2~. i" (T-3o, i, -t- leith), where the lower sign is taken on d. w 8. Monomial representations of N x and Nx~

In the next three sections we shall produce monomial representations (with monomial factors _+ 1) of degrees 98280, 4096, and 24 for Nx. In each case K~ will be in the kernel, so that we can regard these as representations of N~t in the usual way. Observe that to produce such a representation of degree n it suffices to produce a permutation representation on 2n objects for which there is a partition into pairs preserved by the action of the group. The correspond- ing space will be generated by 2n vectors vr of the same norm, which are mutually orthogonal except that vr= - vS when r, s is a pair.

The representations 98280x, 98280y, 98280 z The group Qxl has 2 x98280 short elements x r, corresponding to the 98280 short elements 2 r of A/2A. These 2 • 98280 objects are permuted by Nxl, and so by N x, in such a way that the pairing xr, x r is preserved. Accordingly these groups have a monomial representation 98280~ of degree 98280 on a space spanned by vectors X r with X r=-X~, and permuted like the x r under conjugation. The kernel, K(98280~), can be checked to be the group of order 8 generated by K o and x. We shall later embed Nxl in a larger group G~ in such a way that Q~I is still normal and the short x r are still permuted by Gx~. The representation 98280~ will therefore extend to this group. By applying triality we obtain corresponding representations 98280y of Ny 2, Gy 2 and 98280~ of Nz3, Gz3. 522 J.H. Conway

The representations 4096x, 4096y, 4096= We define d7 = {(0, d, 0), (0,-fad, 0)} d~ = {(O, O, d), (O, O, f2d)} d; = {(0, 0, d), (0, 0, - Cad)} d~ = {(d, 0, 0), (fad, 0, 0)} d3- = {(d, 0, 0), (- fad, 0, 0)} d~- = {(0, d, 0), (0, fad, 0)}.

Then there are 2 x 4096 distinct objects d~, which are permuted by N~, and indeed by N~I since J{'l fixes each of these objects. There is also an invariant pairing, obtained by changing the sign of d. It follows that there is a monomial representation 4096~ of Nx, or N~ on a space spanned by 2 x 4096 unit vectors d~ and d~- which satisfy

(-d)i-=-(dT), (-d)~-=-(d~-) and (-fad)7=d~, (Cad)~-=d~- but are otherwise mutually orthogonal. The kernel K(4096x) of this representa- tion is the fourgroup {t, Yo, z-a, ~ The elements J:2 and ~{'3 both act as -1. Applying triality we obtain representations 4096r and 4096~ of Ny 3 or Ny and N,3 or N~ on spaces spanned by vectors d~-,+ dg.+

The representations 24x, 24r, 24=

We define, for each iefa, sets i I and -ix, ix={(d,O,O)liCd}, -i~={(d,O,O)li~d},

and +i2, +i 3 are defined similarly using triples (0, d, 0), (0, 0, d). It is easy to check that N~ permutes the sets i~ and -i I as follows:

yd, Za: il--*(--1)la~il il, xn: il--~(i~)1,

Table 1. Action of the generators of N on basis vectors. On replacing X. Y~,Z~ by x,, yr, Zr we obtain the action by conjugation on Qxo

g Action of g on the vectors

il dl d~{ Xd. ~ Xd.a (c5 even) (6 odd)

x e i 1 (ed)7 (de)~ X~_I).,a. a or XI_ 1)~a~. a Ye (- 1)~il (~de-)'f (ed)~i X{_ ~).a.aa. or X~_m~ea. a ze (- 1)ui, (de)~ (~de-) + Xn, a.6a, or X~ae. ~

x,=y,= G il (d') 7 (d~)+ Xa~. a (e even)

y~ i 3 (d~)~ (d~)3 Za,. o (e odd)

x.=y.= G (i")1 (d~)7 (d~)~- Xd~.~ (n even)

Notes: u=line[, n= m=16nel, 6'=dine. A simple construction for the Fischer-Griess monster group 523 so demonstrating that a quotient group 2 j 2:M24 acts faithfully and the kernel K(24~) is exactly Q~. The corresponding 2 x 24 vectors, which we give the same names +i t, are permuted like the vectors

( +__ lon I, Oelsewhere) of the Leech lattice space, and so we declare that they have norm ~ (not 1) and are mutually orthogonal apart from the relations -(il)=(-i)1. Again +~2 and ~f'3 both act as -1, since they reduce to y~-za modulo Qx. Applying triality, we obtain representations 24y and 24 z of My2 or Ny, and Nz3 or N~. It should now be clear that the various groups defined in Sect. 6 do indeed have the orders and structures ascribed to them there. Table 1 gives the action of the generators for N on the basis vectors for each of the representations 24~, 4096~, 98280~. w9. The construction of G~

In Fig. 2 we illustrate various quotients of N~. The orders and generators for kernels are marked against the arrows.

]~(2)

N(4o96~) N(2%)

Fig. 2 N*

N(4096x) and N(24x) are the groups of matrices representing N~. The kernels K(4096x) and K(24~) intersect in K 1, which yields the quotient Nxl. Their compositum (K(4096x), K(24~)) yields the quotient onto a group N* of struc- ture 211 : M2 4 =212 : Mzj(y~)" It follows that the group N~I is abstractly identical to the diagonal product N(4096~)AnzN(24~). [When two groups H 1 and H 2 are equipped with homo- morphisms q~l and q32 onto a third group H3, the corresponding diagonal product, or pullback, is the subgroup of their cartesian product consisting of all (ht, h2) with ~o(h0=~o(h2).] Now the two factors in this diagonal product are abstractly identical to the groups of matrices arising in the representations 4096x and 24x. This obser- vation enables us to extend N~I to a larger group G~I, as follows. 524 J.H. Conway

It follows from the theory of extraspecial groups (see Appendix 1) that the largest group of real matrices of determinant +1 that contains Q:`I (as repre- sented in 4096~) normally is the natural double cover H(4096x) of Aut(2[+24), with structure 21 § 24. SO~-4(2)" Now the central quotient of Q~a is isomorphic to A/2A, and the elements of N(4096x) induce the same automorphisms on this as the monomial symmetries of the Leech lattice. So there is a chain of inclusions, say Q~a ~ N(4096~)_~ G(4096,) ~ H(4096~) of matrix groups with structures 21+24 21+24.21t. M24 21+24. COl 21+ 24. S0~-4(2)"

Again, our group N(24~) = 212 : M24 participates in a chain

E c N (24x) ~ G (24x) ~_ H (24:,), say, of matrix groups with structures

212 212:M24 2. Co i S024(]R ).

Moreover, the groups G(4096x) and G(24~) that we have just defined possess natural homomorphisms onto Co I that extend the homomorphisms from N(4096~) and N(24x) onto Ns xl :M24. We can therefore extend the diago- nal product N~I of the latter two groups to the diagonal product

Gxl = G:,(4096:,)Aco 1G(24x).

This group has representations of degrees 4096 and 24 extending those of Nx (map through the left and right factors), and since Q~I is normal and G~I still permutes the short x,, also has a representation extending 98280~. We shall use the same names for these extended representations, and understand that triality is applied to produce similar groups Gy2, Gz3 with similar representations. w 10. The representations 196884x, 196884y, 196884 z

Our group Gxl is actually the double cover of an involution centraliser in the Fischer-Griess Monster. To approach the latter we shall need representations for which ~2 and 3f3 are also in the kernel. We define the representation

196884 x = 300x + 98280~ + 98304~ of G~I to be the direct sum of its representations 300x, 98280~, 98304:`, where

300 x is the symmetric tensor square of 24x 98280~ has already been defined, and 98304~ is the tensor product 4096~ | 24~.

In general, we shall use the same names for representations as for the spaces they act upon. A simple construction for the Fischer-Griess monster group 525

When vectors v and w of 24~ are given by coordinates, vectors in 300~ are naturally represented by matrices - the matrix for v| has i,j entry ~vivj, while that for v|174 has i,j entry ~(v~wj+wzvj). [The factor -~ corre- 1 sponds to the understood factor ~-~ for Leech lattice vectors.]

The basis vectors we shall use for 196884~ are for 300~: the matrices (ii)l and (ij)~ (i 4=j), of norms 1 and 2 respectively for 98280~ : the vectors X, (r short), of norm 1 for 98304~: the vectors d e | i (dsP, i~O), of norm 1.

Here (ii)a is the matrix with a single 1 in the i,i position, 0 elsewhere, while (ij)~ has 1 in the i,j and j, i positions, and 0 elsewhere. The vector X~ has been defined, and d e | i=d~ | is the tensor product of our basis elements d~ and i~ for 4096~ and 24~. All these vectors are mutually orthogonal except when equal or opposite. w 11. The dictionary, and the group G o

In Table2 we produce identifications between the space 196884~ and the similar spaces 196884y and 196884z produced by triality. In Appendix 2 it is shown that these identifications are isometries, and in Appendix 3 that they are invariant under N. In each case K 0 acts trivially, so the quotients Gxo, Gy 0, G~o of Gxl , Gy2, Gz3 by K 0 can be regarded as matrix groups acting in the same 196884-space. The group they generate, which we call G o, will turn out to be the Fischer-Griess Monster. Our philosophy is that the first three columns of the dictionary name various vectors in three different languages L1,L2, L 3. The subscripts and letters by which these languages can be recognised in various contexts are

LI:I,X,x,A L2:2, Y,y,B L3:3, Z,z,C.

Table 2. The dictionary

L~ name L 2 name L 3 name common space name

(ij)~ = Yii § Yi~: =Z~i-Z~ =(ij)~ VA Xij-X~ = (ij)2 =ZisF Z ~ =(ij)2 VB Xij 4- X + = Y~2- Y~+ = (iJ)3 = (iJ)3 Vc (ii), = (ii)2 = (i03 = (ii) VD Xd. i =d + @2 i =d- | =Xa. i Vx d- | i = Ya.i =d+| i = Ya.i Vy d+ @1 i =d- | =Za. i =Za. i Vz x~, =~Z-~-~r,.~+ __1-~Z-,-~z,.~ + =x2~ "~ 1 + 1 + ~Y-~-~x..~1 + =~E-~-.~ +..~ -z+-... =z~.~ 526 J.H. Conway

The rows of the dictionary correspond to a partition of the entire space V into constituent spaces (with dimensions written under their names) v~,v.,vovo, v~,v~,v~,v. 276 each, 24, 49152 each, 4-8576

These are the irreducible constituents under Nxr~o, except that Vo has an invariant vector, namely the identity matrix (in any language). In the dictionary, the symbol-6 means (-1) il61, and the sums in the last three rows are over all 64 c~*-sets e which are represented by even subsets of d. In the three previous rows, it is presumed that i(sd. This presumption could be dispensed with by adding superscript signs - thus X2. i=d -~ | i, Xd.i=d - | i for all d. We abbreviate d~ | i, to d -+ | i. w 12. The algebra

Simon Norton showed by a very simple calculation that did not even require knowledge of the conjugacy classes, that any 196883-dimensional representa- tion of the Monster must support an invariant algebra, and Griess constructed such an algebra in his Friendly Giant paper. This algebra can be greatly simplified by incorporating the fixed vector (represented by the identity matrix I) in a suitable way. We define A,A'=2(AA'+A'A) X,*A=A*X,=(A, 2r| i2,). X r [ +2r(,~)l 2 r (r-s-- +1) X,*Xs=IX,. s (r.s short) (otherwise)

A (q| 2)=(q | 1 2) * A =q| 2A+~trA. q| 2

X ,(q|174 X,-~q_ i , | [2-2(2, 2,)2,] (q| 2) (q' | 2') =(q,q')[(2,2')I+42| 2' + 42' @~ 2]

+ Z ~(q,I r q') [(2, 2')-- 2(2, 2r)(2,2,)]" X,. short r

In these formulae the matrices A and A' are typical elements of 300 x, trA is the matrix trace, and AA', A'A, 2A are matrix multiplications. The letters q and 2 represent typical elements of the spaces 4096 x and 24 x. The image of q under the group element x, is written qr. When r is short, the homomorphism of Sect. 7 takes xr onto an element of A/2A that is the image of two minimal vectors of A itself. These are written 2,, -2, and the algebra is well-defined since the formulae are invariant under changing the sign of 2,. This algebra is visibly commutative, and the above definition is invariant under Gxo. We show in Appendix 4 that (u, v w) is a symmetric function of all A simple construction for the Fischer-Griess monster group 527 three variables (complete symmetry, or associativity with the inner product), and in Appendix 5, that it coincides with its images under triality. It is therefore invariant under G o. The verifications we have relegated to the Appendices are easy but tedious calculations. We advise the reader merely to follow a few samples and take the rest on trust. We hope he does enough to see for himself how easy it is to calculate in this algebra, and to apply Monster elements to it! w 13. The finiteness of G o. Identifying G O

For the particular vector t=4[(ii)+(jj)-(ij)l-(ij)2-(ij)3], which has t,t =64t, (t,t)= 128, we show in Appendix 6 that the images of t span the space, and that the map v--*v.t has eigenvalues 64, 16, 2, 0 only, the eigenvalue 64 appearing just once (at t). We shall deduce that t has only finitely many images under the group H of all matrices that preserve both the algebra and the quadratic form, from which it follows that H and G o are finite. Otherwise for images t' very close to t we can write t'=t +Ou+O(O2), where u is a non-zero vector orthogonal to t, and 0 a small real number. But then t' .t'=t.t+2Ot.u+O(O 2) 64t'=64t + 640u + O(O 2) and since IIt*utl <16Nuil, these equations prohibit the desired equation t',t' =64t'. Now the centraliser of x in G o plainly contains G~o, for which the irreduc- ible constituents of 1968845 have dimensions 1, 299, 98280, and 98304. We shall show that these are invariant under the full centraliser of x in G o. This is because the 98304-space is the just the -1-eigenspace of x, and it is shown in Appendix 7 that on this space the mean square eigenvalue of the map v~v, u is respectively if u is a vector in one of the constituents 1,299, 98280. The 98280 1-spaces (Xr) are now determined as root-spaces for the sub- algebra on 300~, and the 2 • vectors X, as the unit vectors in those spaces. It is easy to see that the only non-trivial element fixing all the X r is x itself. Thus the centraliser of x in G o is exactly G~o, and now the following theorem of S. Smith characterises Go: Theorem. A group G with an involution x whose centraliser has the form 21+24.Col with CG(O2(CG(x)))=(X) either satisfies G=O(G)CG(x ) or has or- der 246 320 59 76 112 133 17.19.23.29.31.41.47.59.71 and has another involution whose centraliser is 2. F 2, where F 2 is Fischer's Baby Monster group. [] 528 J.H. Conway

Since this applies to both G o and H, it also shows that the Monster group G o is the group of all automorphisms of the algebra that also preserve the inner product. The inner product requirement can be dropped, since it can be shown that inner products can be computed from the formula

trace (ad u.ad v)= 81344(u, 1)(v, 1)+9240(u, v). w 14. Axial vectors

The character table of the Monster has been found by Fischer, Livingstone, and Thorne [6]. Our 196884-dimensional representation is the sum of irreduc- ible representations of dimensions 1 and 196883. Character-theoretic calcu- lations showed that the centraliser of an element in class 2A or 3A or 4A fixes a unique 1-space in the 196883-dimensional constituent. [In the ATLAS no- tation used here, nA indicates the class of elements of order n having the largest centraliser order.] In what follows, a particular vector fixed by the centraliser of a given element will be called the axis of that element, and equally of the inverse element. The axis of the identity is the vector 1= where I=~(ii) is the vector i represented by the identity matrix in all three languages. We have 1 v = v for all v. The Monster has two conjugacy classes of involutions, namely 2A (short) and 2B (long), and the short involutions are also called transpositions. The elements xr, yr, z r are transpositions just if r is short, and their axes are then 2A~-8X,, 2B,-8Y,, 2C~-8Z~, where Ar=2,@12 ~, B~=2~@22 ~, Cr=2~@32 ~. For the particular case x,=xii, we find Aij=2[(ii)1+(jj)l-(ij)l] (remember the factor ~!) and, by the dictionary, 2Xij=(ij)2+(ij)3, so that the axis is 4[(ii) +(jj)-(ij)l-(ij)2-(ij)3], which is invariant under triality, as it should be since xij = yi~ = z~j. An element x r or y, or z~ corresponding to a Leech lattice vector 2, of type3 belongs to class 4A, and has axis 8At or 8B, or 8C~ in the above notation. Our triality element belongs to class 3A, and has axis

212 @12+2 @22 + 2 @32-1 + @12 -- 1 + @22- 1 + @32] where 2 is the vector whose every coordinate is 1. In all the cases we have discussed, there are readily visible elements of the centraliser which make it clear that the axis given is the only possible choice, up to scalar multiplication and addition of multiples of 1. Character calculations show that the product of two transpositions is in one of the classes 1A, 2A, 3A, 4A, 5A, 6A, 2B, 4B, or 3C, and John McKay has pointed out what seems to be a suggestive correspondence with the extended E s diagram (Fig. 3). We have marked on this diagram the inner product of the axis vectors of the two transpositions. We have computed the structure of the subalgebra generated by the axes t o and t 1 of any two transpositions a and b (Table 3). These algebras are linearly spanned by the axes vg of various elements g in the dihedral group (a, b), and we write Vg--*g. The table is to be read cumulatively. Thus if ab is in class 6A, A simple construction for the Fischer-Griess monster group 529

3C T 2 1A 2A 3A 4A 5A 6A[ 4B 2B

128 16 13/2 4 3 5/2 2 0

Fig. 3. Inner products of transposition vectors

then (ab) 2 is in class 3A, and so to.t2=4to+4t2+2t4-3u. We use t, for the axis of the transposition a(ab)", and t, u, v, _+w for the axes of powers of ab in the respective conjugacy classes 2A, 3A, 4A, 5A. The cases other than 3C and 5A were calculated explicitly using our definition of the algebra and the sample axis vectors mentioned above. I thank Simon Norton for his more theoretical calculation which filled in case 5A.

Table 3

Classes of ab Algebra and inner product information (and powers)

1 A t o * t o =64t o, (to, to)= 128, (to, 1)=2, (1, 1)= 3/2 2A(1A) to,t 1 = 8(t o + t I - t), (to, t 1) = 16, t~ab 2B(1A) t o * t 1 =O=(to, tl) 3A(IA) to*t 1 =4to +4tl + 2tz--3u, (to, tO=13/2, u-+ab, (u, 1)=9/2 to*/,/ = 10(2 t o - t I - t 2 + u), (to, u) = 45, u * u = 90u, (u, u) = 405 3C(1A) to*t 1 =to + tt-t 2, (to, t0=2 4A(2B, 1 A) t o * t 1 =3to+3tl+t2+t3-v, (to,t0=4, v--*ab, (v, 1)=12 to*V = 12(5 t o - 2q - t 2 - 2 t 3 + v), (to, v)= 144, v* v = 192 v, (v, v)= 2304 4B(2A, 1 A) t o * t a =to+t l-t 2-t3+t, (to, t0=2, t~(ab) 2 5A(1A) to.t l = I-tz-t3-G+w ), (to, tt)=3, w~ab t o * t 2 =~(3to + 3t2--t4--tl--ta--w), (to, t2)=3, -w--~(ab) 2 to*W = 14(t 1 - t 2 - t 3 A- t 4 -}- w), (to, w) = 0 = (w, l) W*W = 350(to + tl + t2 + ta + G), (w, w) = 3500 6A(3A, 2A, 1A) to*t 1 =to+t 1-t2-ta-t4-tS+t+u, (to, tt): 5/2, t ~(ab) a, u--*(ab) 2 t*u =0=(t,u)

We observe that in each case the dimension of the subalgebra invariant under (a, b) is the number of edges which abut the corresponding node in McKay's diagram. Norton has shown that the lattice L spanned by vectors of the form 1, t, t.t', where t and t' are transposition vectors, is closed under the algebra multiplication and integral with respect to the doubled inner product 2(u,v). The dual quotient L*/L is cyclic of order some power of 4, and we believe that in fact L is unimodular. The vectors t, u, v, w of Table 3 all lie in L.

w 15. Transpositions explicitly computed from their axes

The centraliser of a transposition has irreducible constituents of dimensions 1, 1, 4371, 96255, 96256 on which the transposition has eigenvalues 1, 1, 1, 1, -1 530 J.H. Conway respectively, while the map adt: v~v,t (where t is the axis), has eigenvalues 64, 0, 16, 0, 2. It follows that the transposition equals f(adt) for any poly- nomial f with f (64)= f (16)= f (O)= l, f(2)=-l. [We might also use exp( ad t).] By allowing also a term (v, t)t we can obtain a simpler formula for the transposition, namely:

v~v -~ [24(v, t) t + 16 v * t - (v t) * t].

This formula can be used to give explicit constructions for certain elements outside N o. [We remark that our constructions of G~o, Gyo, G~o were not quite explicit, since Schur's lemma was invoked to produce the representations involved. This is true also in Griess's construction.] All we need do is to find an explicit formula for the axis of a transposition outside N o. Consider first the transposition z d(ld[=8), whose axis is 2Cd+8Z ~. Now 1 2C a is twice the tensor square of ~(2ond, 0elsr namely ~(ii)+}"(ij)3, where

V " the first sum is over ied, and the second over unordered pairs {i,j} ~_d. Using the dictionary, we can write this as IIE(n)+~(Xij+X~), where HE~e) is the projection map onto the 8-space E(d) of vectors supported in d. Another application of the dictionary tells us that 8Ze= ~ X d.,,+ summed over the 64 cg*-sets representable by even subsets of d. So the axis of z_ a is

t = ,+ Zxij+ + ZxJ (,)

We analyse the X~ that appear in (,). We note first that the corresponding vectors 2ij, 2~+, 2].~ of A/2A are precisely the images of short vectors in E(d). Now the preimages of these vectors in Q~0 generate an elementary abelian group of order 2 9, and the x, for which X, appears in (,) lie in a group of order 28 complementing x=x ~ in this group. This definition is symmetric under G~o, and so we can apply the same method to produce other transpositions. Let g be an involutory automorphism of the Leech lattice that fixes an 8-dimensional sublattice /~. Then select in the preimage of E/2A in Q~0 a complementary subgroup E to x=x_ ~. Then t=Hg+~X~ is the axis of a transposition, the sum being over all short r for which x~= E. Examples outside N o are now easily computable.

Many of the results in the next two sections are due to S.P. Norton.

16. Norton's inequality

In the remaining sections, we abandon some of our notational conventions, freeing a, b, c .... for use as names of vectors in our 196884-space V, and writing ab for the algebra product a b. We define A(a^b)=ad(a)ad(b)-ad(b)ad(a), noting that since the right hand side is alternating and bilinear, it is a function of the exterior product A simple construction for the Fischer-Griess monster group 531 a/x b. From the character table [6], V/~ V is the sum of three irreducible constituents, of dimensions 196883, 21296876, 19360062527, the 196883-dimen- sional constituent being generated by the elements 1/x b, which are plainly in ker(A). [-For the reader's convenience, Appendix 8 collects such character- theoretic information.] Now if A' and A" are the matrices denoting two vectors of 300 x, we find that ad(A') ad(A") takes

A to 4(A"A'A+A"AA'+A'AA"+AA'A") X r to X r. (At, A'). (A r, A") q@l,~, to q|174 '' +-~ tr A" .q| tr A" .q| from which we deduce that ad(A') and ad(A") commute just when A' and A" do. Hence ker(A), while not being all of VA V,, is strictly larger than 1 A V, so that Im(A) must be irreducible. We shall show later that Im(A) has dimension 21296876, so that the subspace of VA V generated by the a Ab for which ad(a) and ad(b) commute has dimension more than 99 % of the total, and the algebra can be regarded as "nearly associative". Now the function

(cJ (, ^ b), d) = (c a. b, d) - (c b. a, d) = (c a, d b) - (c b, d a) = (a a (e ^ d), b) is plainly a symmetric bilinear function of A (a/x b) and A (c/~ d), which we call (A (a/x b), A (c/x d)). By inspection, the corresponding quadratic form (A (a A b)) = (A(a/~ b), A(a/x b)) takes on some positive values, and since it is defined on the irreducible space Im(A), must be positive definite on Im(A). We therefore deduce Norton's Inequality. We have (aZ, b2)>(ab, ab), with equality just when ax.b = a. x b for all x in V. [For (a2,bZ)-(ab, ab)=(A(a/x b)), and the conclusion from equality is just that ad(a) commutes with ad(b), which is equivalent to A(a/x b)=0.]

17. Subalgebras and their units

Theorem. Every subalgebra d has a unit. Proof. Let 1~ be the projection of 1 onto the space (s4). Then for a, bed, (l~a,b)--(l~,ab)=(1,ab)=(a,b), by complete symmetry and the fact that abed. Now l~a and a must coincide, since they are both in d and have the same inner products with all members b of d. [] Of course the unit of a subalgebra is an idempotent (i2=i). We shall see that many properties of subalgebras are determined by properties of their units. 532 J.H. Conway

Theorem. (1~, 1~)=0 implies d~=0, for algebras d and ~. Proof Let i= 1~, j = 1~, and let a and b be typical elements of d and ~. Then we make several uses of the complete symmetry and Norton's inequality: (i2,j 2) =(/,j)=0, whence/j--0; (i2, b 2) =(i, jbZ)=(ij, b2)=O, whence ib=O for all b~; (a2,b2)=(a2i, b2)=(a2,ib2)=O, whence ab=O, for all a~d, b~. []

Now the complete symmetry implies that ad(x) is a symmetric operator, and so diagonalisable. We write Vo(a) for the eigenspace {xlax=Ox}. Theorem. For an idempotent i, the eigenvalues of ad(i) satisfy 0 <- 0 <_ 1. Proof If x is a corresponding eigenvector, O(x, x) = (ix, x) = (i, x x) = (i 2, x 2) >= 0 by Norton's inequality, whence 0>=0, and similarly 1-0>=0 by considering 1-i. [] The following corollary of Norton's inequality is particularly handy in that it mentions only the algebra multiplication, and allows us to deduce many instances of the associative law from particular ones. Theorem. (aa)b=a(ab) implies (ax)b=a(xb) for all x. (Alternation implies asso- ciation.) Proof By the complete symmetry, the inner products of (aa)b and a(ab) with b are (aZ, b 2) and (ab, ab). Now use Norton's inequality. [] Theorem. For any idempotent i, Vl(i ) and Vo(i ) are subalgebras. Proof If a and b are in Vl(i), then (ii)b=b=i(ib), and so by the previous theorem, i(ab)=(ia)b=ab. A similar proof holds for Vo(i ). [] We note that V1 (i) is the largest subalgebra which has i as its unit, and that by a previous theorem the algebras Vo(i ) and Vl(i ) are mutually annihilating. We write a@b (and say that a and b alternate, or associate) to mean (aa)b =a(ab), and say that an element a is Jordan if it alternates with its square (so that aZ(xa)=(a2x)a for all xeV). Lemma (x, y). xZ@x@y imply x2@y. Proof (X 2 X2) y = (X, X 2 X) y -~- X ()C 2 X" y) = X (X 2 y" X) = X (X y" X 2) ~__ (X" X y) •2 = X 2 (X 2 y). Lemma (x,y, z). x2@y@x@z imply x@yz. Proof x(x . y z) = x(y . x z) = y(x . x z) = y(xZ z) = x2 (y z). The alternating product theorem. If at least one of a, b, c is Jordan, then a@b, a@e imply a@bc. Proof If a is Jordan, we apply Lemma (a, b) to get a2@b, and then Lemma (a,b,c). If b is Jordan, we apply Lemma (b,a) to get b2@a, then Lemma (b,a,a) to get b@a 2, and can then apply Lemma (a, b, c). [] A simple construction for the Fischer-Griess monster group 533

The Jordan product theorem. If a and b are Jordan, and a@b, then ab is Jordan. Proof We apply Lemmas (a,a,b), (a, ab), (a, ab, ab), (a,(ab)2), (a,(ab)2,(ab) 2) and ((ab)2,a,b) in turn to deduce a@ab, a2@ab, a@(ab) 2, a2@(ab) 2, a@((ab)2) 2, and (ab)2@ab. []

Theorem. The subalgebra generated by a collection of mutually alternating Jordan elements is associative. Proof By the previous two theorems, we can adjoin the product of any two of the generators to the generating set. [] Theorem. If S is a set closed under taking squares, then both the alternator S ~ ={ala@s for all s in S} and the annihilator S~ for all s~S} of S are subalgebras. Proof If a, beS ~, and seS, then s2@a@s@b, and so s@ab, proving that abeS. If a, b~S ~ and s~S, then s2b=s(sb)=O, so that s@b, whence s(ab)=(sa)b=O, so abeS ~ Note that a double use of this theorem shows for example, that if c alternates with all of a, a 2, (a2) z ..... b, b 2, (ba)2, ..., then it alternates with (ba)(ab2). For the set of elements that alternate with all iterated squares of a and b is a subalgebra, and hence square-closed. Its alternator is therefore a subalgebra, which must contain (ba)(ab z) since it contains a and b. We feel that the ideas of the last two sections and of Sect. 13 may be useful in the study of other "nearly associative" algebras and their automorphism groups. Finally, we redeem our promise to identify ker(A). If 2, and 2~ are two orthogonal minimal vectors of the Leech lattice, we find XffX~ =0= X,(XrXs), so that X,@Xs. The corresponding elements XrAX ~ of VA V generate a subspace of ker(A) of dimension This leaves only the indicated possibility for ker(A).

Appendix 1. Some remarks about extraspecial groups

The theory of extraspecial groups and the relation to orthogonal groups is well known. We present a brief summary here. Let E=21+2" be an extraspecial group of characteristic 2, with central element t. Then the values of the square and commutator functions r2 and [r,s] are unchanged when we multiply r or s by t, and lie in (t). So there are functions q(r) and (r,s) defined on the quotient vector space V =E/(t), with values in {0, 1}, such that:

r 2 = ~(r), Jr, 8] = t (r' s).

These functions are in fact a non-degenerate quadratic form (mod2) and its associated symmetric bilinear form. [This can be taken as a definition of the extraspecial groups.] Every automorphism c~ of E acts on V as an element of the appropriate orthogonal group SO2~,(2) or SO~-.(2), and we accordingly say that E has type 21++2" or 21+2". [In the case of interest to us, the form is of + type since the )~ form a totally isotropic subspace in A/2A of the greatest possible dimension, 12.] In fact Aut(E) has structure 2z"SO2,(2) for the appropriate S02.=S0~,. Up to equivalence E has just one faithful irreducible representation p, of degree 2". Since any automorpbism ~ of E 534 J.H. Conway must transform p to an equivalent representation, there is a matrix M, unique up to a scalar factor by Schur's Lemma, for which p(r')=M2 ~ .p(r).M~ for all teE.

It follows that Aut(E) has a projective representation of degree 2" extending p. It is known that this can actually be taken as an ordinary representation of a group of structure 2~+2"SO2,(2) which we call the standard double cover of Aut(E). The exact conditions for reality of the various matrices are also known. Suffice it to say that for the case E=2~+ +24 that interests us all the p(r) and M~ can be supposed real, and the standard double cover is obtained by requiring them to have determinant +1, making them all unique up to sign.

Appendix 2. The dictionary defines isometries

We have said that we regard the dictionary (Table 2) as giving names in three different languages L~, L 2, L 3 for the same 196884-space. We show that it works! We split the space into components VA, VB, Vc, VD, Vx, Vr, Vz, Vr as follows. VA, Vn, Vc are represented by the matrices with zero diagonal in the respective languages L~, Lz, L3, while VD is represented by (the same) diagonal matrices in all three languages. Similarly Vx, Vr, Vz are represented in the respective languages by the Xr, Y,, Z, for which 2, has shape (T-3, _ 123), while Vr is represented in all three languages by the X,, 17,, Z, for which 2, has shape (_+28, 0~6). These eight spaces are mutually orthogonal, and are in fact the irreducible constituents for N~y~ except that VD is the sum of irreducible constituents of dimensions 1 and 23 (the former generated by the identity matrix I)_ A glance at the table shows that the vectors (with varying values of the parameters) displayed in its first seven rows are mutually orthogonal vectors of respective norms 2, 2, 2, 1, 1, 1, 1, independently of which column they are read from, and that they are also orthogonal to all vectors in the last three rows. So we need only discuss V r in detail. What is asserted is that Vr has three orthonormal bases

X d. JJ't (~y+~ d.6J, l~Z d.6l + and that the expression for any one of these vectors in another base is a linear combination of vectors of that base with the same d. We can therefore restrict to the 64-space corresponding to one particular d, for which it simplifies to the assertion that the 64 x64 matrix whose 6, e entry is ~ ~, is orthogonal and has order 3. We suppress the easy proof that reduces this to the identity -~-~%-n~-~-~-~,r = I, which expresses the fact that is a quadratic form on the even ~g*-sets.

Appendix 3. The dictionary is invariant under N~y,

Table 1 shows the action of various elements of N on our 'building blocks' 24x, 4096x, 98280~, as deduced directly from the definitions in Sects. 6, 8. We show that these elements have the same effect no matter which of the three languages they are computed in. [Actually, we need this only for elements of N,y~.] For the first seven rows, the calculations are again trivial. For instance, Table 1 shows that the generators behave the same way on X~ as they do on d+| It is only for V r that there is a problem. For the elements x~ (e even) all our basis elements are multiplied by (-1) la~el, since d ~ =(-1)L~'~ld and e commutes with 6. [If ~ is odd, the same thing happens and the letters X, Y, Z are permuted.] Using triality, the evenness of 6, and the definition of + superscripts, Table i tells us that Ye transforms Yd.+~to + Ya~-oand X~.~ to +X~.~a., where 6'=dc~e, for the same sign + =(-1) 16~el. It can now he verified that these are two definitions of the same linear map by checking that the image of Yn.+a has the same inner product with all the X~a- whether computed directly in L 2 or in L, as a sum of basis vectors. A simple construction for the Fischer-Griess monster group 535

Appendix 4. The complete symmetry of the algebra

Defining (u,v,w)=(u,v,w), we compute the value of (u,v,w) at all triads of basis vectors. The results turn out to be invariant under all six permutations of u, v, w. (In almost all cases this is immediate from the formulae.) In fact all non-zero structure constants are found among the permutations of:

((i j) l, (i j) l, (ii)) = 4 ((ij)l,(jk)~,(ik)l)=4 ((ii),(ii),(ii))=4

((ij) l, Xr, X,) = ((ij) i, )L | 12r)

((i i), Xr, X,) = ((i i), )~, | t 2,)

(X,,X,X,)=+I if r-s-t=_+l (q| q| (q| i, q| l/8 (q| i, q| 1

(q| q,@z. _1 1 2

(q| q~| X,)= --~1 Ar(i)'2~r(j),

where 2,( a denotes the i'th coordinate of 2,. Since these answers are obtained by taking the inner product of any one of the three basis vectors with the algebra product of the other two, the complete symmetry is established.

Appendix 5. The algebra is the same in all three languages

This Appendix is the core of the proof that our construction works. We show that the algebra product u v, or equivalently the symmetric trilinear form (u, v, w) = (u * v, w), does not depend upon the language in which it is computed. It suffices to take u, v, w from among our basis vectors for Va ..... Vr. The number of cases is reduced by the following remarks: (1) all permutations of u, v, w are equivalent. (2) cases related by triality are equivalent. (3) We can replace any case by an equivalent one under Nxr z. We indicate use of this in the phrase "by symmetry". (4) If any element of N~y z negates u but fixes v and w, then we have (u,v,w)=O in all three languages. We now use these ideas to show that it suffices to consider the 6 main cases: I: Vx*V x II: Vr* Vz Ill: VT* Vr IV: Va* VA V: VB* Vc VI: Vo* V, For suppose (u,v,w) cannot be proved to be 0 using (4). Then if one of u, v, w lies in a component Vx or Vr or Vz so must another (consider the involutions x, y, z), and (using (2)) we refer to case I if these two constituents are the same, and otherwise to II. We may now suppose that all of u, v, w are in Va, liB, Vc, VD, VT. If one of them lies in Vr, so must another (and we refer to case III), since any basis vector of Vr is negated by some x6 (6 even), which pointwise fixes all of VA, VB, Vc, Vo. Otherwise we can suppose that all of u, v, w are in Va, Vn, Vc, VD. If two of them are in components Va or V~ or Vc we refer to IV or V according as these components are the same or different. Otherwise two of them are in Vo (case VI). The rest of this Section discusses these cases in turn. 536 J.H. Conway

Case I (XX). In language Lt, by symmetry we need only consider the case Xd. ~* X~(i6d), which can only be non-zero if xa.~xj=xd.~j is + 1 or short. These cases are:

Xi*X i =21| Xi* X ~ =Xi~ (i4~j)

Xd.i*Xl :Xa (j~d, Idl= 8)

X~dI*X~=X+i. ,3 (i,jed, iW-j, ldl=8). In Language L 2 symmetry permits us to consider only the cases (d+|174 and for non-zero cases some r which is I or short must interchange 1 + and d +. We need therefore only consider subcases: (1): [r=l,kl,~2kl](l+|174 ) (2): [r=l, kl, 12kl](l+ | | (3): [r = (d. 6)+] (d + | i) * (1 + | (ir Idl = 8) (4): [r = (d. 3) + ] (d + | i) * (1 + | (lEd, Id[ = 8). The following calculations will show that L 2 yields the same answer as L r (Recall that i 2 denotes a basis vector for 24r.) Calculation (1) 1-term: (i2, i2) I + 8 i | 2 i = ~(ii) + ~ ~ (k k) k*i ik and Oik terms: [(i2, i2)-2(i2,2~)z](Yi~ + ~-)= -3(ik)i other kl and f2kl terms: [(i2, i2)--O](Ykl + Ykt+ )- 1 5(k l)r These terms neatly assemble into the matrix (with leading entry ii):

1 -3 1 1 =~i(~l~i, g -3 1 1

Calculation (2) 1-term: (i2,jz)I +4i@zj+4j|189 2 i j-term: [(i2,jz)_2(i2,)@(j2,20)]yij=O_2. _1_.172 ij --_•162 2 ~ij 12ij-term: [( i 2,h)-202,' %)(h,~+ . z~j)]0+ ~j+_ -0--2.~. 1 I2- ~ = -~Y~j1 + other kl and 12kl terms all 0. The total is + Yo - Y~+] = + (ij)3] = X~j. Calculation (3)

(d. 3 + term: [(2'-]2)i " -- 2i( 2' ).+"d'h)(/2''~d'6)] ' + Yd.,5=(lz,J2)Ya.e,+ ' ' + If i=j, this is ~Yd'~=X~, and otherwise 0. Calculation (4) (d. 6) + term: [(i2,j2 ) - 2(i2, 2d+o)(j2, 2d+~)] Yd~.

Ifj*d (hence j4:i) this is 0, and for j=i it has a factor ~_2.~i.~,1 and so is also 0. In the remaining case, the coefficient of Yd~ is 0--2. + +~_~1_1 ~ 61j. The total is therefore ~-~-~ij Yd~ = X+d.~j.

In all these cases, the L 3 calculations need not be performed, since they can be obtained by the interchange trick: interchange ~2 and -12, superscript + with superscript -, and reverse the order of loop multiplication. Case 11 (YZ). In language L1, by symmetry we need only consider the cases Yj* Zk =(l + | | k), for which the relevant r are just the ie~.

i term: [Ut, k i) - 2(il,,13(kt, 23] X, A simple construction for the Fischer-Griess monster group 537

The coefficient of X, is, according as i=j=k i*j=k i=j+k i,j,k distinct:

~-21 \(--3/2--58!- 32 ~-21 \Sl(-l~z-+3-~- 0-2(-~)\8i(1] +3_~ 0-2(~)(~)=~-.1-1

This result can be expressed in the form

-5 +3 -1 (X,,Yj, Zk)=3}- or 32 or 3~- accordingas[{i,j,k}l=l or 2 or 3,

and since this is symmetric in X, Y, Z it holds equally in L 2 and L 3. Case III (TT). In L 1 we must consider X~.~ * X~+~, which is non-zero just when (d. 6)-(e. e) is 1 or short. This can only happen when de=l, yielding an answer in Va+Vs+Vc+Vo, or when de=f, [ft=8, yielding an answer in Vr. This remark is independent of language, since translation does not affect d or e. We therefore consider the subcases TA (to which TB and TC are similar), TD, and TTT. Subcase TA. By symmetry we need only consider X e* (ij)~ =(Aa, (ij)~)X~, which is X e if i,j~d, and 0 otherwise. In L 2 this becomes (~ ~ Yg-~) (Y,~+ ~; ).

Unless i and j are both in d, this is 0, since Ye.sY~j§ (4-) is not short. If they are both in d, we find the answer

!Vy+$Z.a d'Jij--81"-a -!~y+ d'gJ ~ say, which is X a. 5 5'

The interchange trick gives the same result for L 3. Subcase TD. In L1, we find

Xe*(ii)=(Ad,(ii))X d, which is e or 0 according as i~d, ir In L 2 this becomes (~E Yn~),(ii), which is -~V!y+~2d., = or ~0=0,

the same circumstances. A similar proof holds for L 3, Subcase TTT. Let def=l, where Idl=lel=lf[=8. Note that dc~e, dc~f, ec~f are all congruent modulo ~, and so can be regarded as the same cg*-set, 7, say. We have d. e =de. 7=f. ~. Now consider (in L~) the product X~- ~ X~ ~, which is X a.n.~.~+ or 0 according as d. 6. e. e is or is not short. We find d. 6-e-e=(-l)le~'51d .e.6e=(-l) t~alde-yfe=(-1)l~51f . y6e.

We can express this in terms of the trilinear form by saying that

(X~.5,+ X ~.,,X~)=(-1)+ l~l or 0 according as yb~p is 1 or not. Now [5t=]ec~f]+tfc~6[, and h, fl=le~fl+4-1fc',3], which yield so that the above value, when not 0, is -~-~,5- By symmetry it can also be written as -,-r,=-r162 and so in the totally symmetrical form (X~,X,.~,+ X r.,p)=-~-~%%~+ - y~- ~ if 73etp=l, 0 if not.

We now show that it has the same value in L~.

Using Xd.++__1 5 --~ -- ~, -- ~6' Ye.~',+ etc., the desired L2-value is found to be: h'

-- tp ,(yd.5%4- r+, 538 J.H. Conway

summed over all 6', e', q~' for which 73'e'~0'= 1. This simplifies to

when we substitute the L2-value -~'-,'-,'-r~'%e-r~' for the value of the trilinear form in it. We shall use several tricks to evaluate this sum. First we note that in A/2A all triples of vectors of shape (28016) that add to zero are equivalent under the monomial group to the triple (2 4, --24, 04, 012), (04,24, --24, 012), (--24,04,24,012).

It follows from this that any case for which the Ll-value of ~tX+n.~, ~.~,Y+ Xj~) is non-zero is equivalent by symmetry to the case (X~.~, X+~, X~) Taking 6=~=q~=7 in the above formula, we find the L 2 value to be ~2~1=R, say, where R>0 could be computed by finding how many triples ~', e', q~' have ~6'e'~o'= 1. But more simply, by 'cubing' the above argument, we find R3= 1, since on translating successively from L a to L 2 to L 3 to L1 all non-zero values get multiplied by R each time. The argument actually proves that any value which is non-zero in any of the three languages is the same in all of them. The remaining values must be zero in all three languages. Case IV (AA). In L~ we find, from the Jordan product rule A * A' =2(AA' +A'A) that:

(ij) 1 *(ij) 1 =4[(ij)l]z=4(ii)+4(jj) (i,j distinct) (ij)~ * (jk)l = 2(ik)~ (i,j, k distinct) (ij) ~ (k l) ~ = 0 (i, j, k, 1 distinct).

In L 2 the corresponding calculations are

(Y~j+ Y~j+ ) , (Y,j+ Y~+ )-)~j|__ ,~,j+2~j+ | ~-~j+

-2 2 2 = + =4(ii)+4(jj),

2 2 (Y~+ Y~-) * (Y/k+ YS+)=2 Y~k +2 Y~- = 2(ik)~, (Yq+ Y~])*(Yk,+ Yk~)=O.

The interchange trick shows that the same values are obtained in L a. Case V (B C). We must compute

(ij) 2 * (ij) 2 (i, j distinct) (ij) 2 , (j k)a (i,j, k distinct) (ij) 2 *(kl)a (i,j, k, l distinct). In Lx these are respectively

(X,j + X~) (X,j -- X~) = 2,j | 2,j - 2,+ | 2,~

= -2 2 + 2 2 =-4(ij)~,

! I

(X,j+ Xg)*(Xj~ -- Xjk..... ) -- Xik "~- Xik --Xik --X~k--O, and (X~+ X+).(x~ -x~)=0+0-0-0=0. In the trilinear notation we can state these results as

((gh)~,(ij)2,(kl)a)=-8 if {g,h}={i,j}={k,l}, andotherwise0. A simple construction for the Fischer-Griess monster group 539

Since this is a symmetrical statement, it is valid in all three languages. Case VI (DD). Here, no matter which language is used, we find

(ii),(jj)=4(ii) or 0, according as j=i or j~-i.

Appendix 6. The eigenvalues of v-~v t for a particular t

Define f(v)=2v, v-8 v. Then as r varies, the (transposition vectors) f(Xr),f(Y,),f(Z,) span the space, since for example from f(Xr) and f(X_,) we can recover the vectors A,=2r| 1 2r (which span VA+VD), and X r (which span VB+Vc+Vx+Vr). Also, we have f(Xi)=f(Yij)=f(Zi)=t = 4 [(ii) + (j j) - (i j) 1 - (i j)2 - (i j)3]. So it will suffice to show that each f(X~) is a linear combination of eigenvectors of v~v, t, where t=f(X~)=f(Xr), say, that all the eigenvectors used have eigenvalues 64, 16, 2, or 0, and that the only eigenvector of eigenvalue 64 that arises is t itself. How many cases for X~ must we consider? Unless, X~=Xr or X_r, there is an element of Qxo that negates X s while fixing X,, so that we need only consider one of each pair X~,X_ r But up to sign, the relationship of X~ to X, is determined by that of 2~ to 2r, which is controlled by the inner product 0-,, 2~), up to sign. So a sufficient list of cases is: s=r, s= -r, (2, 2~)=0, (2, 2~)= -1, (2,, 2s)= -2.

By calculation, we find that in each case there is a space spanned by A=A~=2~| 2., X=X. A'=A~=2~| 2~, X'=X~, and such as may be necessary of

A" = (2~ + 2~) | (2, + 2~), X"=X~.~,

that is closed under algebra multiplication by A and X. Below we exhibit eigenvectors (with their eigenvalues) that suffice to spanf(X~)=2A'-8X' in the various cases:

s=r: A-4X(64). s=-r or (2.2~)=0:A-4X(64), A'-4X'(0). (2., 2~)= -1: A-4X(64), A+4X(0), X'(2), 3A'+A"+3X(O), A+2A'-2A"(16). (2, 2~)= -2: A-4X(64), A'-4X'-A"+4X"(16), A'-4X' + A"-4X"- + X (O).

[These examples are also useful in computing the structure of some of the algebras generated by two transposition vectors in Sect. 14.]

Appendix 7. The mean square eigenvalue of certain maps

We consider the maps v~v*u restricted to 98304~, for the three particular vectors u=l, u=(ij) 1, u=Xij. On the 24-spaces (d•174 for each fixed value of d +, we find that this has the matrix

4 1 0 34 --3 [ 4 ]i01jl410 , ___~ 1 4 ' 0 1 "'. .

in the three cases. The mean squared eigenvalue is correspondingly 16 =(u, u), ~ =~,(u, u), 6-~= 3(u, u). 540 J.H. Conway

Since the constituents of dimensions 1,299,98280 are irreducible under Gxo, each of them admits up to scalar factors just one invariant quadratic function, and so the above values must hold for all u in those constituents. This characterises the three constituents as invariants of Cn(x), and so of C~o(X).

Appendix 8. Some information about characters of the Monster

We write the irreducible characters as 1, 2, 3, ... in order of their degrees, and use a 'multiplicative' notation for linear combinations of them.

Tensor products: 2| 2@4=2.3.4.5.6.7.8.9.10 2| 2@6=2.3.4.5.62.7.82.9.10.12.13.15

S ymmetrised tensor powers: 22- =3.6 22+=1.2.4.5 23-=3.6.7.8.12.13 23* =22.3.42.5.62.7.82.9.10.15 23+ = 1.22.3.4.52.6.7.9.10.14 32-=3.6.8.12 32+=1.2.4.5.7.9.11

[Steve Smith has made a much more extensive table of character decompositions.] The permutation character on transpositions is 1.2.4.5.9.14.21.34.35, and the first few head characters are 1, 0, 1.2, 1.2.3, 12.22.3.4, 12.23.32.4.6, and 14.25.33.42.5.6.7. This information has been supplied by S.P. Norton.

References

1. Griess, R.L.: The Friendly Giant. Invent. Math. 69, 1-102 (1982) 2. Parker, R.A., Conway, J.H.: A remarkable Moufang loop, with an application to the Fischer group Fi2,. In press (1985) 3. Conway, J.H.: Three lectures on exceptional groups. In: Finite Simple Groups. Higman, G., Powell, M. (eds.) London: Academic Press 1971 4. Tits, J.: Remarks on Griess's construction of the Monster, I, II, III, IV. Preprint 5. Frenkel, I., Lepowsky, J., Meurman, A.: A natural representation of the Fischer-Griess Monster with the modular function j as character. Preprint, Mathematical Sciences Research Institute, Berkeley, California, 1984 6. Fischer, B., Livingstone, D., Thorne, M.: The characters of the Monster group. University of Birmingham, 1978 7. Griess, R.L.: The Monster and its Nonassociative Algebra. To appear in the proceedings of the conference: "Finite Groups: Coming of Age", Montreal 1982

Notes. A very much more extensive bibliography appears in [1], and should be consulted by those interested in more than the mere construction of the Monster. It seems clear that [5], which appeared after this paper was written, makes a considerable advance in our conceptual understand- ing of the Monster. Reference [7] has many interesting remarks about identities and configurations in certain subalgebras.

Oblatum 6-VII-1984