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LIBRARY
Michigan State
University

 

 

 

This is to certify that the

dissertation entitled

A 2-LOCAL APPROACH TO CONWAY'S SIMPLE GROUP
THROUGH THE 2-MODULAR GEOMETRY OF THE LEECH LATTICE

presented by

P. R. Hewitt

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mathematics

 

 

Major professor

gmdfiwl AW

Date April 2L1988

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

 

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A 2-LOCAL APPROACH TO CONWAY'S SIMPLE GROUP
THROUGH THE Z-MODULAR GEOMETRY OF THE LEECH LATTICE

By

P. R. Hewitt

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1988

7&3 \

Vito
/' 015 "

ABSTRACT

A 2-LOCAL APPROACH TO CONWAY'S SIMPLE GROUP
THROUGH THE Z-MODULAR GEOMETRY OF THE LEECH LATTICE

BY

P. R. Hewitt

In this dissertation we examine the simple group 01 of
J. Conway, and in particular its 2-local geometry which
arises from certain of its 2-modular representations.

We proceed from the hypothesis that we have a group 6
with an involution 20 whose centralizer 8 in c is an
extraspecial 2-group of width 4 extended by the full
orthogonal group OBI-'2). We then examine the fusion of 20
into 8 \CL(8). Next, we add the hypothesis that 20 fuses
into 02(8) and construct a flag-transitive, rank-4
simplicial complex A for c. We prove that the normalizer cc
of a connected component of A contains 8 and fuses 20 into
02(8).

We then give a nearly complete enumeration of the point
suborbits in 60. Finally, we use this information to examine
representations of so over F2 that are given locally by
generators and relations for onl-modules. In particular, we
show that the existence of an adjoint module for 60 leads to
a module locally isomorphic to the Leech lattice modulo 2.

The techniques we employ throughout most of the
dissertation are geometric and combinatorial. In studying
the representations of so we use freely the language of
sheaves and homology, but in fact make no essential use of

this theory.

ACKNOWLEDGEMENTS

It is a pleasure to thank those whose unflagging
support has led to the completion of this work. Among these
are J. Hall, my thesis advisor; S. Smith, who first
suggested the problem; and U. Meierfrankenfeld and
B. Stellmacher, whose comments improved this manuscript

immeasurably.

iii

TABLE OF CONTENTS

List of Tables

0. Introduction
1. Generalities on groups of Fz-type
2. The adjoint geometry for groups of type 01
3. The adjoint representation
(and related representations)
4. Concluding remarks
Appendix

List of References

iv

m> ¢~ P‘ <

36

47
48
49

Table
Table
Table
Table
Table
Table
Table

\JO‘UI4FUONH

LIST OF TABLES

possibilities for involutions of 8
classes of involutions in f(V) (or 0(7))
lifting of involutions from t to 5
point suborbits for R}

B-orbits in P

local composition series' of HOQSA)

suborbits for the “a involutions

10
11
13
29
35
42
48

0. INTRODUCTION

The purpose of this work is to apply the geometric
representation theory of M. Ronan and S. Smith to the group
.1 of J. Conway. This is a finite simple group that lies in
the gray area between the sporadic groups and the finite
algebraic groups. For example, the simplicial complex
determined by its maximal 2-local subgroups is locally the
truncation of a building over F2 [Ronan-Smith 8]. It is
natural, then, to try and push the analogy as far as
possible. This point of view we adopt in constructing, for
example, various candidates for the ‘adjoint module' of 01.

The working hypotheses for the thesis are that we are
given a finite group that contains an involution whose
centralizer has the same shape as that in ol. (Cf. (1‘1) and
(2&1) below.) The two main results of this thesis are:

(l) to show how a complex which is locally isomorphic
to that for .1 can be constructed naturally from
this class of involutions (cf. (2‘1) and (2‘1§));

(g) to examine representations of this complex in
projective spaces over F2: the first of these is
closely patterned on the adjoint representations

of the algebraic groups over F2 (9f. §1.
especially (§*§)).

The first of these results may be summarized in the

following.

THEOREM Let c be a group of type -1.
(i)A.t__s_tlea logmm bt aficenthessnjssemefi

zohezetbsmmnt NWMMMliz—m
astheirnetLraIWE-l.

(infiaslmgefles-WZ-MMAMM
mmmm-zmfiamgfim
summing. 3522;942:5an
mmwz-mmm-l.

 

In §1 we present the general group-theoretic and
geometric definitions used throughout the thesis, and we
establish some of the basic results that intertwine the
group theory and the geometry. Most of these results are
well-known.

For background on the foundations of diagram geometries
we refer the reader to [Aschbacher 2] or [Tits 15]; for the
specific 2-local geometries involved here, [Ronan-Smith 8];
and for the basics of the geometry of groups of Fz-type,
[Aschbacher 2] or [Timmesfeld 14]. We use freely the
language of sheaves and homology as to be found in
[Ronan-Smith 9,10], although the material we present is
completely elementary and requires no depth from this
theory.

In §2 we begin with the main technical lemma: we
examine the fusion, for a group of type 01, of a Zacentral
involution into its centralizer (cf. (ggg)). We then produce
the critical quads ~ these are elementary abelian subgroups
of order 2‘ on which the normalizers induce the symplectic
groups J:(F2) (cf. (2‘1)). These help lead quickly to the
complex alluded to above.

It is the class of quads which to us demonstrates most
clearly the ambiguity of -1's status. On the one hand the
class of quads is sufficiently rich in structure so as to
lead to a nearly complete description of the full 2-centra1
involution class. It turns out that the permutation rank of
.1 on these involutions is 11: this is more than that
encountered in classical groups over F2 ~ five, typically ~
and less than that for, say, 2. Janko's group J; ~ 33 ~ or
the group 96 of B. Fischer and R. Griess ~ perhaps around
150. On the other hand, the presence of the quads creates
problems in defining what should be an analogue of the
adjoint module for an algebraic group. Indeed, unlike the
case for algebraic groups, the essential defining relations
for the ‘natural'iadjoint module for 01 are not implicit in

the ‘Oz-geometry' (9f. (3‘1)). In particular, we are unable

to establish the existence of a ‘useful' module for an
arbitrary group of type -1.

Thus in 51 we add as hypothesis the existence of one of
the choices for ‘adjoint module', and then determine the
internal structure of this and related modules. The main
result of this section is first to produce a small,

‘natural' module, and then to dissect this module rather

completely (cf. (1Ԥ)).

l. GENERALITIES ON GROUPS OF Fz-TYPE

(lJ)D.EEINlIIQE§ AND NQIAIIQN A finite 81’0“? F 13 said to be

of Fz-tyng in case there is an involution 20 in 6 whose
centralizer 8 satisfies:

(1L2) Q :- 9*(6) is extraspecial.

Note that in such a group <zé> is the centralizer of Q.
Thus, the center of any Sylow 2-subgroup of 6 is generated
by some conjugate of 2°; equivalently, any Sylow 2-subgroup
of 8 is a Sylow 2-subgroup of C. We will further assume
that 2*(6) - {1}. This rules out merely the case 6 - O(C).8.
Should we have occasion to consider groups under (1‘2)
without this extra hypothesis, the groups will not be
referred to as groups of Fz-type.

We denote by t :- B/Q the Fitting factor of B, and by
A :- Q/<z&> the central factor of Q. Recall that A is an
elementary abelian 2-group of even rank 2n, say, which
affords a faithful, nondegenerate orthogonal Fit-module,
induced by conjugation. Call n the width of Q, or more
generally of 6.

Let 3 be a group of Fz-type, with notation as above.
Denote by P :- 20; the class of 2°, and call these ~ or,
often, the groups <zo‘> ~ points. For any point 2, we will
denote its centralizer by 8‘; and we continue this
subscripting with Qz :- 9*(8t), t; :- fig/Qt, and
All :- Q:/<z>. Say that points z and z’ are gelling]; when
2’ 6 Q2. The following fundamental result ~ in a mudh more
general form ~ can be found in [Aschbacher 2, (17.5),
pp. 125-126].

(IJ)LMIf§isefF2-mgthensfllinemgisa
mm.mde.fizvz’6Ponthen
€:-<2.2’>sa§i§_fiea:

(a) L’ c P;
(g) at :- <0le e L'> s Ne“); w

<9.) 7/59,“) a 2203).

PM If q 6 Q:, then z’q - z’ or z’z, whence q 6 New). The
claims will thus follow from the symmetry of collinearity in
that we may choose q (in the above) to be an element of
Q: \ 8y.

If the width of Q is 1, then either 8 a D8 or else
6’ a: Qa.63' In either case it is straightforward to check the
symmetry.

Assume that symmetry does not hold, so that z e Qt,,
but 2’ 6 Q2, for certain z, z’ in P. Thus N (L) s 8 (2). We

5 3

use this to argue that z 6 86(2’)’ a contradiction to the
*
hypothesis that 9 (62,) - Qt”

Consider the groups 8 :- 8Q (2’) and 8’ :- 6Q (2).
z 2'

Write 8 a <z’>xQo, where Qo:5 Qz is extraspecial. The
asymmetry yields [5! n Q2, ,3] S <z> n Qt, - {1). This
implies that 8 n 02, 5. 2(3) n Q.’ - <z'>. Also

[3’ ,3] 5 Qt, n 8 - <z’>. Hence <z> S ¢(8.8’) s L. This gives
N (3.1?) s 8’; that is, N (8.3’) s 8.3’ . In particular

Q: Qt, 8
fl’ - 02" or z e 86(Qt’) ~ a contradiction, as noted”

(1‘5)QEEINIIION Let L be the set of foursgroups as in the
.lemma above. We refer to these as ltggs. We denote by

F :- (P,L) the involution geometty gt C. The gittantg
between two points is their distance in the collinearity
graph on P. Let Pd denote the set of points at distance d
from 20, and for general 2 e P let Pz :- P n Qz be the
neighbors of 2. Finally, let Pam :- {z e Pal |zzb|1- n), a
C-stable set.

Note that PIIJ {20} is a subspace ~ that is, any line
that contains as many as 2 points from the set in fact
contains 3. More generally, if R is any subgroup, then
P(R) :- P n H is a subspace. We use also the notation
L(R) :- {L e LIL s X}, and F(X) :- (P(K),L(R)).

*
The demands that c be finite and that Z (3) - 1 are
requisite to use the following fundamental result. In their

stead we might demand merely that L be nonempty.

(L2)IHEQBEH (Cf. [Aschbacher l] and [F. Smith 12].) If C Le
esz-smanleissmm. martian-19161.:
imarnhis 52 one of 13,03). ~2. or ”$202). :1

(mum (from E. Iignnesfeld's [14, (5.1), pp. 163-164])
Let 6’ setisfx hypothesis (11), end let t e 6’ \ Q be en
immun- The Missing held.

(1) [Qot] - 6Q(QO) - 2(Qo)! m Q0 :- o(<tozo>)-
(it) If 80(t)/<zo> v‘ 8““), tl'gn t Le eonjugate via Q Q

(:2 .

0
(1.1.1) If 6Q(t)/<zo> - 6”“(t), then [Q,t] 1; elementary
abelian, egg t6 n £0 £2 t:Q U (tzo)Q. Moteove;
|NG(CQ):88(t).Q| dLVidee 2.

mo: (1) Now HAM-'1! - IM:6,,<c)I - 580(00):. Also.
[t,q]qo - [rqo,qqo] e [t<zo>,q<zo>] - <[t,q]> for any q e Q
and q() e 00' That is, [Q,t:] s C’Q(Qo). Thus, by an order
argument, [Q,t:] - €Q(Qo). (M. Since [Q,t] is normalized by
<Q,t> it must be that 2:0 6 [Q,t], even when 20 is not itself
a commutator of the form [q,t] for q 6 Q.) However, as
[Q,t]/<zo> - [Al,t] C 8‘“) - Qo/<zo>, necessarily BQ(QO) -
[Q,t] s 2(Qo). That is, [Q,t] - 80(Q0) - 2(QO).

(11): If q 6 Q is such that qt - qzo, then t‘:q - (:20.

(iii): Assume now that 80(t)/<zo> - 6"“). From (1) we
note that [Q,t] - 2(C’Q(t)) - 80(C‘Q(t)) is at least abelian.
Assume that q 6 Q is of order 4. Choose q’ so that
q I [q’,t] meg <zo>. Thus qt I tq’ nee <20), whence t
inverts q (in the dihedral group <t,t:q> a De)' This gives
q<zo> e 6"“) and q 6 600:), a contradiction. Hence [Q,t]
is of exponent 2.

Consider tq e t8 n tQ. The previous paragraph yields
8Q(tq)/<zo> - 8’“(tq) - 86a) - gQ(t)/<zo>, so that
tq e tBQ(GQ(t)) - t[Q,t] c t u (:20) .

Finallg, letqg e 6‘ with gQ e 88(tQ). The above says

that t‘ e t or t 20, and either case gives

3 E Ne(<tszo>) co. 0

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M (1): Let the three pairwise collinear points be 2, y,
and x. For q 6 Q2 in the centralizer of y but not of x, y is
collinear with xq - xz e P, and q induces the transvection
of de(<x,y,z>) whose center and axis are <z> and <y,z>.
This gives (1).

For (it), let y be collinear with each of x and 20.
Since :er0 e Qy is not ,an involution, it must be that y -
(x20): - [x,zo]. Thus, y is uniquely determined by x and 20
in this case. Moreover, the number of such x for a fixed
y 6 P1 is simply the number of points of Qy that do not
centralize 20.

Finally, (LL10 follows from (1), since otherwise x and
x’ would be collinear, and x would be at distance no more

thani+lfromz.
OD

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PRQQE For (1) we argue as in (1.1.1). In the first case w is
collinear with x'0 - xy, and so <w,x,y> - <w,x,xy> is a
plane. In the second case, if it 6 P3 then either w e L” \ 6’,
from the above; or else w and 20 induce distinct
transpositions in <x,y>', so that <w,zo> induces a copy of
(53 there. The former possibility says that
[20 ,w] - (z ow): e Qy’, so has order either 2 or 4 ~ giving,
respectively, a singular or nonsingular point of fly. The
latter possibility gives (wzoa) 20 .w o s Q”, inverted by
both at and 20. However neither w not 20 centralize xy, so
that (wzo)6 - 1

Moreover, if x’ - (wzo)3 is a point then x’ is
collinear with x - (xy)zo and y - (xy)'.

(1.1): If V is collinear with some x 6 P2", then the
statements follow from (1,). If V — x — y - 20 is a path
joining w to 20, with x e 6’, then [w,zo] - (wzo)2 e Qx, and

again the statements follow. E1

2. THE ADJOINT GEOMETRY FOR GROUPS OF TYPE 01

(2‘1)DEE1ELILQN A group 6 is said to be of type 01 in case 6
is of Fz-type (as in (1‘2)) and 8 satisfies the following:

(2‘2) Q - 9*(8) is an extraspecial 2-group of width and
Witt index 4, and t :- G/Q is isomorphic to 0:62).
Naturally, the simple group -1 of J. Conway is of type -1.

(Limiflamlibsefms-l-
(1)1hesetL2flinesisnetemm-

(u) n: c is a 11112.net: s, :- <Q,Iz e c’> s ”6“" and
7/3,, (t) 9' 22(F2). 89.1:__reove each mm is ins—<1 ced hr
L

aneintsallinmaithitsmterentbegine.

(iii) '3 acts fies-Wan v on P. IL_ddee . P1 is a'sinsle
B-etbit, 2f length 2.135.

2399: Clearly (t) and (1;) follow from lemma (1‘1), in light
of theorem (iii). For (L11) observe that c is transitive on
points, by design. Now 6 ~ acting as I ~ has a single orbit
on the singular points of M. Thus, 8 has a single orbit on
the foursgroups of Q that contain 20, and P1 induces a

2-cycle on each of these. 0

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ll

PRQQE In the notation of [Aschbacher-Seitz 3], the
involutions of t a 0:62) fall into 5 classes: a2, the
Siegel (long root) elements; c2, the products of distinct,
commuting transvections; a‘ and a2, those whose commutator
subspace is a maximal totally singular subspace; and c‘,
certain involutions that lie in az.c2. The ‘8' refers to the
fact that the commutator subspace is totally singular, and
the ‘c’ to the fact that commutator subspace is totally
isotropic but not totally singular. The subscript gives the
dimension of the commutator subspace.

The nomenclature serves also to describe the classes in
the full symplectic group $(V). The class of a symplectic
involution t can be described as in the following table,
meaning that the condition listed for V(t) :- {v 6 7|
<v,vt> - 0} suffices ~ with d :- qu.([V,t]) ~ to determine

2

the class of t:

Table 2
_e e___lasses ef M1213. is 9(1’) (er 0(V))

 

slees seeditiesent
ad V(t)-V
bd V(r) is a hyperplane, d is odd
c 7(t) is a hyperplane, d is even

 

Of course, d is even for t 6 0(7).

The involutions of 2 all lift to involutions of E :-
B/<z&>. More generally, one can show that each involution of
0:3(F2) lifts to an element of the same order in the
automorphism group of an extraspecial 2-group Q’ of width n
and type 6 using the Frattini argument. Indeed, each

respects a decomposition of Q’ (or Q’ODB, of the same type

12

but of width n + 1) into a central product of subgroups
isomorphic to Q8’ whence one obtains that the element
normalizes a fixed-point-free, elementary abelian
3-subgroup. (This argument comes from U. Dempwolff's

[4, (3), p. 453]; one could in fact use the result given
there by embedding Q’ in Q’01/4.)

Now, the involutions of a coset EN are precisely
EC (E), where E is any involution of 5. Moreover,

E - E[N,E]. Thus there are |6‘(E):[N,f]| N-classes of
involutions in th. In particular those of types a‘, a1, and
cu ~ those for which 9 is free over F2<t> ~ lift to a unique
N-class of 8. An order argument now forces 85(t) to have
shape [N,t]-Bz(t), for each of these types. _

Next we consider the action of N :- NE(tN) on the
module N :- <t,€m(t)>/[N,t], for t of type a2 or c2. Let
(1) denote the following:

- - * ' _

(T) 0 ——> 5m(C)/[I“.tl -—9 AI -—-) <t> ——> 0
In the case a2, (’1) is split for 02(N*) since this group
induces 910235)) e 85113.2 (ilere n:(rz) :- <82 c o:(rz)>.)
However, for the case c2 0 (N ) induces D$‘(F2)) at 116, and
it is possible that (f) is nonsplit

(8x6; (<E>,8‘(E)/[N,E]) ,. 0). We consider the split and
8
nonsplit cases separately. In either of the split cases we

now assume that the M-class of E is fixed by 02(N*). For the
nonsplit case we choose any E in the coset. In the chart
below we list, for either of these t, the orbit lengths
under 02(N*) for those elements of N* that map onto E under
«; the:CN(N*)-class lengths of involutions in EN; and
finally the subgroup N :- 02(8§(E))’ expressed as

6,4?) - <02<c§(E> )/6,,<'c’>>.

l3

 

18111.9. 3.
The lift—ins 2f 111221232111 item 1 $2 5
02(N*)-eleea-
02(N*)-2.tbi£- lengths in
Z-elaes lengths in Ex" mm) N
82 1 8 22“ . 2142.433
3,3 (nonsingula; 3.8.3.8
memes)
9 (singnlat 9.8
Delete)
c2 1 8 2294.25,.l
(split) 15 15.8
e2 10 (gm-2 10.8 21""-2‘32
guadrie)
(nen- 6 (neg-1 6.8 2“"-2‘0’(r2)
£211; uad )

 

In the split cases, 65(E)/8‘(E) - 6£(EN). In the nonsplit
cases this quotient is 260:0’2). Let N s Ne(<t,zo>) be the
preimage of N in 8. Note that N - 02(N).
We will make repeated use of the fact that, for each
class, [02(N),N n Q] covers
[02(N),N (1 Al] - m.aad.([fll,t]), and thus equals 01([Q,t]).
Next we show that the c2 involutions do not lift to
involutions of 8, unless (1) is nonsplit. Assume, on the
contrary, that (t) is split and t e 8 is an involution that
maps to E. From lemma (lefi) we conclude that 8Q(t) w
Nb(<t,z&>). The latter group is the central product, over an
element of order 4, of [Q,t] a 2x4 with a group

21+5 a 4021“. of symplectic type. However N as above

l4

normalizes no subgroup of Nd(<t,zo>) of index 2. Hence 80(t)
is not properly contained in Nb(<t,z&>). The only possible
conclusion is that these cz involutions cannot lift to
involutions of 8 when (t) is split.

Rather than determine which of the remaining
5-involution-classes do in fact lift to involutions of 8 we
concentrate on the consequences, for each class, of finding
an involution in the given coset.

Consider an involution t that maps into c2 in t.
Necessarily s is nonsplit, and so Né(<t,z&>) has shape
NQ(<t,zo>)-260:(F2). Now, |”8(<t’zo>):88(t)| is at most 2,
but lemma (Lt) forces |NQ(<t,zo>):€Q(t)| - 2.
[NQ(<t,zo>),N] contains 01([Q.t]) and covers the
4-dimensional orthogonal factor of NQ(<t,z&>), and so in
fact |[Q,r]:8[Q’c](t)| - 2.

For the class c‘, [Q,t] is not elementary abelian, so
that |N§(<t,z&>):80(t)| - 2. This says that for t of this
class t[Q,t] is a single Q-class of involutions, of length
32.

For t of class a2, [NQ(<t,zo>),N] - NQ(<t,zo>), a
central product, over <z&>, of [Q,t] and an extraspecial
group of width 2. Hence NQ(<t,z&>) - 60(t) and so there are
at most 3 classes of involutions in t0:

-—- the two Q-classes of t[Q,t], each of length 4, which
are exchanged upon multiplication by 20, and possibly
fused in 8;

'and
—— t.{involutions q of 8Q(t) \ [Q,t]}, a class of length
8.9.

Finally, for t of class a‘ (or a2), [Q,t] is either the
indecomposable orthogonal module or else a direct sum of
(20> with the natural symplectic module for :P‘(F2), so that
€Q(t) - [Q,t], and t[Q,t] is exactly 2 Q-classes of
involutions, exchanged upon multiplication by 20, and

possibly fused in 8.

15

We have now verified that the entries of Table 1 give

 

all of the classes possible for involutions of 8, their

lengths and centralizers, and their class in t. D

(25_5)§QBQL_LA_BXLet€hee£$m'1-Ifzen512’ ere
eemmuting pointe then the aetion pf 2 en Qt"L§ isomorphic
mastefz’ 2110:.(Eeareseee__rei_ssse n Memetzmie
2’ QBEJ—Lexhanedmgselementeffid

 

23993 If the points are collinear, this is clear. If not,
then Table 1 identifies 8; n 8t,, and the possibilities are
essentially distinct for distinct classes of involutions in
82 B 82’ . C]
(2.1.6.)___Y.COR0LLAR lit. 9’ 13. ef Sm '1.

<1) 1.: w e 6 is e p__n_oi c _ethe aqua/(Q n on) s ec<w>/6Q<w).

(ii)Apo_ir_lto_f6’\(PZUQ)m1§_teeesise.e’.§.§’.et
8.

(111) 1; w e 8 1e e point pt t-clese a‘ e; a;, then 80(w) 1e
the 1ndecemposab1e otthegonel mednle fie; 8 n 6'.

23993 (1) follows from corollary (21;) and the appropriate
Isomorphism Theorem, since ti’Q(w).Qw s (6’ n 8') .Q'.

(11) and (111) Observe that if z e P \ {20} then
(Q n Q')’ - P1 n Px. Hence, by (1,), 80(w) g rem/80w) for
w e 6 at distance greater than 2 from 20. This can happen
only for w of class a‘ (or a;) or c‘in t, as one sees by
inspection of Ieh1e 1. Thus if w is of t-class a‘ or a1,

then 80(w) must be indecomposable. D

(M)WL2§C£Q£EM'1-fleieuefln£__l§h°-

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mints 91 P2,-

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16

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elm gt 2, ene conversely eflh coset xQ pf fleh e Siegel
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interseseien .Ls. either {1}. 3 92125.21: 5 use. In
pett1eu1a: £19 commuting peinte et distenee 2 item ene

anethe; Lie together en e un1gue guad.

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(areeieelx thoseefelfisczss?) lame new
9311111125; 11th 1te singular center 1n 9’ (i.e. , the pteduet
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mmmmmmmmm). Infect.

:7, :- <Qz|z e at’> s New), eng 79/89,”) as us.

 

(11.1) EXGPZZQQWEC’OPthen [Wfi'le-{ll- _1_:

elsewePz thenweaPx2

’ o
m The transitivity on P1 follows from lemma (2_,_1).
Lemma (15.6.) gives that for each y 6 P1, the set P2 ‘ n Qy is
conjugate ~ yin Nc(<y,zo>) ~ to P1 \ 800'), which is a
single 6’ n Byoorbit, as t at 0:072) is rank 3 on the singular

points of Al. Thus, 6‘ is transitive on P2 ‘.

17

Similarly, since 8 n 8’ is transitive on the
involutions of Q n 8’, either all of these lie in Qy, or
none of these do and the set is conjugate via Ne(<y,z&>) to
P292 0 Qy. Now the elementary abelian 2-group Q n Qy has
order dividing 25, so only the latter case can hold. The
flag-transitivity now gives that PL} is a single B-orbit.

We have shown that there are no triangles of
collinearity. A fortiet1 there are no planes. Thus, by
(21311), we conclude (11).

The first and last orbit lengths are now clear. The
length of 1,2,2 is 2.70.1‘tl’1/#(P1 n P), for any fixed
x e PLZ. A comparison with lehle 1 yields that the only
possibility for x is for it to have class az in t and
satisfy one of the situations 1 or 2 of Iehle 1. The second
paragraph of the proof yields that <x,z&> is conjugate into
Q, so that both x and xz0 are points of PLJ' and the eeeene
case of Ieh1e 1 holds.

In particular we have ascertained the following facts.
|88(x)| - |8|/#P2'z - 21833, and the foursgroup Q/E’Q(x) has
two (regular) orbits in x[Q,x]. There are exactly 8 points
of P2,2 in xQ; and <x,zo>' C P but is n9_t a line. Finally, 11:
and 26 have 3 common neighbors. This is (1) and (111).

Call these common neighbors y, y’, and y” - yy’. The
diameter of the geometry F($) induced on the group
9 :- <x,y,y’,y”,z&> ~ perforce an elementary abelian group
of order 16 ~ is easily checked to be exactly 2, with each
pair of noncollinear points joined by at least 2 paths. For
example, the distance (in P(9)) between points of 9 n Q is
certainly at most 2, as is the distance between Z0 and any
point of 3. Moreover, each point of 9 nl’z'2 - x[Q,x] is
collinear with a unique point of any line in P that contains
20. Finally, one uses these observations to show that any
two points x’, x” of x[Q,x] are jointly collinear with at
least one point z of [Q,x]. Indeed, if P1r‘I;’ 2 z’,z” and

P nP,Qz’z, z”z, thenz-z’z”-z’zz”z.
1 x 0 0 0 0

18

As a consequence, P(9) is the seeeesie glgsute of x and
25 ~ the smallest subspace that contains x, 20, and all
points lying on all minimal paths joining any pair of points
of the subspace. This is because the above gives 3 common
neighbors in 9 to any pair of noncollinear points of 9 ~
accounting for all of the common neighbors in all of F. It
is immediate also that F(9) is isomorphic to the
9‘(F2)-quadrangle. Henceforth we will denote by 92", the
unique quad containing two commuting points z and 2’ at
distance 2 from one another.

If 9 and 9’ are quads with [9 n 9’| z 8, then there is
a point 2 that is the radical, in each of 9 and 9’, of any
eightsgroup in 9 n 9’. This means that 9 - 9‘1 and 9’ -
9“, for certain x and x’ in 8:, each acting as a Siegel
element on A2. As a result, x I x’ neg Q', and so 9 - 9’
(from the definition of a quad above). This finishes (1y)
and (y).

Consider now a path w — x — y - 20, where x 6 P2. . We

2
assert that exactly one of the following holds for w:

(a) hm?” ] - {1}.

(h) |wzo| - 4, and w acts as an involution of class 02

on 9 - 9xz , with singular center x.
' o

By the previous lemma, we know that the order of |wz°|
is 2or 4 (remember that each point x of P22 defines a

Siegel element on M, and 20 defines a Siegel element on each

A) so that quzo) - chqu’zo])' Further. y '

<[Qx,z°],zo> - <x,[Q,x]>. Thus w centralizes gene point of 9
at distance 2 from x if and only if w centralizes eeeh point
of 9 collinear with x (since if an element centralizes both
a hyperplane and a point outside the hyperplane, it
centralizes the whole space). These conditions are thus
equivalent to [9,w] - {1}. In particular, w cannot induce a
transvection on 9. Now consider w collinear with x but not

commuting with 20. First, {1} " [9,w] s 9.[Qx,zo] - 9.

19

Second, w normalizes each line on.x in the eightsgroup

9 n Q;, and centralizes a unique one of these, say <x,y’>,
which must also be its commutator on 9. If w were to induce
an 812 element on 9, then 20' - zoy’ (since y’ is the unique

point of [9,w] collinear with 20). This contradicts the fact

that w e 6”, satisfies C’Q (w)/<y’> - 6" (w). Consequently,
y’ 7'
the class of w in Ne(9)/Bc(9) S 66 is as claimed. The

singular center of an involution t of type c2 in any
symplectic group is the radical of 7(t). Since w normalizes
each line on x, x must in fact be the singular center for w
acting on 9. Thus (e) and (h) ~ which are mutually
exclusive ~ exhaust the possibilities for w. Finally, each
Qz is generated by its involutions, so the identification of
3y is complete.

The above ensures that [9,x’] - [9’,x] - {l}, where

9 - 9x: and 9’ - 9£,z , whenever x and x’ are collinear
'o '0

points of P22' Corollary (21g) implies that the only Siegel
elements of 6* :- 8z(xQ) that lift to points in 86(x) must
normalize some nondegenerate 2‘ in A* :- Q*/<xQ>, where
0*. :- 0203*). Now the only such Siegel elements that also
centralize 9 lie in Q*, and thus {0} # [M,x] n [N,x’] -
<y<z&X>, say. In particular, as every quad is geodesically
closed, <x,x’,z&> s 9 n 9’, and so these quads are equal.
This is (y11), and the proposition. D

(gy§)g§ngng§(1) We have shown that each point of P12
generates with 20 a ‘fake line' ~ a foursgroup all of whose
involutions are points, but which is not a line. We call the
fake lines of the proposition hyperb011c 11nee; and if the
need for emphasis arises we will refer to the lines of the
involution geometry as the e1ngn1et 11nee. The hyperbolic
lines are precisely the foursgroups that lie uniquely in
some quad. If h is a hyperbolic line then one sees that its

normalizer induces 22032) a 63.

20

(11) We will see in the proof of (L12) below that the
normalizer of a quad induces all of 9‘(F2), with a
transvection induced by a point at distance 2 from its

center in the quad.

(MMQEEINITION Call any path 2’ - z — z”, where L’ - <z,z’>
and L" - <z,z”> are lines, a corner, of width:

min{|x’x”| lx’ eL’ \<z>, x" 6L” \ <z>).

 

Note that the width of the corner depends only on 2 and the
lines L’ and L”.

(2_JQ)COROLLARY _Th_e involution geometty m e gm pf type
-1 eenta1ns ne irreducible pentagone. Mete p;ee1ee1y, 1f
zo—zl—zz-za—z‘b-zfi-z0 _ieeeyeleefi 5 M11151;
pp1nt§ (subscripts read modulo 5 ) m .n_o_ gene; ef 311th
l, m £9: a_l1 i g must he M 21 1e c0111nee; 11th
z1+2z1+3' In particular, 11 x 6 P2", then x ene x‘o ere the

pnly points pf P2 4. collineat w1th x.

 

 

PM If 22 6 P2" then so is 23, by part (11) of the
proposition. However, zlz2 - 2:0 and zaz3 - 2320 are
collinear. Thus 21 and z“ are collinear, since now
<21” I‘ O> must lie in a quad. This contradicts the fact
that there are no planes, since the hypotheses preclude
the possibility that <zl,zz> - <za,z‘>.

Thus all 21 lie in 6’. The proposition now gives that
all lie in some quad, in which one checks easily that all

pentagons are as described. El

(LADWIetceeftm-ldetwegmxeg
he eellineer. Qse ef the fa w holes.

(a) x e P“. <wa - x (y - (x209) em Iwzol - 3.
(1.») x e P“. (w)2 - 1 (y - (x209) eta («'20)2 e P2.2'
(9.) xePzz. en_d [m9 l- {1}.

, 3,20
(g) x 6 P2 2, (wzo)2 - yx, where <y,x> - [w,9x ],

2
O

 

MYEQOQX-

21

M Assume first that x 6 P2", and let y - (xz°)2. By
(1,141) either (b) holds, or else (my)2 - x and |wzo| - 3
or 6. The last case gives (wzn)3 - zo'.wzo e P n Q”, whence
<x,y,(wzo)3> is a plane ~ a contradiction.

If, on the other hand, x 6 P2 2, then the remaining

cases are lead to (e) and (e) by application of proposition

(2,2,v1-v11) and lemma (1,8,1). For if (e) does not hold

then L :- [91‘ z ,w] is a line, and x 9‘ (wzo)2 e L \ Q.
' o

D

(2.13% Let 5 he 9_f. _rp_t e '1.

(1) P3.2 - P3 0 6’ 1e e single B-etleit. The product o_f zo
MemmwefPa'zienotem; thesis.
P n 2093.2 - w. m e w _age ee en 1nvp1ut1on efi tm a“
9_n Al, enel eatisf1es th_e cond1tions 1n em 5 pi we 1.

(11) 11 x 6 P2 2 then 88(x) nee 2 etb1ts 1n P3 (1 Qx, viz.
the 2.36 ppints that centrelize 20 eng the 2.96 whoee
ptgduet w1th zo ha_e orde; 4.

(iii) Bee—h w 6 P32 1_ film; _with 15 Mint g 1."2
.3 .

P3 2 has length 253 5.7.

2.

PRmE We prove (g) first. Corollary (Li) says that the
action of 880:) -6’nf.’ in P nQ -P \ [Q ,2] is
x 3 x x x 0

isomorphic to the action of 88(x) in P1 \ [Q,x]. Here
1+2.6
2
singular points of 6”“(x) \ [Al,x] ~ 36 in all ~ and
Al \ C’M(x) ~ 96 in all.

(1) and (111): The above ensures that P3 2 r‘ a. From
corollary (L11) any neighbor x 6 P2 of a point w 6 P3 2
must in fact lie in 6’. Corollary (L_6_) implies that the

.3322 acts with a single orbit in each of the sets of

involutions in case 1 of Tab1e 1 are not points. However

w 6 6’0 (20) \ [Qx,z°] , whence wz0 is not a point. Hence, by
x

22

the corollary again, we have P3”2 meg Q - a‘ (say), with the
conditions of entry 2 of Iehle 1. Thus if w is any point of
PM, then #183.2 - 16.3780 - 2°3’5.7 -

8.1575.2.36/#(P' n Pz,2)' This gives the parameters of
(111). Note that a comparison of the size of the sets

P. n P2,: and 02(8 0 8') n P2,2 yields containment here.
Hence each minimal path joining w and 20 lies entirely in
the elementary abelian group 02(6' (1 L"), of order 211. CI
(LIDLE—WALercbeefm '1. Fathesenneceedmmest
efl‘thets_1L_0 tains 20. mcomenemelizexinfiefl‘o. The
WM. 60 -<6,,D> menyfl :-Nc(L), zoeLeL.
c ieetme-I 2* (c) - (11. eneroieiseeeieint
BEQEEEEX-

ERQQE Since 20 is central in a Sylow 2-subgroup of C, I} -
(P'o, L o) is the unique connected component of 60 normalized
by0 20 . Thus, 6 s 60 and so go is of type -1 ~ provided that
2* (60 0)I- {ll (part of the hypothesis (211)). In fact P2” C
Po, so that 2060 2 P1; whence P60) - zogo, F0 is the
adjoint geometry for 60, and 2* (co ) - (1). It is now clear
that co 2 <B,D>, whenever D - N€(L), L a line on 20. A

Frattini argument gives that <C,D> - €0.[j

(2.45:)8eneefsutbseesmethstfiseenneeteshrenleeimfi

bxfioifneeefieu.
We note that this hypothesis follows from (211) and the

broad classification [5] of D. Holt of transitive groups in
which a 2-central involution fixes a single point. However
the connectivity will be used in this section only to
establish (2,1§,v), whereas Holt's result is immeasurably
deeper than this.

23

deifiwmgmefmd-

(i)meinqueti_onsemnreeserelsmar_i_el_xesdu 1
ee__et_slnne eeesnlexAhhtaeerth 1e them:
1 2 4

11
A O

I

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linesahesseds.t_§e_ei__lxe ec ve .Meefmellere
WMZ-mefimmozw'nfidsherez
ehsz’areepmtihgpeihtseteimnee3freumemeher-
gelltherieeve e eitxpellhexealhenemelizerefahes
Kieasplihwxensio ethxthehsthiehstmmz,.m
the—imam Hie—ducesehxisthetefmz‘entheuim
faetetefthe)@lexeo_de.lhese2meeur(l¢)iea
hear-hereses es the 759 221.1195 9.: P00.

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_eanothe liegeLmiQEhesKmu newness.
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thet eonta1ns he pe1nt.

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fie; eeeh simplex 1e isomorphic tn the ttnneet1en pt the
eleeeisele-womt thettheresideeleiamsw et.
smattlietefehgs.yhiehsm_§rethetamefehe
inn-2M over F2 (cf- [hm-Shim 8] fer nether
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A 21th the ineidenee geometty en the vept1ee§ 21 A.

(ix)1hes$hhflimeefelm_i_e_ssmle ceillheeesetedfl.
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-1
fpllpwe:
91 e 21*8o0‘;(r )
92 a 22*“. [22(r2)xr‘(r2)1
4+6.2
9, 9' 2 -[9‘(F2)><£2(F2)]

r a: 211-111
11 24

D:

 

 

 

 

24

there 9,62) S 1230') (as the stehilizet ef en ml) is e
(neneentral) tum eevet pf 9‘62); Ihpe egg; vetten v hey
eereefleredeslwv). ens! flv-9(?v).1heetehiliz-etete
ehenhet bee e se1fnerna11zing m 2-§nbgroup ef 1nc1e35 3
ehiehieeMZ-ethsmeieelleffi-

(2)1eelltheeheeffle14mmehlthem. lieee.
sweet. enehesee (eeehresemeeeeeF2-me<h11efetite
nomal1zet) the adjoint sheaf, egg _denete th1e by 5“. 1
genote 110(5“) by 9, egg tefet te thin 5.3. _t_he adjoint module
(cf. [Renee-Smith 9.101fet details en sheet limits! fez
g_h__tieseoet ).I_eseLd ebehtheeetefhnerhelieliheezbl
1‘h :- (P,Lh) the hyperbolic geometry; the hy I‘ :-

+
(P,L U Ln) , the augmented geometry.

Beesllthetifl" - (P’.L’) iseseemettxeithliheset
size 3. tlm th__ere is e enixeteel deut(I")-meeele 110(1") :-
FZP’M’ geeetetedhxthepeihteeneflhlegtetheeetihies

W5? :- {RX :- X ZleC L’}.
LeX zeL
Finally, 1n the following gi_a,gr_a_n pf canon1ee1

surjectione:
FP -—>" H(I‘)
2 0

«.1 l ”
7 7+

h
H0(Fh) —9 ”0(P+) 7-) Q

nieehieemetehmmdflrmhgsl.
2

23005‘ Let x 6 P2 2 and w 6 P1‘ \ P2 2 commute with 20. Such a

w is chosen in 80(20) \ [Qx,zo], and is forced to be an

involution of 6’ of type a‘, owing to proposition (Li) and
lemma (1,12). Set 1: :- 02(88(w)). x has shape 21”"6
to (2__4_) ~ and contains [Q,w] - 8%(w), 2a maximal elementary

abelian subgroup of Q, of order 2 . H/[Q,w] is the natural
orthogonal module for N8(J¢)/R at 0:62) at 2‘62), and

~ due

25

J(\ [Q,w] is composed entirely of involutions: the 2‘.28
Né(K)-conjugates of w, the same number of conjugates of wzo,
the 23.35 conjugates of x, and the 24.35 conjugates of xy
for any y e [Q,w] \ [Q,x]. Thus 3 is an elementary abelian
subgroup. Set N :- N¢(X). Observe that since the definition
of X is symmetric in w and 20, so is the definition of N.

We show next that 20” - 9(8). It thet; follgws that N is
irreducible in X; indeed, we show that 20 - 20 R, and so
9 - <Q |z e P(R)> acts irreducibly in X. To see this note

X
that P3 2 is a single 6- orbit, so that the orbital (zo, w)6

is symmetric (or, self-paired). Thus 2“” 2 wflna. Now Q s N,
so for x 6 P2 2 n P as above we have that wa C rif; 20”.

This in turn gives the claim, in that that 20 can be
conjugated y1e 7x to any point of R with which it is
collinear. It is not difficult to see that F(K) is
connected.

Consider now the complex A“, defined to be the
(flag-complex of the) points, lines, and quads that lie in
R. The residual geometries in An of 20 and of any line on 20
are easily checked to be (truncations of) respectively
920:2) and 96362). (This is done entirely within 8 by
noting that the lines and quads in X that lie on 20
correspond exactly to the point- and line-stabilizers in
(N n 6’)/X a: £‘(F2) as this acts in [Q,w]/<zo>.) Thus AM is a
flag-transitive complex over F2 that satisfies the residual
diagram D11 of D:

D: u fl 4 Cl

11

 

This complex is plainly embeddable (in H) over F2, so that
the main result of S. Smith [13] yields the identification
N/R a ”he and R 8” Golay code (1.e., the simple factor of

 

dimension 11 obtained from the span of the octads). We note
that this extension is necessarily split, although we could
deduce the splitting from that of N n 6 over X.

26

Note that, in Illa, the normalizer of a quad induces
9‘62), with each transvection induced by a point at
distance 2 from its center.

The residual connectivity of A is a consequence of the
connectivity of F ~ _f. (211$) above.

To prove (11), we suppose that R and K’ are hexes, with
P(X n K’) 2 20, say. Now 8 induces a permutation group on

the collection H: of hexes containing 20, with Q acting
O

trivially. 8 is transitive on P32, hence transitive on Hz .
' o

The normalizer of any one of these hexes is, megglg Q.
merely the stabilizer of a maximal totally singular subspace
of A. Since this action is rank 3, and since there are pairs

of distinct hexes in H; intersecting variously in <z¢> or a
O

quad, we see that these are the only possible intersections
for a pair of distinct hexes with a point in common.

The remaining statements of the proposition ~ save
possibly the isomorphism of (y) ~ now follow
straightforwardly from the claims already established,
together with the information known on AR' at least once the
residual geometry for 20 in this complex is determined. This
can be done by using the correspondence of the lines, quads,
and hexes on 25 with, respectively, the singular-point-,
totally-singular-line-, and totally-singular-4-space stabi-
lizers (the 4-spaces corresponding to wt) in t.

For (y) observe first that there are natural

surjections:
H(l‘)—1"L»H(I‘)—1:» z+81——)z+8 l—euua' )
o h o + 9’ h + ea;z '

Thus if HO(I‘h) - 0, then H0(I‘+) - 9 - 0. Assume that

H6(Fh) I O. The relations 8b and the flag-transitivity of C
in Pb give that «h is injective on the set of points in any
singular or hyperbolic line, quad, or hex of F. Fix a quad
9. R

L
connectivity of A now gives that ker(1h) - 8+«h is a trivial

I 5%, nee 3h, for any singular lines LHL’ e L(9). The

module for c, of dimension at most 1.

27

To finish (y) consider 9’ :- P(9)11'h'yh U {0} 5 HO(I‘+).
Now 9’ is closed under addition: 22’ e P(9) and 22’ I z + 2’
mpg 8+, for distinct z and z’ of P(9) (use (211)). Thus 9’
and 9 are isomorphic Nt(9)-modules. Finally, the argument
above that provides the hexes of A can be repeated to
complete a proof that the sheaf {3nd has a nontrivial image
in the constant sheaf H6(P+); use this map to invert 1+. D
In order to finish the enumeration of the B-orbits in P

we need the following.

(211Q)PROPOSIT10N Cons1det en U9 3 8 thet netnaligee e get
91 9 nntne11y nonperpendicnla: singnlat pe1nte 1n N. 11 U s
8 is e netetel “s-eehsteee ef this "9 then 6“ =- 56(fl) e R}.
the eieele 8:222 2f M- hell ens Z- lathe.

We first prove a short lemma.

(2,1Z)L§hhe Considet e at zo-— y - x— w — v, whete

xePz‘,wefi’y,ve8x\<w,x>.W_e_t_henheveweP3 an

3’ .—

v E P .
4,5

EBQQE QB LEMMA We have V 6 P33 by (2111).

The a2 involution VQ‘ in 8x normalizes a unique 2-space
in A‘ and containing y<x>, and vQ: induces a transvection
there. Thus v normalizes a unique quad 9 on <x,y>, and is
collinear with a unique point w’ of I - <x,[y,v]>. As v is
connected to 9 it induces the “6-involution with singular
center w’ and axis a. On the other hand 20 is connected to 9
and induces the Ufi-involution with axis L I <x,y> and
singular center y. As L and a are disjoint, e - (v20)5
centralizes 9 (e1. the Appendix).

Now 2 - yvzo - w’ 2o" is collinear with neither y nor
20 6 Q2 (as in the Appene1n). Unless e
is 1, z is the unique point z’ of 9 with e 6 Qt" Neither 20

2V V
w’, ands-170.20

nor v centralize 2, so e - 1 and |vzo| - 5.

By (2111), v cannot be in P1 or P3, whence v e P‘. D

28

PBQQE QB 23929511193 First note that there is exactly one
Oc(Fz)-class of sets of points that are maximal, with
respect to inclusion, amongst the sets of mutually
nonperpendicular points. The normalizer of any such set is
isomorphic to “9 (use Witt's Theorem on extending partial
isometries).

Next observe that the centralizer A of a corner
2’ — z — 2” at z of width 4 is the centralizer int?z of a
subgroup 113 of Q, hence has shape 21+60;(F z)' 02(14) acts
simply transitively on each of P! \ 8: and2 P{,\ 8:. Now a
point of either set can be viewed to correspond to the
subgroup 020:2) stabilizing the point; abstractly this is a
complement in d to 02(11). There are in fact two classes of
complements to 0201) [Pollatsek 7, (5.2), p. 415]. Th
respective representatives of the classes act on the 64
lines on 2’ net perpendicular to <z,z’> ~ as well as the 64
for <z,z">'~ with orbit decompositions of l + 28 + 35 and
8 + 56, respectively. A consequence of this proposition is
that (to be proved in the proof of (2112) below) is that a
complement that fixes a point in.P£.\ 6; teen net fix a
point in P2,, \ 6’2.

N 9-8 811 2“"0'r °h9*n-
ow .- Cu(z°)- e()' _ 52). mt ()

R :- Q n 6“ - 2?". The copy of “5 chosen centralizes a path

20 -y'—-X'- w, where x 6P2 u and w'e P 33. In particular,
2:68 0 8 I <zo:>. That is, en is of type XJ/Ja. An old result

of Z. Janko [o6] asserts that one of the following must hold:

cue“); 2t
6“ has exactly one class of involutions (and 6“ has the

character table of J ).
3

The second case is impossible since the involutions of D \5!
are of type cz on M, and (21h) rules out these involutions

as points on 6. This finishes the proposition. 0

For future reference we list the D-orbits in P1! :- zocll

and the parameters for the lines between them.

29

1213.129.
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3O

£3992 This proposition is really a compilation and
refinement of the results on P3 contained in (Zyll) and
(2‘12), and one should have these statements firmly in mind
as one reads the following. In fact the bulk of the proof is
devoted to proving the statements about Pma' It might be
worthwhile to look ahead at this point to Ighlg i at the end
of the section. This gives a global view of the E-orbits in
P.

(1): Let w 6 PL‘. Corollary (Zyll) says that w is
collinear with x 6 P2 where either x e 8 and w induces an

”6-involution on :P - :1”x 2 , or else y - (xzo)z 6 P1 and

0
w 6 8y. Consider the first possibility. In this case w e Qx

acts nontrivially on the unique quad :f on <zo,zo'> s Qy.
Moreover w centralizes a unique line L in f n Qx.
Necessarily L - <y’,x> for some y’ of P; n Q. Thus, 2

o
induces an ”6-involution on f,, . The first possibility

reduces to the one just considered by reversing the roles of
w and 20.

If 2 is the point of L - 9 n 9’ that is collinear with
neither w nor 20, then a point z’ 6 P2\ 8 must normalize f,
9’, and L, and conjugate 20 to one of the 4 points of ?’
collinear with yz. If this is not w one can replace 2’ by
another point of P; or by a product of this point with a
point of 62 that induces a transvection on 9’ with center 2
and obtain thereby a point that conjugates 20 to w.

We have shown that any point of PerI; lies in the
quad 9;”30, perforce the unique quad on w that is
normalized by 20. This finishes (i) and yields the
parameters for P3” in (ii).

We will derive the statements about P3.3 simultaneously
with (iii). This is all that remains to prove. The
combination of (gyll), (gyLZ), and the first part of this
proof show that there are but 3 B-orbits in P3, and provide

the statements in (ii) regarding P32 and Paa'

31

Consider the set of paths z°-— y - x - w, for a fixed
w e Pma' The stabilizer of any one such path is a copy of
"3’ as was noted in (2‘15). In the same proof we saw that
the set of lines <x,w>, for x e Pszon,one of these paths,
determines a set of mutually nonperpendicular points in A".
Thus there are at most 9 such paths, and in any 8 n Bw-orbit
of these paths, a point stabilizer is induced by a copy of
”8. However, by (zyle), there are at least 4 of these paths.
We conclude that there are exactly 9, with 6’ n 6" an 119
acting naturally.

Finally, any v e P" must centralize some 1: e P" n P2".
From (Li) we conclude that P" C P2" U P3.3 U P¢.5' We
have observed that each such point is connected to a quad to

which 20 is connected. This finishes the proof. D

(2.19)£ROPOSITION Leg 6 he ef hype 01.

(i) vaeowarmewePa'z. mm‘ffi
cehtaihee in ehe unigue he; eh w egg 20; e; elee v e P‘J.
_1_ she latte; w x :- [v,zo] - (v20)2 6 P" n P2,2’ whehee
w e $;Jn ehe ehigue ghee eh v £h§£ Le hoggalized hy zo.

(inheiveP'fgsemewePaJ.mxz-P'anehe?
he he ehe unique ghee eh w ghee ie ngggalized hy zo. Aeeeme
v e <w,x>. Exactly eee e; ehe followihg three cases eeh
2222;.

 

 

v dees hot cehtgelize x, egg v 6 P45;

v doee eenggalige x, bet he; i, egg exectly one 9: v

ehe vw Lies ih Psa’ yhile Eh; other lies Le P46;

 

v does centralize f, ahd v e P“.

 

32

(iii) P, mists 2f M): {921; 81121355:

P”. 9: 1mm c 21°337; ml: 9.: mm mm h_as 10
neighbors _i_h P3 3’ 25 in P3 ‘, end 200 in P‘ 6;

P‘ 3, efi length dividing 2“s.7.135;

1422

P”. 2fl_.g_hent 2 357; Aeahpainshmhaa3
neighbors 13 each 9; P43 egg P3 ‘, 96 in P‘
.42936igPH;

5

1032

P , leengch 357; eachpeihgeheLehee3

4,4
neighbors in?3 2, 1921hP‘ 6, egg 36131:”.
(imifvePH—mthe __:.ethe $253line§envmatsenmg
Imagsmzmz’”; mzzazmml
9.21M from PHQQZ from P‘ 6.

M We show first that if v e P‘, then v20 is one of
exactly four possibilities, with digressions to finish off
(i) and (ii).

From (2_,__l§) we may choose a quad 5° to which both v and
20 are connected. Say y 6 P1 0 :P and w e P" n 9. Assume
first that both v and 20 act nontrivially on :P. If their

product induces a 5-cycle in ”a then in fact v e P as

noted. Otherwise their product induces a 3-e1ement with
fixed points in 116 (a 3-cycle, as in the W). and
moreover 6$(v,zo) consists of a single point 1:, say. In Bx
it is easy to check that {(vz°)3,(vwzo)3) - <x>. This
finishes this first case and also gives (11). Moreover this
says that if v e PM5 then x - (v20)3 and v generate a
hyperbolic line; and the quad 9", contains all of the
neighbors of v in PM. Thus #2” - 2933527.2.25/3.

If 86(9) contains either v or 20 then we claim that
|vz°| - 4. We establish this together with (1). Say 20
centralizes :P, and let I! be the hex on w and 20. Now 20 e 6"
is of type a‘, so that v’ :- v20 e Qw and v’ commutes with

v. Since I! 2 BO (20), v - v’ would imply v e R. Rather it

33

must be that x :- vv’ - [v,zo.] e [Q',zo] s it. Thus :1” :-
Yv : is the unique quad on v that is normalized by 20.

Moreover, v induces one of the 2-central Illa-involutions in

Km: , whence x - 20".2o is at distance 2 from 20, and
collinear with w.

As this argument shows, if v is any point collinear
with w but not in P2.2 U P3,z’ then v 6 Ph‘.

From the above v 6 PM‘ is connected to precisely 3
points of P3 32 ~ _11. the 3 neighbors of v in 3" ~ whence
#P‘ 4. - 26 33 5. 7. 240/3. Assume now that z e P, but 2 is not
in 9’. Either z centralizes :f’ or else 2 induces an
its-involution with v as singular center. Since the center x
of 20 (20 as a transvection on a?) is not collinear with v,
|zz0 | is a multiple of 6 whenever z e 860’).

Now count 2neighbors to see that each point v e P 4. is

connected to 2 223 points of P3 4. and 2 I53 points of P s

“9
while each point w 6 P3 4. is collinear with 23 32 points of
P , 25 from each of P and P , and 2 from P . For
4, 4 6,3 ,6 4.5
example, ifveP‘, setx-(vz),and$-:fx. Now

a n e -2“"“32 has just 3 orbits on L. the 3 in w, the 36

not in" .‘P that centralize :f, and the 96 that do not

each point v e P 6 is connected to 22 32 points2 of P

centralize :19. On the other hand if w 6 P3” then any
neighbor of wthat centralizes 9",sz must lie in PM"
Hence, the 36 lines of Lv that do not lie in 9’ but that
centralize :1“ must contain one point each from Pan.’ We claim
that the remaining neighbors of v ~ those that do not
centralize 9 ~ all lie in Pus' Again it is enough to see
that w has at least one neighbor in P"

For this begin with the observation that v20 and x‘1
are collinear points of 9, whenever u e P” acts nontrivially
on 3’ (ef. the Appendix). Set w - v'mx"; this is a point of
P3.2 that is collinear with each of v and x, and is
centralized by u as well as 20. Hence there is a quad :P’ on

u and a hex J" on 20 whose intersection contains w.

3A

In A”, then, we can find a unique hex R" containing 9’
whose intersection with R’ is a quad 9". Now 20 e K’ must be
connected to :P", as must be u e X”. Necessarily, u e Pms’
as we have seen in this case.

With this one calculates that the centralizer 6’ n I?" of
a point v of PLB ~ lying in the normalizer of both of the
hyperbolic lines <v,x> and <x,zo> ~ has shape
3 x 21“[£2(F2)x £2(F2)] and hence has exactly 3 orbits in
PV: those of lengths 3 and 36 just mentioned, and the one of
length 96 consisting of points that do not centralize :Pv'x.
Each of these lines contains, besides v, a point each of
1,4,5 and Pms'

Now counting the points of P‘ 5 in three ways one sees

that #P‘ 5 - 216335.7/n, say, where each point v e P‘ 5 is

collinear with 2:2 points of P3 3, 512 of P3 ‘, and 23512 of

Pms' If v is collinear with w e P3,3’ then 8g(v,w,zo) a
11‘ x 115 5 119 s 03%), stabilizing a (4,5)opartition of the
9-set. Since n is at least 1, 2 divides the index of
86(v,w,zo) in 8§(v,zo). This+leaves only a subgroup 1132 as
a possibility for 6’ n 6’” 5 03(F2)’ whence n - 5.

The fact that n - 5 implies that if <v,v’> is a line,
with v e P".6 and v’ 6 Pas” then vv’ 6 P‘ 5' This finishes

the last of the P‘ 5 parameters, and thus the proposition. [:1

(2,20)W(i) A consequence of the results of this
section is that if :P is a fixed quad, N is its normalizer,
. and P is the set of points connected to :P, then the Hecke

:f
algebras 8ndN(l-'2P$) and Endgfl'zP) are isomorphic (as

Fz-msss).

(1,1) It can be shown that in ol each line on a point

v 6 P4 3 contains 1 point each of P‘ a and P3 4’ This gives
the remaining parameters for P, and demonstrates that P has

diameter 4. Moreover if K. :- P‘ U P‘ s’ then K. has the

3
property that each line that contains a point of K; in fact
contains exactly 2. This will be noted again in the next

section.

35

13119.2

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3. THE ADJOINT REPRESENTATION FOR GROUPS OF TYPE 01
(AND RELATED REPRESENTATIONS)

magma“ Vin—insema amflm -1
(cf. (2‘1)), giph phg assumption (2plg) phpp 1p; pdjoint
geometzy 1g gpppepted.

Retaip pp; notation pf §§l-2. Much pi phig Lg gixpp 1p
(LA), (LA), Q51 (Lg). Throughout we will mean by $204)
the subspace of 8nd(‘) consisting of symmetric matrices ~ as
opposed to the appropriate gpppigpp of 8nd(d). Although this

involves a choice of basis, 82(31) will arise only in

 

situations where the action theron is induced from a

symplectic representation.

(3‘1)DEE1NIILQN§ ANQ NOTAT ON We give another description of
the homology module H6(P’) for P’ - (P’,L’) a geometry where

the lines have 3 points each (pf. (2,15,v)). Consider the

(deuLP’-)permutation modules FZP’ and FZL" Define maps:
*
FZL’ —0-)F2P’, 1 i——-)Xp , and FzP’ -g->F2L’,p 1———-)21 .
p61 13p

Identify each of these modules with its dual through the
usual inner product; this identifies 0* with the dual of a,
as the notation suggests. In this setting, HB(P') -
cokow(a*). The surjectivity of a is equivalent to the
injectivity of 0*, in turn equivalent to the nonexistence
of a set K’ # O of points such that every line meets K’

evenly. P’ \ K’ is an example of a hypprplppg ggppipp, as
considered by M. Ronan. We use this in the following.

(M)W H0(Fh) v‘ 0.

(3‘3)3§MA35 Using (2plfi) we conclude from this hypothesis
that 9 - HUGS“) vi 0. At least in ol, the set P \ K. -

P \ (1,4,3 U Phs) is a hyperplane section for I‘, as was
mentioned in (2‘29). Thus HB(P) # O in this case. What
remains to do is establish the existence of a hyperplane
section P’\ K; for the hypepbolip geometry Ph. We could then
apply (gpli) to conclude that 3 fl 0.

36

37

Whatever K? might be, it is pp; K.. In fact, there is
pp B-invariant hyperplane section for the hyperbolic
geometry, as we show in (1‘11) below.

The anomaly of having two distinct hyperplane sections
is not unusual in geometries where there are both ‘singular'
and ‘hyperbolic' lines. For example the symplectic spaces
over F2 (or, analogously, any field of even characteristic)
have as hyperplane sections, in addition to the linear
hyperplanes of subspaces perpendicular to a given point, the
set of points of an orthogonal quadric. These are ‘linear'
only in the universal (homology) module for the hyperplane
section ~ 1.g., the orthogonal modules of l dimension
greater.

The near-hexagon on the octads for flg“has a hyperplane
section consisting of the set of all octads at distance no
greater than 2 from a given octad. However, the
representation of the near-hexagon in the ll-dimensional
factor of the Golay code does not realize these hyperplane

sections in linear hyperplanes.

(1.3)W M112 119151.128 Lat S be the set 015 quads and

H the set of hexes. The shadow geometry

A(H) :- (H,{H9|$ e 3}) over H is a partial linear space,
where we regard as H-lines the sets My :- H n A, for 9 e S.
The H—lines contain 3 points each.

Fix a hex K* and define K* as the set of all hexes X
that lie on paths of the form R* - z - X’ — 9’ - H, where
<z> - 16* n u' is a point,-:f’ - u' n u is a quad, but
9’ n in- {l}. K* can be shown to be the complement to the
unique N* :- N6(R*)-invariant hyperplane section of A(H).
For 6 - o1, K* corresponds to the set of cobrdinate frames
of the Leech lattice A, taken mpdplp 2, that are pp;
perpendicular (mpgulp 2) to the frame (that corresponds to)
R*. Verifying that H’\ K* is indeed a hyperplane section is
tantamount to sorting the hexes into N*-orbits, of which

there are 6: one at distance 1 from K*, and two each at

38

distances 2 and 3.

Rather than establish the nonvanishing of the module
H;(A(H)) as above, we show below how (3‘2) implies that this
module is nonzero. This approach involves only some
elementary local calculations in various homology modules.

Y. Segev has studied the geometry A(H) extensively. He
uses elegant geometric arguments to obtain delicate
information about the geometry. He then constructs a
concrete isomorphism of A(H) with the ol-geometry on the
codrdinate frames of A. A corollary to his work is the
existence of hyperplane sections in A(H). There is some
obvious overlap of our projects, although our aims are
somewhat different.

I thank Dr. Segev for useful conversations that we had
at the Noordwijkerhout meeting ‘Groups and Geometries', and
for providing me with a draft of his paper while I prepared

this manuscript.

 

(&)LM(1)AW5M§SM€Q§M:
Q 6* 1 11.
55: (r7 j :I________o
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mzsmmwgmMfim-m
Wmmmmgfim; 5+Q§n9£§§fll
ogphogona]. module M phg M 02072) a £‘(F2) 1,3 1’2,
6611666661602“) ttthgtttttittmathtrz-mum
mute 41mm: c 2U; Ifmmgmgttt
(3.1mm m 921% 11 m ”2.? 3.66 11* mm sect 6+ in .8.

 

/\
totally-31112133.: 3412M m-spass. ass! Lama 26
lg a 6-gipepsiona1 orthogopa; subspacg.
(ii) if? :-/\2(s) mmntmc-mss—flre.

themmme.lnmslfi.flo(fis)#0.

39

PRQQE We use the term S-ppipp to mean an element of 8; an
S-lipg is a shadow so,“ :- S n A5,," where L e L(R) and
K e H. Notice that the partial linear space A(S) has ‘lines'
in correspondence with the 7 points of 96303), and so a
universal cm-representation of A(S) is not exactly as in
(_._1_5_._2 V) or (1.3).

Given the definition of 53 above for the faces of a
fixed chamber of A it is not hard to cheek that the
definition extends equivariantly to all of A (pf.
[Ronan-Smith 10, (4.2), p. 142]).

We finish by showing that 55 has a nontrivial image in
25,. The point is that ”0(58) is the limit of the system {35,
so that {38 —) 6’ 2: ”C(28) must factor through H0658). See
the Regiprpcity of [Ronan-Smith 10, (1.2), p. 139]:
Ito-(6.21) a “0-,;(H0(§).1) :23: as am: shsef 922.: A ans! =1 an!
Fae-modulg.

Recall that (2,15,!) allows us to regard 9 as generated
by P, subject to the defining relations determined by L and
Lh. Fix :1? e S. The element zlAz’1 + zzAz’2 is the central
element of AZ(:P) whenever the two pairs {21,21} are
orthogonal hyperbolic pairs. Denote this by {y' Now if K is
a hex and f s R is a quad, then Ne($,u) induces a group 6 a
26$‘(F2) s 1112‘ such that H is uniserial for 6: :f s R~ s It ,
where X~ is the linear hyperplane containing the points of R
that are connected to 9. Thus 6 has a single fixed point in
(@(K). Perforce this is {y' which then lies in a copy of
‘X* 5 A20!) (the dual of the kernel of the cocycle for the
extension of R* over X that lives in the permutation module
of dimension 24).

The remaining terms of the subsheaf of Re can be
constructed straightforwardly from the terms for the quads
and hexes. See the proof of (;Ԥ) below. Note that it is
unimportant what actually appears in 8 at the vertices P of

A: 53 will map-to 28 in any case. B

4O

(anagram Human) .- o.

(11) mm P0010096, a e A mtg 3 she tttm 6W 9.: th_t
WMMA:

6’ 4 z 1
: g L

 

 

39;; l depptgs a tpiviai ppdplg; philg 2 ppp 4 depote
pgturai modules pf phggg gipgpgippg £91 phg respective
paraboiip iactors isomorphig _t_g 122(Fz) gag £‘(F2); 8’
indicates pp orthogonai module Eng; is twisted by p triality
automopphism fgpm pp; orthogoppi moduie A; ppg 4 1;
contained ip 8’ é§ a totally singpla; subspace.

 

P3003 (i) We show that 8%9 represents the geometry A(H).
That is, we show that for each R e H there is a fixed point

e 899 for N (R) such that e - 0 whenever

u C R K’ H"
{H,R’,H") is an H-line.Indeed for each R e H, the image of
58'“ 9 5A.“ a 8nd(ll) in 389 has a l-dimensional fixed-point

space for Nc(R); say this is F26

6 +6 +€

X'
It is straightforward to verify, exactly as in the
proof of (éifi) below, that 91? , i 6 I, contains the
1

following composition factors:

1 l 8

2 2 286+ .

4 4 662 2
11 ll ...

I

Thus, for example, there is a uniserial P‘-submodule 3‘ of 9
that contains fimrll and Eur; and has composition factors

Bur‘ a 4, 692, and 2 (6 is the F‘-semilinear module for

9‘62) 3 66, as before). Similarly, in 8 we find the

*
uniserial P‘-submodule 8‘ containing 11 of shape:

41

l 26 or 27 ...
2 6+
*
4 1 6 82 482 ...
*
11 11
I

*
In particular Ron?‘(F2,8‘®9‘) =1 Hoar‘m‘ ,8‘) - 0.
Now consider the H-line {H,X’,H"} on the vertex v; e 8.

Since 6 + e + e

R X’ H" e 8‘03‘ is centralized by 9‘, the
remark above forces ex + t + e

X' R" - 0. This concludes (1).

(ii): Let A be HB(A(H)) ~ nonzero, from (1) ~ and let
931 and V“be, respectively, the images in A of the
subspaces F2“ and Fzfly of the permutation module FzH. Define
V2 - X 71192. 72 cannot vanish, lest A - 2 7n; vanish. Thus
72 is generated by the 15 points 71:32 subject to the
relations determined by the lines of the natural module for
£2 a: t‘(F2) in which fan/1‘2 fixes a point ~ a presentation
for this 4-dimensiona1 natural module.

Similarly the sum 71 of the images of 731 under
elements of 8 - 91 must be the natural module for 6 (as this
acts through its quotient 02(2)) in which 91.11 fixes a
point (One could either repeat the argument of (2,1§,v), or
else invoke the general theorem of [Ronan-Smith 9, (4.1),

pp. 338-339]). D

(1‘1)REMAE§§(1) The notation 5A is meant to suggest a
connection with the Leech lattice A. Indeed for each a e A,

5A;a

of the fixed points for ”a 5 cl. We regard this as saying

is just the subspace 33.0 of A :- A mpg 2 that consists

that the pair 6,5A is ‘locally isomorphic to' the pair
01,5X . In proposition (;i§) we strengthen this to say that
A :- ”0(5A) ‘is locally isomorphic to' A (pf. the precise
statement in Ipplg 6 below).

42

(ii) The Mpg that HOGA) ’5 0 would, similarly, lead
to the nonvanishing of H3(P) through a nontrivial image of P
in 52(Ho(5A)). This is noted also in corollary (1.2).

(M)Wmcbsfl£m-I.MEA£02MQ§

definedinlsms (1.6). MMMMraboli 91915

salifiameries mH0(5A)l, 1.0mm;
1

taplp.

Tabie 6
.__g ipcai composition sepies' pf ”0(8A)

 

1 8' 8" 8’
* *
2 a 264 294 a
*
a 2 6 662 6 2
*
11 1 11 11 1
I V 17/" 91/7 W1 Z/‘y

 

REMARK Since Conway's group 01 is a group of type 01 this

result is a ‘local uniqueness' theorem for such groups.

mm; Set A - 110(5 A). Lemma (3,6) (that is .- 0) provides us
with identifications §A°v -—;+ 7*:5 A, i e I. Starting with

1
these the chart gives a Pi-series 71 s '1 s 11 s ill1 s 21,

i e I. The entries in the chart identify the composition
factors by their dimensions ~ a blank indicates 0 ~ possibly
in conjunction with other notational information which will
be explained as the proof develops. The proof is carried out
in twelve steps, one for each nonzero factor not given as a

sheaf term.

43

mp 1 Define '11 :- Z V‘Pn. This is a Pll-submodule. In

111/711 there is a sheaf over AV as follows. First,
11

If 57 s I , and 7/7 is a l-dimensional

11 6 11 6 11

P -submodule of V /Y . Next, 7 s V , and Z V? -
6,11 11 11 6 z 6 2.11

V , so that V /V s I /V is a natural 3-dimensional

2 z 11 11 11

module for P2 11' Finally, the P1 n-conjugates of I!"

generate the subspace of the orthogonal module V1 that is
perpendicular to V11. We summarize these calculations in the

following table:

1
6+ - (2 7491,11)flll - VII/v11 S vl/Vll
3 - (X v4’2,11)/v11 - VZ/vll

1 - 1r4/vll

6+ is the 6-dimensional orthogonal module for £‘(F2) a:
0:62).

The homology of this sheaf is the simple Ilia-module
denoted by 11 ~ the Md moduie (pf. [S. Smith 13]). The
fact that A i 0 forces the image, in 111/711, of the
residual homology module to be nonzero. Hence Uu/V11
contains a copy of the Todd module. Since Vn/V11 is
generated by the subspaces (V‘flnfl’n in this copy of the

Todd module, these are perforce equal, and we have verified

the first of the twelve entries: _t_zhg W M 7/7
9___ver I'IEEMMQMM’I-
Observe that '11 2 V , and that '11 n 71 has a

11
1’1 n-series l:1:6+. (We abbreviate a composition series by
such a sequence.)

All of the other steps follow this same line, and so we

finish the proof with but a sketch of the calculations.

44

gm 2 W‘ :- 2 72?‘ leads, in exactly the same way as
mp 1, to the sheaf:

*
2n6 -2
*
06

6 denotes the fi-semilinear module for $‘(F2), as before.
(The convention will be that the symbol for the dual of a
module will be the symbol for that module with a superscript
w’ .) Note here that only the subgroup 02(11‘) - 03(11‘) of ‘1!"
acts trivially. Also, 6 contains the 92 ‘- and

' *
’1 (subspaces 2 and 4, respectively. As Ho(‘ 2) a: 6 , the
second entry of the chart is verified.

V SI SI ,and' n7 -2:4.
2 6 11 6 1

Stgp 3 '2 :- X 7192 leads to

*
o4

*
4 n 202 - 2
(iv—<1 4 - 202
*
203
* 2 3*
Since 4 n 4 - 2, the 91 z-subsheaf 5:: 0—0—0
generates, in the homology of the above, a factor of
*
20Ho(§‘*) a: 204 . Again, the simplicity of the residual
homology module, together with the nonvanishing of A, gives
~ *
Vz/Vz a 204 .
v 2v,v; andv nv -1:3:203*.
2 1 6 2 11 g

D 2
202

mp _4_ 1‘ :- Z In?‘ leads to:

2n4-1

45

H (1 z) is a copy of the S-dimensional orthogonal module
0 0:0

for find-'2) at 050:2). This and the fact that 2 n 4 - 1 imply
that the homology of the above is a factor of 502. However,
there does not exist a copy of the symplectic 4 in 502 (for
the parabolic I" 11), so that this homology module is

402 a :z/v‘.

§Lsp _5_ 11 :- Z ['11’91,11]?1 - X 1291 leads to:
*
4

2

'k
4

whence to ill/V1 a: 8”, a triality-twisting of VI.
12',U;andflnv -l:6:4.
1 2 6 1 11 +

All of the major steps of the proof are now complete.

fipsps §-§ "Dualize" the arguments in spsps 3, 2, and i, in
that order, to obtain:

“Hz/V2 a: 204; III/'4. a: 6; and {Um/U11 =1 11.

Spsps 24.2, First define Z1 :- X 11291, and verify that
2/138’.Z is then stable for?,?,and? (22?
1 1 1 2 6 11 1 1
generates a constant sheaf for each).

Now 21 s A is stable for each of the parabolics and
contains the generators for A, hence A - Z11 - Z‘ - 22 -

21' CI

 

(3.9)003011fl1m is 13 self-__1dua 19:. 6.
(ii) 111s singular lines pf, 6' s95 gusdratisaiiy Q A:
[A,L,L] - 0 whenever L e L.

(iii) 52(5) r_p.r___§_e esen s m _QJLLLa i t 2mm £21: 6-

m (i) The Ms _6_ shows that there is a nonzero
map 5A -—-) A*, and so the reciprocity applies.

In particular 6 is represented in :Pp(A), so that
(without loss) 6 leaves 32(A) invariant.

46

(ii) and (iii) Let L - <z,z’> be a singular line. From
the proof of (M) observe that [A,zo] - V‘ S III - 68(2).
Now 2‘, s 7316(6) s v: s :13. The c-map FzP —9 32(8) S
8nd(A), z F—9 l + z (of trace 0) factors through H6(F),

since 1 + z + l + z’ + l + 22’ - (l + z)(l + z’) - 0. D

(3,10)gsnsgg Counts of the vertices of A and of the image of

these vertices in A shows that the map is indeed an

embedding.

As a final result we indicate how a proof of the
nonvanishing of the adjoint module that includes a

description of a hyperplane section will not be so easy.

(LAW—1312161114212 Ema-wmmm
MWMO 121'ng -1.

23992 Any B-invariant subset of P must be a union of various
orbits Pm“. Now there are hyperbolic lines that lie
entirely in P1, and hyperbolic lines entirely in Pas; and
for any point v not in P‘s, there is a hyperbolic line that
intersects v6 in precisely v. This gives the result.

To see these claims, first note that if y,y’ are in P1,
<y,zo> !‘ <y' ,zo>, then <y,y’> c P1 is a hyperbolic line.
Consider now w 6 P3 3 and x e P” n P2 ‘. Choose v,v’ e

P; n 8‘, [v,v’] - l, and such that <v,w>, <v’,w>, <vv’,w>,

and <x,w> are pairwise distinct. We have then that
<v,v’> c P‘5 is a hyperbolic line.
For x 6 P2 2, there is a v e P‘ such that <v,x> is a

hyperbolic line with vx e P“. Similarly if w 6 P32 resp.
P3“ there is a point v 6 P65 such that <w,v> is a

hyperbolic line with vw e P‘.3 U P‘s. And finally, if

v E P resp. P U P there is a point v’ e P such
6,6 6, 6,6 4.5

3

that <v,v’> is a hyperbolic line with vv’ 6 P‘s. D

4. CONCLUDING REMARKS

We became interested in .1 while trying to characterize
the Friendly Giant 96 of B. Fischer and R. Griess. 96 is of

Fz-type, with 8 of shape 2“2‘

-1. We have been investigating
an inductive approach to the adjoint module 86(F) for 96,
trying in particular to establish that this does not vanish,
based on the existence of the adjoint module for -1. As yet
we have not succeeded in this, although we can demonstrate
that if H3(P) is nontrivial then 96 admits a ‘nice'
2-modular representation. This representation is given by a
sheaf locally isomorphic to the sheaf of 02-fixed-points in
the Griess-module modulo 2.

The approach of the present paper does produce for a
group of type 96 a 2-local geometry as described in
[Ronan-Smith 8]. However it seems completely hopeless to
enumerate sufficiently the B-orbits in P to give the desired

nonvanishing as was done here. Indeed, the number of these

suborbits is at least:

wm/mz z 41.514...
S. Norton has suggested that there may be around 150
suborbits. We have enumerated most of those out to distance

4, and there seems to be no end in sight!

47

APPENDIX

APPENDIX

CALCULATIONS IN THE lS-POINT QUADRANGLE

It is perhaps easiest to calculate in us as it acts on
the 15-point quadrangle by viewing the quadrangle as the
transposi-tions of 65' Thus the singular center of (12)(34)
is (56); its axis is {(12),(34),(56)}. One sees that the
class of st ~ for involutions s and t with axes and singular
centers 1, m, p, q ~ is determined by the incidence

structure on {p.q,1,m}:

m1
Lbs—cmgsubr mmuemms

 

P ' q P q P ' q P q q P
o———- o—————o or or
1 - m 1 - m I 1 I m I 1
m 1 m
1 (12)(34) (1234)(56)
P P P
o————- o————-
I 1 1 1
m
o o
q q m q m
(123) (123)(456) (12345)

 

48

LI ST OF REFERENCES

LIST OF REFERENCES

. M. Aschbacher ‘Finite groups in which the generalized
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‘Overgroups of Sylow subgroups in sporadic
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49

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50

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M'11111171111111!flifilflfflflflfflfliflfliflfl'Es

9