MICHAEL A S C H B A C H E R *

F L A G S T R U C T U R E S O N T I T S G E O M E T R I E S

1. I N T R O D U C T I O N

This paper is concerned with the interaction between group theory and incidence geometry• Our notion of geometry comes from Tits. In [3] Tits defines a 9eometry over an index set I to be a triple F = (V, z, *) where V is a set of objects, z: V ~ I is a type function, and • is a symmetric incidence relation on V such that objects of the same type are incident if and only if they are equal. In the theory developed in [3], interest is focused on the flags and their residues; of most interest are the residually connected geo- metries admitting a flag-transitive automorphism group•

Let G be a group and ~ = (G i :ieI) a family of subgroups of G. Define F(G, Y) to be the geometry over I such that for i~I the set of elements of type i is the coset space G~\G, and with G~x. Gjy precisely when G~x c~ Gjy is nonempty. Evidently G is represented as a group of automorphisms of F(G, Y) under right multiplication, G is edge-transitive (in the sense that for each {i,j)~_ I, G is transitive on incident pairs (x, y) of type (i,j)), and F(G, ~-) possesses a flag of type I. Moreover, if F is a geometry possessing a flag of type I and G is an edge-transitive group of automorphisms of F, then there is a family Y of subgroups of G and an isomorphism of F with F(G, Y) commuting with the actions of G on F and F(G, ~-).

This construction is of interest for at least two reasons. First, given a family ~- of subgroups of a group G, we can hope to obtain useful information about G by studying its action on the geometry F(G, ~ ) ; we may even be able to show G is determined uniquely by using geometric methods• Such a classification theorem appears in [4], for example. Second, we may be able to go in the other direction and use group theory to study geometry. For example, in [1] and [2] a moderate knowledge of the subgroup structure of certain groups, together with the theory in this paper, is used to establish the existence of certain interesting geometries.

To even get started on either of these two lines of investigation, two questions must be answered: What are the flags of the geometry? What are the isomorphism types of the residues of the flags ? Certain flags are readily visible Namely, for J c I let G = n . G Then S. = (G "j~J) is certainly

• - - J j ~ " j " o j "

a flag of type J. Moreover, the geometry F(J) : F(Gj, ~'~I) is naturally asso- ciated to S j , where ~ j = (Gj~11:iEJ') and J' = I - J. If G is flag-transitive on F(G, ~ ) then every flag is conjugate to an Sj. In another direction, we

* Partial support supplied by the National Science Foundation

Geometriae Dedicata 14 (1983) 21-32. 0046-5755/83/0141-0021501.80 Copyright (~ 1983 by D. Reidel Publishing Co., Dordreeht, Holland, and Boston, U.S.A.

22 M I C H A E L A S C H B A C H E R

might hope that the residue of Sj is isomorphic to F(J). A few moments thought shows G is flag-transitive on F(G, ~ ) precisely when F(J) is iso- morphic to the residue of S s for each J __q I. Thus, if we can force flag-transiti- vity our troubles are over, Unfortunately counter-examples show that flag-transitity can fail even under strong connectivity assumptions.

This paper introduces the notion of a flag structure to deal with the questions raised in the last paragraph. Using the theory of flag structures, two results are proved which establish the flag-transivity of edge-transitive groups in two situations of sufficient generality to apply to the problems encountered in [1], [2] and [4]. The result with the nicest statement appears below as a theorem; the second result is more technical and is summarized in Lemma 3.8.

A diagram on I is a tuple

D = ( D j : J c _ I , ] J I =2)

such that Dj is a nonempty family of geometries on J. The graph of D is the undirected graph with vertex set I and i adjacent to j if J = {i,j} is of order 2 and some member of D s is not a generalized digon. The diagram D(G, Y ) of (G, ~ ) i s the diagram on I with D(G, ~ ) j = {F (J')}.

We prove:

THEOREM. Let G be a group, I a finite set, and Y = (G i :i~l) a family of subgroups of G. Assume:

(i) for each subset J of I of eorank at least 2, G s = ( G s,41~ : i ~ J' ) , and (ii) the connected components of the graph of D(G, ~ ) are strings. Then (1) G is flag-transitive on T(G, W). (2) F(G, @) is residually connected. (3) For each J ~_ I, the map

(Gju(ii)z~'-'~Giz i~J ' , zEGj

is an isomorphism of F(J) and the residue of Sj.

A graph on I is a string if we can order I = {1, 2, . . . , n} so that the edges of the graph are { i , i+ 1}, 1 <<.i<n; such an ordering will be termed a string ordering. Condition (i) is equivalent to the assumption that F(J) is connected for each subset J of I of corank at least 2. The most interesting geometries seem to satisfy condition (ii). For example, if G is of Lie type and ~- the set of maximal parabolics containing a fixed Borel group (truncat- ed by one member if G is of type D or E ) then D(G, ~ ) has a string graph. If G is a sporadic group of GF(2)-type distinct from He, and ~,~ is the set of maximal subgroups containing a fixed Sylow 2-group, then it is shown in [1] that conditions (i) and (ii) hold.

The discussion above is confined to geometries with edge-transitive

FLAG STRUCTURES ON TITS GEOMETRIES 23

groups, but similar results hold for geometries with more general auto- morphism groups. Such a situation is discussed in Section 4 and illustrated with the example of the geometry of points, lines, and quadrangles of the near hexagon defined by PE~6(3).

Lemma 3.8 addresses the question of when G is flag-transitive in the case where its diagram does not have a string graph. At the end of Section 3 we give an example of a pair (G, ~ ) for which the connectivety assumption (i) of the main theorem holds, but such that G is not flag-transitive on the geometry F(G, ~-).

We close this section by recalling some notation and terminology. A flag of F is a subset of V in which every pair of elements of V is incident. Let T be a flag. r(T) is the type of T, V r = {veV:v*t for all t~T} V r = V r - T, F r = (V r, ~l Vr, *l Vr) and F r = (Vr, ~[ Vr, * I v r) are geometries over I and v(T)', respectively, and F r is called the residue of T. The geometry F is connected if the graph (V, *) is connected and F is residually connected if the residue of every flag of corank at least 2 is connected and the residue of every flag of corank 1 is nonempty. 5:(F) denotes the set of flags of F. For i, jEI, x~ V, V~ denotes z- 1(0, Fi(x ) consists of the members of V i incident to x, and N.. is the graph on V. with x adjacent to y if F;(x) n F;(y) is non-

~,J empty. A rank 2 geometry over {1, 2} is a generalized digon if x * y for each x e V 1 and y ~ V 2 .

2. FLAG STRUCTURES

In this section F = (V, ~, *) is a geometry over a finite set I. Aflag structure on F is a presheaf

Z = (Y , ( 2 r : r e J - ) , (~Or,s : T , S s J - , T _~ S))

where Y _~ 5 : ( r ) and (FS1) Z r = (U r, F r, .r) is a geometry over I for each T e J . (FS2) For each T, Se~- with T ~_ S, Cpr, s :2 r ~ Z s is an injective morphism

of geometries. (FS3) For all T, S, R ~ - with T~_S~_R, q~r,R=q)r,s(Ps,R and Or, r =

identity. (FS4) ~be3--, Z e = F , T~_(ur)cpr , and v . u for each veU r and uercPr 1,

where (Pr = q°r,~" We abuse notation and write X for the inverse image in U r o f X _~ (ur)cpr ;

that is, we think of the restriction maps ,or, s as inclusions.

(2.1) Let T, S 6 Y with T ~ S. Then T~_(Ur)q~r,s ~_ (uS) r, and TeS:(Zs). Proof. This is a consequence of (FS4), (FS3) and the fact that q)r,s pre-

serves incidence.

For T e Y let ~- r = { S e Y :S _~ T} and for S e Y r let S r = S - T. Observe

24 M I C H A E L A S C H B A C H E R

(2.2) For each T ~ - - , Z(T) is a flag structure on (ET)T, where J-(Z(T))= {Sr :S~J -r} and for R, S ~ -'r with S~_R, Z(T) s~---(Zs)r and tpST,R =

(P S,R]Z(T) S T "

A flag structure Z on F is rigid if J = 5P(F). The next lemma shows that the only rigid flag structure is the obvious one.

(2.3) Let Z be a rigid flag structure and S, T ~ Y with T~_ S. Then q~T,S :ZT (ES) r is an isomorphism. In particular, q~T :ZT --~ FT is an isomorphism.

Proof. By (2.1), ~Pr,s:Zr~(Es) r is an injection. Let xi6(uS)r , i= 1,2, and R i = T w ( x i } . Then R i~50(F)=3 -, so R i ~ -T and q~T,s:Ri-~Ri. Thus ~Pv,s :Zr-~(Zs) T is a bijection. It remains to observe that if x~(*S)x2, then R = R 1 u R2E•(I" ) --- ~-, so R E y T and hence Xl(*T)x2.

A flag structure Z on F is dense if for each T~J- and each X~5~(Z r) with Tof corank at most 2 in X, we have X ~ J . It is evident that

(2.4) IrE is a dense fla9 structure and T6~-- then Z(T) is a dense flag structure on (ZT)T.

(2.5) I f Z is a dense fla9 structure on F then every flag o fF of rank at most 2 is in ~'-.

(2.6) Let E be a dense fla9 structure on F and X a flag of rank 3 in F. Then the following are equivalent:

(1) Xe~- . (2) X ~ ( X r ) f o r someflag T o f f of rank 1 contained in X. (3) Xs~9~(ZT)for each flag T o f f of rank 1 contained in X. Proof. This is an easy consequence of the definition of density and (2.1).

(2.7) Let X be a dense flag structure on F and X a flag ofF. Then the following are equivalent:

(1) X e J . (2) XE~(Xr)for each T e J with T c_ X. Proof. Part (1) implies (2) by (2.1). Assume (2) holds; we prove (1) holds

by induction on the rank of X. By (2.5) we may assume X has rank at least 3 and T e J is a subflag of X of rank 1. By (2.4) and induction on the rank of X, X T e J ( Z ( T ) ) , so X e J - by the definition of J(Z(T)) .

A flag structure X on F is residually connected if (Xr)r is connected for each T e J of corank at least 2 and (xr)r is nonempty for each flag T of corank at least 1. The structure X belongs to a diagram D on I if for each subset J of I of order 2 and each T e J - of type J', (XT)reD s. The minimal diagram D(Z) of a dense flag structure X is the diagram with D(Z)j consisting of the isomorphism classes of all (zT)T as T varies over the flags of 3-- of type J' . Notice that as Z is dense, such flags exist.

The next two lemmas are essentially Lemmas 6.1.2 and 6.1.3 from Tits

F L A G S T R U C T U R E S ON T I T S G E O M E T R I E S 25

[3], except they are stated here for flag structures ra ther than geometries.

(2.8) Assume F admits a residually connected dense flag structure Z. Let J~_I with IJI >12 and x, yeV. Then there exists a path (/)~ : 0 4 i ~ n ) in (V, *)from x to y with z(/)1)6J for each i in the range 0 < i < n.

Proof. Choose x and y to be a counter-example with distance d = d(x, y) in (V, *) minimal. As £ is residually connected, F = (Z°)4 ' is connected, so d is finite. Clearly d > 1. Let (q: 0 4 i ~< d) be a path f rom x to y. By minimali ty of d there is a path (Ui:0 ~<i~ < m) from /)1 to y with z(ui)eJ for 0 < i < m. Now if z(vl)cJ then x,/)1,ul . . . . . u m = y is the desired path, so assume "c(/)l)~J.

Let T = {/)1 }" By (2.5), Te~--. Then Z(T) is a residually connected dense flag structure on (ET)r and by (2.5), x, u I ~(UT)T . By induct ion on the rank of I, there is a path (wi:0 ~<i ~<k) f rom x to u, in (ET)T with z(wi)eJ for 0 < i < k . Then x = w o,w 1 . . . . ,w K = u l , u 2 , . . . , u m = y i s the desired path.

(2.9) Let E be a residually connected dense flag structure on F belonging to a diagram D, and suppose I = 11 + 12 is a partition of I such that I i is the union of connected components of the graph of D for i= 1 and 2. Then x I *x 2 for each pair of elements xifrom V with z(xi)e I ~ .

Proof. We may assume I i is nonempty for i = 1 and 2. If ]Ii[ = 1 for i = 1 and 2, then the empty set is a flag of corank 2 in ~-- with E ~ = F, so the lemma holds by the definition of the graph of a diagram. Thus we may take 11 of rank at least 2. By (2.8), if x~e V with z(xi)eI i there is a path P = (/)i : 0 <<.i<<.n) f rom x~ to x 2 with z(/)~)eI 1 for 0 < i < n. Choose n minimal. If n = 1 we are done, so take n > 1. Let r = {/),- 1}, J = z(v,-1)' and E = DIs. Then E(T) is a residually connected dense flag structure on (ET)T belonging to the diagram E. By (2.5), v _ 2 , ye(Ur)T , SO by induction on the rank of I , / ) _ 2 * Y" This contradicts the minimali ty of n.

We have the following corol lary of L e m m a (2.8).

(2.10) I f F admits a residually connected dense flag structure then ~i,s is connected for each subset {i,j} of I of order 2.

(2.11) Let Z be a dense flag structure on F such that Z(T) is rigid for each flag T o f f of rank 1. Then the following are equivalent:

(1) Z is rigid. (2) 3"- contains all flags of rank 3. (3) Each flag X of rank 3 contains a subflag T of rank 1 with X~SP(zT). Proof. Par t (1) implies (2) and (3) trivially. By (2.6), (3) implies (2). Assume

(2) and suppose X E S ~ ( F ) - - J . By (2.5) X is of rank at least 3. Let X = { x 1 . . . . , x }, T = { x l } and T i = { x l , x i } , i > 1 . By (2) and (2.1), T i ~ T j ~ ( Z T ) , so as Z(T) is rigid X = u T / e J -T. Thus X E J , cont rary to the choice of X.

26 M I C H A E L A S C H B A C H E R

(2.12) Assume F admits a residually connected flag structu,e belonging to a string diagram and order I according to a strin 9 ordering. Let {i,j, k} ~_ I with i < j < k and x ~ V , r~{ i , j , k} , with x i * x ~ and x j * x k. Then x i * x k.

Proof. Apply (2.9) to (12T)r, T = {xj }.

(2.13) Let E be a dense residually connected flag structure on F belongin 9 to a diagram D such that the connected components of the graph of D are strings. Then Z is rigid.

Proof. Let T be a flag of rank 1 and E = D Ix(r)' Then E ( J ) is a dense residually connected flag structure on (12T)r belonging to the diagram E, and the connected components of the graph of E are strings, so by induction on the rank of F, Z(T) is rigid. Let X be a flag of rank 3. By (2.11) it suffices to show X~SP(E r) for some T of rank 1 in X. Let X = {xi, xj , Xk}, with z(xr) = r and i < j < k, where the components of the graph of D are string ordered. Let T = {x .}. Then ~(xi) and Z(Xk) belong to different components of the graph of E, so Joy (2.9), xi(* r)xj. Thus X s 5a(Zr), as desired.

(2.14) Let 12 be a dense flag structure on F and assume ~0T,S:Zr~(zS)T is an isomorphism for each T, S 6 ~-- with T ~_ S. Then 12 is rigid.

Proof. Let T be a flag of rank 1. The hypothesis is inherited by the flag structure 12(T) on 12(T) r, so by induction on the rank of F, 12(T) is rigid. Let X be a flag of F of rank 3 containing T. By (2.11) it remains to show X~5~(zT). But by hypothesis ¢Pr :Zr ~ F r is an isomorphism, so the remark holds.

(2.15) Assume the hypothesis of (2.9) and let T i be a flag in ~--(Z) of type Ji with di ~- Ii, i = 1, 2. Then T 1 u T2 ff~-(~] ).

Proof. The proof is by induction on the rank of T 1 w T 2 . The lemma is trivial if T 1 or T 2 is empty, so let T and S be subflags of T~ and T2, respectively, of rank 1. We have T 1 E~ --r and T 2 ~ j s , and as I2 is dense, R = T u S e Y -s by (2.9). Now by induction on the rank of T 1 u T2, RsU(T2)seJ-(12(S)),

so T u T 2 = R U T z e ~ --s. Hence, T ~ T 2 e J - r also. Now by induction on the rank of T 1 w T2, ( T ~ ) T ~ ( T w T2)TGJ-(~.(T)) , so T 1 u T2~J- .

3. E D G E - T R A N S I T I V E G E O M E T R I E S

In this section G is a group and ~ = (G i :i~1) is a collection of subgroups of G. Define F(G, ~ ) to be the geometry on I such that for iEI, the set of elements of type i is the coset space G/Gi, and with Gix * G~y precisely when G.x c~ G.y is nonempty. Evidently G is represented as a group of auto- t .J morphlsms of F(G, ~ ) under right multiplication. Moreover, we have the set of flags

Sj, ~ = {G jx : j 6J } J ~_ I, x e G

F L A G S T R U C T U R E S ON TITS GEOMETRIES 27

with Sj,~ of type J, and G transit ive on the flags Sj,~, x~G. So as every flag of rank at mos t 2 is of this form, G is edge-transi t ive on F(G, ~ ) , and G is flag-transitive precisely when every flag of F(G, ~ ) is an Sj..x. Moreover , S I ~ is a flag of type I. Evidently the converse is t rue also; that is:

(3.1) Let F' = (V', z', *') be a geometry admitting an edge-transitive group G of automorphisms, and assume T is a flag o f F of type I. For ie I let tie T be of type i. Let G i = Gt,, ~ = (G i :iel), F = (V,r, * ) = F(G, ~ ) a n d define

~'V'--* V

tig~--~Gig i~ I ,g~G

Then c~ is an isomorphism of geometries commuting with the actions of G on F and F'.

Thus the g roup geometr ies F(G, Y ) are precisely the edge-transit ive geometr ies possessing a flag of type I. Fo r the rest of this section let F(G, i f ) = F = (V, ~, *), and assume I is finite.

(3.2) Let H = ( G i : i ~ I ) and U = {Gih :heH, ieI}. Then U is the connected component of each Gi in (V, *). In particular F is connected if and only if G = H .

Proof. Let W(Gix ) be the connected c o m p o n e n t of G~x in (V, *). We have l e G i c~ Gj, so G i* Gj. Thus W(Gi) = W(Gj) = W for each i,j~I. Hence G i <~ NG(W ) for each i, so H <<, NG(W ) and thus U ~ W. Conversely, let g~G, h~H, and Gig*Gih. Then Gigh- l*Gj so there is x~G~nGigh -~. N o w gh- ~ = yx, y E Gi, so g = yxh ~ H. Hence G~g ~ U. It follows that W = U.

If H = G then, by definition of V and U, V = U, so that F is connected. Converse ly if F is connected then V = U, so for g~G and i e I there is hEH with Gig = Gih. Then as G i ~_ H, gEH. Thus G = H.

Let ,Y- = {Ss, ~ :J ~_ I, xeG}. For T = Sj, s,Y- define Z T = (U T, z T, .T) to be the geomet ry on I with (zT)-I (J)= T, t*u for each t e T and u e U T, and (UT)T the g roup geomet ry

F(J) = F(Gj, f f s )

where G j = ~jEjGj a n d ~s=(Gj~{ i ) : ieJ ' ) . Notice that if S=SK,ye3 - then S __c_ T precisely when K ~ J and ye(Gs)x.

In that event we define

q~T,S" UT ~ US

(G~)x~--~(GK)y k E K

(Gj)xr---~GKw{j } j ~ J - K

(Gj~(ii)z ~--+(G~i))z i6K, z6 Gj

To simplify notat ion, in this p a r a g r a p h let A = G~:, B = G j , and for L __q I,

28 M I C H A E L A S C H B A C H E R

let A L = A c~ G L and B L = B c~ G L. As B L c_ A L ' q? T S is well defined. Further if z e B then zeAjc~(Ai)z for each j 6 J and ie l , so ( u T ) ~ O T , S ~_~ ( u S ) T" with T ' = Tq)T, s. If Z, w e B and r, s eK ' with (A )z = (A)w, then r = s and z w - l e A c~B-= B , so (B)z = (B )z. Therefore, qOr, s is an injection. Finally, if(B)z* (Bs)W there is ue(Br)z c~ (B)w. As B ~ A and B <. A , uE(A )z c~ (A )w, so (A ) z* (A)w . Hence, (Pr,s is a morphism of geometries. If T~_S~_R then evidently q)T,Sq)S ,R = q)T R and (PT T ~T identity.

Let Z = E(G, i f ) be the 'triple (T,'(Z • r e T), ((Pr,s" T, SexY-, T ~ S)). We have shown:

(3.3) Z(G, o~) is a dense flag structure on F(G, ~) .

(3.4) E(G,W) is residually connected precisely when Gj--(Gsu{i~: ieJ ' ) for each subset J of I of corank at least 2.

Proof. T = Sj ,xeJ- is of corank at least 2 precisely when J is of corank at least 2 in I and, by (3.2), (E r) is connected precisely when Gj = (Gsu{i ~:ieJ ' ) . If J :p I then Gixe(Er)r for i e I - J. Hence the lemma holds.

(3.5) The following are equivalent: (1) G is flag-transitive on F(G, ~) . (2) Z(G, ~ ) is rigid. (3) For eaeh flag T = Sj in J .

q~r :F(J)--* F T

(Gj~i~)z~--~Giz ieJ', z eGj

is an isomorphism of geometries. Proof• G is transitive on the flags in Y- of type J for each J ~_ I, so (1)

is equivalent to (2). (2) is equivalent to (3) by (2.3) and (2.14). Let D(G, ~ ) be the diagram on I with Dj -=- {F(J') }, for J a subset of I of

order 2. Then as G is edge-transitive on E(G, ~) , D(G, ~ ) is the minimal diagram of E(G, ~) .

(3 6) (i,j) is an edge in the graph of D(G, ~ ) precisely when G . , ¢ G G . • { i , j / . i ' j '

Proof. G .... = G. G. if and only if G. is transitive on the j-objects in (t ,J~' , l ' J ' l t .

F( {i, j}'). As G .... is edge-transitive on F( {i, j}'), this holds precisely when each "tt,J~' . . ! /-object is incident to each j-object in F( {i,j} ).

(3.7) Assume Z(G, ~ ) is residually connected and I = J + K is a partition such that J and K are the union of connected components of the graph of the minimal diagram of Z( G, ~ ) . Then G = G j G K .

Proof. Let S = St, 1, and T a flag in J - of type J. By (2.15), T w S e J . In particular, for x e G and T = S j , x , T w S e Y - , so T w S = S~,~ for some yeG. Then Sj,~ = Ss,y and St, a --Sr,r. The latter implies YeGic and the former implies x E G sY. So x e Gsy ~ GsG K and thus G = GsG K.

F L A G S T R U C T U R E S ON TITS G E O M E T R I E S 29

Lemma (2.13) and the discussion in this section establish the main theorem. Thus we have good control over F(G, i f ) = F when the diagram D(G, o~) = D has a string graph. We next establish a lemma which gives flag-transitivity for arbitrary D when the action of the groups G i ~, i, j e I , on F(i) is suitable. The lemma has some obvious generalizations, b~t the version here suffices for applications in [1] and [2]. For i~I let D(i) = D(G~, ~ ) .

(3.8) Assume E is residually connected, G i is flag-transitive on F(i)for each iEI, but G is not flag-transitive on F. Then there exists a flag of rank 3 in &~(F) -~--, while if T is a flag of rank 3 in SP(F), then either of the following imply TE~- :

(1) There exists an ordering (i,j, k) of z(T) such that j and k are in distinct connected components of the graph of D(i).

(2) There exists a subset A of z(T) of order 2 such that." (a) for each i~A there exists an ordering (j,k) of z ( T ) - {i} such that

G. . has just two orbits on the coset space Gi/Gi, k, and (b) G"~ GiG ~ for r~z(T) - A and for some i~A.

Proof. As G i is flag-transitive on F(i) for each i s I , E(S) is rigid for each flag S of F of rank 1 by (3.5). So (2.11) implies ~-- does not contain all flags of rank 3. Let T be a flag of rank 3 not in Y, and let te T. By (2.11), T¢Se(Et). Hence (2.15) implies that (1) does not hold. Assume the hypothesis of (2). By edge transitivity we may take T = {G,, Gi, GkX } with i~A and x~Gi, j. As G has two orbits on G/G , every member of F (G) is conjugate under i , j i i ,k k i G i,j to G k or GkX, so as both are incident to Gj . . . . in F we conclude Fk(G ) c Fk(G1). By transitivity of G i on F~(Gi), we also have F~(Gi) ~ Fj(GkX ). By symmetry, we have F(t)_~ F (Gi), where A = {i,s}, r e z ( T ) - A, and tET with z(t)= s. Thus Fr(t ) = Fr(Gi). Then since fqi,s is connected by (2.10), we conclude F(t) -- Fr(Gi) -- V. Hence, G = GiG ~ = GsG, contrary to (b).

We close this section with an example. Assume I = {1, 2, 3}, and either: (a) for each ordering (i,j,k) of I, there is a normal subgroup K i of G i

contained in G I such that GJK i ~- L3(q) and that Gi,/K~ and Gi,k/K i are the distinct maximal parabolics of G]K i, or

(b) G ~ L3(2) and o ~ is a family of three distinct subgroups of order 21. Consider (a) first. Then F(i) is the geometry of points and lines of the

projective plane of order q. In particular, the hypothesis of (2) in (3.8) are easily verified, so by (3.8), G is flag-transitive on F and hence by (3.5) the residue of each object of F is a projective plane. Hence, in the language of [3], F is a geometry of type M, where M is the Coxeter diagram represented by a triangle. It is not known if finite flag-transitive geometries of this type exist; Timmesfeld [43 has shown none exists if q is even.

Next consider (b). This example is due to Franz Timmesfeld. Let H = Aut(G) - PGL2(7), and observe H acts 3-transitively on the set f~ of subgroups of G of order 21. Hence Gi= (Gi,j, Gi,k) for each ordering (i,j,k) of I,

30 MICHAEL ASCHBACHER

so F(G, ~,~) is residually connected. Again each residue F(i) is the projective plane over GF(2), so if G is flag-transitive on F, then F is a geometry of type M whose Coxeter diagram is a triangle. However Steve Smith has observed that G is not flag-transitive, since each /-object is incident to exactly seven of the eight j-objects, and hence each flag of rank 2 is contained in six flags of F of rank 3, but in only three flags of J - of rank 2.

4. O T H E R GROUP GEOMETRIES

In this section G is a group, ~ is a family of subgroups of G, P -- (~ i "ieI) is a partition of o~, and ~ i = (Gt :teAi) is indexed by A~. Let F(G, ~ , P) be the geometry on I such that for i~ I the set of elements of type i is the disjoint union of the coset spaces GIGs, t~Ai , with G x* G f f precisely when G xc~ Gf f is nonempty and ~ = fi if G and G~ are of the same type. For j_c I let A(J) be the set of subsets of us~sA ~ = As containing a unique member of As for each j e J. We have flags

Ss,o, x = { G x : ~ O } J__ I, 0cA(J), x e G

and subgroups G o = c ~ o G . We also have families ffs,0 with partition Ps,o = {~s,o,i "ieJ'}, where

~s,o,i = (Gou(t) : teAi)

and the group geometry

F(Go, ~ s,o, P s,o) = F(J, 0).

Define Y to be the set of flags

J- = {Ss,o, x : j c I, 0EA(J), x e G }

and for T = S j , o ,~Y- define E T= (uT, z T,*T) to be the geometry on I with ( z r ) - l ( J ) = T, t*u for each t e T and u e U r, and (Ur)r the group geometry F(J, 0) For S = S n, e J ' , we have S _~ T precisely when K _~ J, f~ __ 0, and y e Gox, in which case define

~OT, s : uT~--~U s

Gtx ~ GtY t E

Gtx ~ Gn~it) t ~ 0 -

(Gou(ti)z~--~(Ga~(t))z t e A j , i eK ' , z ~ G o

Then as in Section 3 we define E = Z(G, i f , P) to be the triple

( J , (YY : r s J ) , (~oT,s : T, S e J - , r ~_ S) )

and we have

FLAG STRUCTURES ON TITS GEOMETRIES 31

(4.1) E(G, ~ , P) is a dense flag structure on F(G, ~ , P).

(4.2) Z(G, ~ , P) is residually connected precisely when

G o = ( H : H ~ Y s , o )

for each subset J of I of corank at least 2 and each 0cA(J).

The minimal diagram of Z(G, ~,~, P) is the diagram D(G, ~ , P) where

D(G, ~ , P)j = {F(J', 0):0EA(J')}

for each subset J of I of order 2.

(4.3) (i,j) is not an edge in D(G, J , P) precisely when for each O~A({i,j}'), each a e A i, and each ~eAi , G o = (Go~(,~)(Go~a~).

Here is an example illustrating this construction. Let G be the group Pf~2(3) extended by a reflection r. Let TeSyle(G ) with Cr(r)eSylz(CG(r)) and let Gi, 1 ~< i ~< 3, be the maximal subgroups of G containing T. We may choose notation so that GI ~ A 6 / E 3 2 , G 2 = CG(Z(T) ) , and G 3 _----- $6 /E16 .

Here, E2. denote a product of n copies of the group of order 2 and X/E2, represents a group H with normal subgroup K ~ E2, and H/K ~-X. Let G3, , = G 3 and G3, 2 = C~(r) ~ Z 2 × Pfls(3). Let Y~ = {Gi}, i = 1, 2, ~ 3 = {G3,1,G3,2}, ~ = u ~ = l ~ i , and P = ( ~ 1 , ~ 2 , ~ 3 ) . It is easy to check using (4.2) that X = Z(G, ~ , P) is residually connected and using (4.3) that D(G, ~ , P) has a string graph, so by (2.13), X is rigid. Hence, by (2.3), the residue F x of a flag X of type 1 or type 3 is isomorphic to (XX)x, from which it is easy to check that F belongs to the diagram D where D~, 3 consists of generalized digons, O1, 2 consists of generalized 4-gons, and D2, 3 consists of linear spaces. In Buekenhout's notation this is represented as:

L ~ 1 o 0

2 3

Indeed the rank 2 geometry F(G, {G1, G 2 } ) is a near hexagon in the termino- logy of Shult and Yanushka, and the rank 3 geometry F(G, ~ , P) is the geometry of points, lines, and quadrangles of this near hexagon.

REFERENCES

1. Aschbacher, M. : Parabolics in groups of GF (2)-type (Preprint). 2. Aschbacher, M., and Smith, S. : 'Tits Geometries over GF(2) Defined by Groups over GF(3)'

(to appear in Comm. Alg.). 3. Tits, J.: 'A Local Approach to Buildings' in The Geometric Vein (the Coxeter Festschrift),

Springer, 1981, pp. 517-547. 4. Timmesfeld, F. : Tits geometries and parabolie systems infinite groups (Preprint).

32 MICHAEL A S C H B A C H E R

(Received: June 28, 1982)

Author's address:

M. Aschbacher, Department of Mathematics, California Institute of Technology, Pasadena, California 91125, U.S.A.