On amalgamation of rank 1 parabolic groups

On Amalgamation of Rank 1 parabolic Groups

by

F.G. Timmesfeld

§ 1 Introduct ion.

Abusing the notat ion of parabolic subgroups, we cal l a f i n i t e group P a rank n

parabol ic group of char.p, i f and only i f P = oP'(P/Op(P)) is a perfect cen-

t ra l extension of a f i n i t e simple rank n Lie- type group in char.p or one of

the fo l lowing exceptions:

PSL2(3 ) and 2G2(3 ) i f n = I , p = 3

PSL2(2 ), PSU3(2 ) and Sz(2) ~ F20 i f

o r

n = 1 and p = 2

A6, z 6 = Sp(4,2), G2(2 ) ' = U3(3), G2(2), 2F4(2 ) ' and 2F4(2 ) i f n = 2 and

p = 2 .

I f P is a rank n parabolic group in char.p, then a Borel subgroup B of P

is j us t the normalizer of a p-Sylow subgroup. I t is obvious that B n oP'(P)

projects onto a Borel subgroup of the Lie-type group P. In th is paper we are

mainly concerned with the embeddings of a "possible" Borel subgroup B into

rank 1 parabolic groups. For th is end remember that a group B is p-closed i f

i t has a normal p-Sylov; subgroup. Wekare now already in the. posi t ion to formu-

late our:

Geometriae Dedicata 25 ( l 988) , 5 - 70

© 1988 by D. Reidel Publishing Company

Op(Pi)' ~i : oP'(Pi)

fo l lowing holds:

6 F .G. TIMMESFELD

Theorem 1. Let B be a p-closed group and {Pi [ i E I }

parabolic groups of char.p. Suppose there exists a fami ly

monomorphisms:

Xj :B ~ P i ( j ) ; where j ~ i ( j ) is a map from J in I

such that , x j (B) is a Borel subgroup of P i ( j ) fo r each

and N i = [Mi ,oP(~i) ] for i E I .

a set of rank 1

{Xj l J EJ} of

j E J. Let M i =

Then one of the

(1) IJl ~ 2

or (2) There ex is ts a pair j ~ k E J such that

× i l ( N i ( j ) ) ~ xk l (M i (k ) ) .

Theorem 1 is somewhat abstract. To make i t more e x p l i c i t we state some

"concrete" coro l la r ies and also an equivalent more "geometric" version of

Theorem 1, which ac tua l l y was the or ig in of that theorem. The proof of Theo-

rem I depends on the c l ass i f i ca t i on of weak BN-pairs of rank 2 by Delgado-

Stellmacher in [4 ] , a forthcoming paper of Delgado describing the f a i l u re of

fac tor iza t ion modules for the groups with a weak BN-pair of rank 2 and on

certa in "amalgam-type" arguments.

Before we can state the more "geometric" version of Theorem I we need

some fur ther notat ion.

We say the group G sa t i s f i es Pn i f and only i f the fo l lowing holds:

(1) G = <Pi i i E I > , I l l = n; where the Pi are pairwise d i f f e ren t rank I

parabolic groups of the same char.p.

(2) The Pi have a common Borel-subgroup B.

We say G sa t i s f i es P+ i f i t sa t i s f i es in addi t ion: n

ON AMALGAMATION OF RANK 1 PARABOLIC GROUPS

(3) Let Pij = <Pi'Pj > for i * j E I. Then either Pij is a rank 2

parabolic group of char.p with Borel subgroup B or Pij = PiPj and B =

P in Pj.

I t has been shown in [14] and [17] that the notion of groups satisfying P+ n

is "more or less" equivalent to the notion of a classical, locally f in i te

Tits chamber system C of rank n, with discrete transit ive automorphism group

G. I f G satisfies P+ n' then the diagram ~ = ~(1) is defined in the usual

way. (This is also the diagram of the corresponding Tits chamber system C !)

Generalizing this concept one can define a graph F(1), i f the group G just

satisfies Pn' in the following way:

(a) I is the vertex set of F(1).

(b) Let M i = Op(Pi), Pi = oP' (Pi ) " Then ( i , j ) is an edge of F(1)

i f and only i f

M i n Mj is not normal in ~i and in ~ j .

I t w i l l be shown in (2.3) that i f G sa t i s f i es P+ i and j are connected in ' n '

F(1) i f and only i f they are connected in ~(1). Now the "geometric" version

of Theorem 1.

Theorem 2. Suppose the group G sa t i s f i es Pn" Then F(1) contains no

t r iang les .

Theorem 2 can be considered as a genera l izat ion of the non-existence-theo.

rem for c lassical loca l l y f i n i t e Ti ts chamber systems of general ized t r iangu-

la r type ( i . e . mij ~ 3 for a l l i # j ~ 3 !) with d iscrete t rans i t i ve automor-

phism group, which is in a rb i t ra ry charac ter is t i c a consequence of [12], to

a rb i t ra ry chamber systems which are jus t bu i l t -up from rank 1 ce l l s , which

resemble rank 1 groups of L ie-type. I t also covers most of the geometries

8 F.G. TIMMESFELD

which could be obtained from rank I and 2 p-local subgroups of some sporadic

group G.

The above mentioned papers [14] , [17] and [18] together with the work of

G.Stroth (see these proceedings!) show that the problem of c lass i fy ing (at

least loca l l y ) c lassical l oca l l y f i n i t e Ti ts chamber systems with discrete

t rans i t i ve automorphism group is near to completion. Theorem 2 of th is paper

and Corol lary 2 of [19], where a pushing up-type resu l t was obtained under

the condit ion Pn and r(1) connected, can be considered as the beginning of

a more general theory, which should include the geometries of the sporadic

groups. In my opinion, the main problem here is to f ind sui table condit ions,

which allow one to show that a p-local geometry of a simple group is e i ther a Ti ts

geometry or j us t has those rank 2 residues which occur in the sporadic groups.

The connectedness of F(1) might be a f i r s t ingredient of such condit ions.

The f i r s t coro l la ry we state is ac tua l ly a sharpening of Theorem 1 in

the special case, when M i = [Mi,~ i ] for a l l i C I.

Corol lary A. Let B be a p-closed group and { P i l i C I} be a co l lec t ion

of rank I parabolic groups of char.p, sa t is fy ing :

( . ) M i = [Ni ,~ i ] fo r a l l i C I

where Pi = oP'(Pi ) ' Mi = Op(Pi)" Suppose there ex is t monomorphisms

Xj :B ~ P i ( j ) ; j = I . . . . . n, i ( j ) E I

such that x j (B) is a Borel subgroup of P i ( j ) " Then e i ther n ~ 2 or

x iZ (M i ( j ) ) =×k l (Mi (k) ) fo r some pair j * k ~ n.

Of course Corol lary A can be stated also in language of groups sa t is fy -

ing Pn:


Corol lary A'. Suppose G = <Pi [ i E I>, n = l i t ~ 3

in addit ion

( . ) M i = [Mi,~ i ] for each i E I .

Then M i = Mj for some i * j E I .

sa t i s f i es P and n

I f we specia l ize Corol lary A' to the case where a l l the Pi are permuted

by the automorphism group of G we obtain a coro l la ry , which also can be ob-

tained from the Baumann-Glauberman-Niles theorem.

Corol lary B.

subgroup B, M = Op(P)

( ,) M = [M,~]

and l e t A = Aut(B).

Let P be a rank 1 parabolic group in char.p with Borel

and ~ = oP'(P). Suppose

Then IA:NA(M) I ~ 2.

F ina l l y a s im i la r coro l la ry in the more general case, when M • [M ,~ ] .

Corol lary C. Let P be a rank 1 parabolic group in char.p with Borel

= oP'(P), M = Op(P) and N = [M,OP(~)]. Suppose there ex is t subgroup B,

automorphisms mi' i = 1,2 of B such that

±1

N $ M for i = 1,2. -1 - I

~1m2 ~2ml Then N ~ M or N ~ M.

Some remarks on the h is tory of Theorem I o r (equiva lent ly Theorem 2!) .

The f i r s t resu l t in th is d i rect ion was the nonexistence proof for groups sat is- +

fy ing P3 with P i , j ~ L3(2) for i , j ~ 3 by A.Chermak in [1] . This has been n . •

+ Pi generalized in [10] to groups sa t i s fy ing P3 with ,J ~ L3(P i j ) for i , j ~ 3 +

and in [16] to groups sa t i s fy ing P3 with P i , j a rb i t r a ry rank 2 parabolics in

char.2. The same resu l t in a rb i t ra ry charac ter is t i c is a consequence of the

10 F.G. TIMMESFELD

main theorem of [1~. The only known resu l t which moves s l i g h t l y away from

the +- condi t ion is the nonexistence of groups sa t i s f y i ng P3 with Pi ~ L2(2)

and P i , j ~ A7 proved in [10].

Since fo r our Theorem 2 we do not need to know the s t ruc ture of P i , j '

i t is obvious that Theorem 2 is a fa r reaching genera l iza t ion of a l l these

resu l ts . But in fac t i t should be viewed as a resu l t about possible embeddings

of a p-closed group B in to rank I parabol ics Pi" I f more than two such em-

beddings are given, one can take fo r G j us t the free amalgamated product of

the Pi over B and one obtains cer ta in r e s t r i c t i o n s on the embeddings. I t

should also be mentioned that A.Chermak has c l ass i f i ed the groups sa t i s f y i ng

the conclusion of Coro l lary A over F 2 in [2 ] .

§ 2 contains the proof of the equivalence of Theorem I and 2 and of the

connectedness in r (1) and A(1) i f G sa t i s f i es P+ In § 3 we s ta r t wi th n"

the proof of Theorem 2. I t contains some fu r the r notat ion and propert ies of

groups wi th a weak BN-pair of rank 2. From § 4 on we assume by way of contra-

d i c t i on tha t G sa t i s f i es P3 and r(1) is a t r i ang le . We show that

G i = <Pj,Pk > ~ N(Z) in the constrained case fo r some i = o,1 or 2; where

Z = aI (Z(S)) and S is the p-Sylow-subgroup of B. § 5 contains the main re-

duct ion. Here we assume that G is the free amalgamated product of Po' PI '

P2 over B, G 2 = <Po,PI> ~ N(Z) and that we are in the constrained case. Then

by [ 6 , ( 7 . 9 ) ] G O n G I = P2 and we can show, considering the coset graph

F = F(Go,GI), that a cer ta in parameter usual ly ca l led d or b is smaller or

equal to 2. § 6 contains the f i na l cont rad ic t ion in the constrained case. Here G.

the fac t that d ~ 2 shows that S is "small" Especia l ly i f Z i = <Z l>, G 2

i = o,1 and V 2 = <Z onZ1) >, then V2/Z is a GF(p)G2-module on which M o

or M l acts quadra t i ca l l y . But by (3.11), (3.12) such a module does not ex is t .

ON AMALGAMATION OF RANKIPARABOLIC GROUPS

F i n a l l y § 7 contains the cont rad ic t ion in the non-constrained case, which is

easy but does depend very much on the s t ruc ture of the parabol ics in groups

with a weak BN-pair of rank 2.

F i na l l y a remark on the references. Except the already mentioned papers

of Delgado and Delgado-Stellmacher a l l other papers are e i t he r only quoted in

the in t roduct ion or to obtain resu l ts on the s t ruc ture and on GF(p)-represen-

ta t ions of rank 1 or 2 Lie- type groups.

11

12 F.G. TIMMESFELD

§ 2 Proof of the equivalence of Theorem 1 and 2 and of connectedness in

r (1) and 4 ( I ) . . . . . . . . . .

We use in th is section the notation introduced in the introduct ion: F i r s t we

show:

(2.1) Theorem I and Theorem 2 are equivalent.

Proof. F i r s t suppose that Theorem I holds but Theorem 2 is fa lse. Then, pro-

ceeding by induction on n, we may assume n = 3 and ?( I ) is a t r iang le . Let

Xi = i d : B ~ Pi ' i = 1,2,3.

Then, by Theorem i , Nj ~ M k for some pai r j • k ~ 3. Hence, as Nj

Mj n M k and [Mj ,oP(~) ] ~ Nj, we obtain Mj n M k ~ j since Pj = oP(~j)B.

But then j and k are not connected in ? ( I ) .

To prove that Theorem 2 implies Theorem 1 l e t G be the free amalgamated

product of the groups P i ( j ) ' j E J over B. Then there ex is t monomorphisms:

Pt ~ G, j E J X j : P i ( j ) 1( j ) -

sa t i s fy ing :

(*) ~jXj = ~kXk for a l l j , k E J

considered as mappings from B into G. Hence B' = Xjxj(B ) is the common

Borel subgroup of a l l the pt and l ( j )

B' = P' n P' for j * k. i ( j ) i ( k )

Hence G sa t i s f i es Pn' where n = IJ I . Now, to prove that Theorem 2 implies

Theorem I , we may assume n m 3. Hence there ex is t nodes j , k E J, which

are not connected in r ( J ) .

To s imp l i f y notation l e t P'. = P~ and M'. ~' NL be the correspon- j 1( j ) j ' j ' J

ding subgroups of P'. j E J Then we may by symmetry assume M'. N M' ~ j ' " j k ~ j "


We have to consider the two cases:

(~) M Ik$ JM'.

(B) M' < M'. k = j

Suppose f i r s t (a) holds. Then, by the structure of the rank I Lie-type groups Pt pL

~j = M~<(M~)j K J>" NOW oP(~) _~ <(M~) J> and, as Mj' n M'k ~~'Pj and

I I [Mj,M k] s Mj n M' k we obtain

N [ : [M] , o P ( P ~ ) ] ~ m [ N m I j j k"

This implies I j (N j ) ~ Xj(Mj) n Ik(Mk) and thus, since isomorphisms respect

intersect ions:

Nj ~ Mj n x]mlk(Mk) = Mj a Xjxkl(Mk )

by (~). Applying X] 1 to this equation we see that part 2 of Theorem 1 is

sat is f ied for the pair i ( j ) , i ( k ) ; j # k E J.

13

I f now (B) holds, then N~ ~ M~ and arguing as before we obtain 3

- I X]I ×k (Ni(k))~ (Mi(j))' J ~ k~a.

This shows that Theorem 1 is a consequence of Theorem 2.

(2.2) Lemma. Let X/M, M = Op(X), be a perfect central extension of a

rank 1 Lie-type group defined over GF(q), q = pn. Suppose X acts on the

set ~, I#J ~ q, with kernel K. Then one of the fol lowing holds:

(a) M ~ K

(b) X = M. K

(c) q = I~I = 9 and

In any case [M,oP(x)] ~ K.

X/K--- ZZ 3× PSL2(9 )

Proof. I t is obvious that the second statement holds, i f one of the cases

(a) - (c) holds. We prove that one of the cases (a) - (c) holds by induction

14 F, G. TIMMESFELD

on l~ l . For th is assume tha t (2.2) holds f o r the ac t ion of such a group

on a set ~ with IAI < I~ I . We f i r s t show:

( , ) Let a E ~. Then X = M. X or M ~ X .

Suppose by way of c o n t r a d i c t i o n tha t (*) is f a l se . Then

X>MX > X .

As IX: Xal =< q and M is a p-group, we obta in

n - I 1 , IX/M: (XaM)/MI < ; = p

Since X/M is a rank I L ie - t ype group def ined over GF(q), th i s is by [ g ] ,

[13] and [ 7 ,11 (8 .28 ) ] impossib le. So (*) holds.

I f now M ~ X a f o r each a E ~ then (2.2) (a) holds. So we may assume

M $ X B f o r some B E ~. Then X = MX B and XB/M B ~ X/M, where M B = MnX B.

Consider ing the ac t ion o f X~ on ~ = ~-{~) by induct ion assumption one o f the

f o l l o w i n g holds:

(a ' ) M~ ~ K n X B

(b ' ) X~ = M~(KnX~)

since (c) cannot hold f o r X~, as 141 < q.

Now in case (b ' ) we have X = MX~ = MM B. K = M. K and so (2 .2 ) (b ) holds.

So we may assume tha t (a ' ) holds. In th is case e i t h e r X B acts t r i v i a l l y on A

or X~/M B ~ A 6 and 6 ~ IAI ~ 8 by the same quotat ions as above. In the f i r s t

case X B ~ K and again (b) holds. So assume the l a t t e r . Then q = 9 and

X/M ~ PSL2(q).

Now, as I~I ~ 9 and X = MX~ e i t h e r IM: MBI = 3 or 9. In the f i r s t

case obv ious ly X B ~ X and (2 .2 ) (c ) holds. The second case is impossible

since X / K ~ q . This proves (2 .2 ) .


(2 3) Proposi t ion Suppose G sa t i s f i es P+ for some n > 2. Then • " n =

i , j E I = {1 . . . . . n} are connected in A(1) i f and only i f they are con-

nected in £ ( I ) .

= S p . . = n S g, g c Pi '" where Proof. Let Mi, j 1, j ' J '

oP ' (P i , j ) and denote by:

- : ~ i , j ~ i , j / M i , j "

S E Sylp(B), ~ i , j =

I f now i and j are connected in the diagram A(1), then P i , j is a perfect

central extension of a rank 2 Lie-type group. Hence by [17 , (3 .1 ) ] Mi ~ Mj

and by the descr ip t ion of the s t ructure of the rank 2 Lie-type groups in

[17,(3.2)] Mi n Mj is not normal in Pi and -{j. Since Mi, j ~ M i n Mj

th is shows that i and j are also connected in r (1 ) .

15

So the converse remains to be proved. For th is assume M i n Mj # ~ i and

~ j , but P i , j = PiPj with B = P i n Pj. Suppose f i r s t oP(PK) ~ C~k(Mk)

for k = i or j . Thus, as Pk/Mk is a rank I Lie-type group, we have

Pk = CPk(Mk)" ~"

This immediately impl ies M k n M Z e ~ k , where {k,Z} = { i , j } , since

M l n M k ~ S. Hence we have

(*) oP(Pk) $ C~k(Mk) for k = i or j .

Now by symmetry wi thout loss:

I P j : B I = q + l ~ r + l = I P i : B I , q = p~, r = p6.

Hence i P i , j : P i l = IP j :BJ ~ r + l .

Consider the act ion of ~. on 1

= { P i x ] x E P i , j ) - P i -

Then, as i ~ i = q ~ r , (2.2) appl ies. Hence

X Ni = [Mi 'oP(Pi ) ] s Pi for a l l x C P i , j "

16

i~ow we obtain:

F. G. TIMMESFELD

N =TTN x < npX x running over P i , j

so that N = N i = I , since N- is a normal p-subgroup of

in -Pi" But th is is a cont rad ic t ion to ( . ) .

P i , j contained

(2.4) Proof of Coro l lary A and A'.

I t is c lear that (2.1) also shows that Coro l lary A and A' are equivalent . So

we only prove A' .

Assume now that G sa t i s f i e s Pn for n > 3 and that M i = [Mi ,~ i ] , i c l .

As Pi = oP(P'i)S th is impl ies also that N i = [Mi,oP(P'i)] = M i fo r each

i c l .

Now, as n _-> 3, Theorem 2 implies that there ex is ts a pai r i , j c I such

By symmetry assume that the f i r s t possi- that M i n Mj ~ i or M i n Mj ~ j .

b i l i t y holds. Let P i , j = <Pi 'P j > and Mi, j = Sp. 1, j

- : P i , j ~ P i , j = P i , j / M i , j °

and denote by:

Suppose f i r s t M i n Mj < M i . Then e i the r Mj = Mi, j < M i or Mj # M i and

[oP(~i),M i] = M i n Mj, s ince [Mj,M i] < M i n Mj and s ince oP(~i) <Mj Pi < = ~ > .

Now the second case obviously contradic ts the assumption.

In the f i r s t case we have Mi < ~ and is B - inva r ian t . Suppose ~ j /Mj

is not isomorphic to 2G2(3m ). Then the s t ruc ture of the rank 1 L ie- type group

~j /Mj impl ies Mi = S' n Z(S) except in case PNj/Mj ~ U3(2 ) or Sz(2) and

IS: Mil = 2. Since S/M i is elementary abelian we obtain in any case that

~ i /Mj is an extension of L2(pn), pn = IS/Mil by Mi/Mj, which is perfect and

central in the f i r s t case. Now both p o s s i b i l i t i e s obviously cont rad ic t

[Mi,~ i ] = Mi, since in the second case Mi/Mj ~ ~2 or ~4"

I f f i n a l l y

con t rad i c t i on as above or M. 1

Mi is not elementary abel ian

cannot act n o n - t r i v i a l l y on a

again a con t rad ic t i on to [M i


2 2(3m ) /Mj ~ G then we obtain e i t h e r Mi = Z(S) g S' a

= #(S), I S : M i l = 3 m and IMil = 3 2m, but

• Now in th is case ~ i /M i ~ L2(3 m) and thus

not elementary abel ian group of order 3 2m,

,oP(~ i ) ] = M i .

I?

This shows that M i = M i n Mj ~ Mj. Since we have to show M i = Mj

to prove Coro l la ry A ' , assume by way of con t rad ic t i on tha t M i < Mj. Now the

hypothesis N k = M k f o r k = i or j obviously impl ies

C~k(Mk) ~ Mk fo r k = i or j .

Let now H be a p'-complement to S in B and assume i ~ Ho = CH(Mk) fo r

k = i or j . Then, as Nk = Mk' the 3-subgroup lemma impl ies

[Pk'Ho ] ~ C~k(Mk) ~ Mk

and so by the act ion of a p ' -group on a p-group [Pk,Ho] = 1. Now wi th symme-

t r y we obtain immediately Ho = CH(Mz)' { i , j } = {k ,# } and Ho = C H ( ~ i , j ) '

where i,j = > and, as P i , j = ,jH, Ho This shows that, i f

^ is the natura l homomorphism from P i , j on P i , j / ( M i , j H o ) then

(*) CPk(Mk) ~ Mk fo r k = i and j .

Hence the pa i r P i ' Pj is a weak BN-pair of P i , j in the sense of Delgado-

Ste l lmacher [ 4 ] . But thus i t fo l lows from the main theorem of [4] tha t

M i $ Mj.

(2.5) Proof of Coro l la ry B.

Let ~ i ' i = i . . . . . n be a set of coset representa t ives of

ml = 1 and assume n ~ 3. Let

~ : B ~ P i = 1 . . . . . n

NA(M ) in A wi th

18 F.G. TIMMI~SFELD

be monomorphisms w i th (X.

a i : b I -~b, b E B.

Then by Coro l l a ry A'

a i M = aiZ(M) : aiZ(M ) = M aj

a i a j I f o r some pa i r i • j ~_ n. But then E NA(M),

proves n < 2.

a c o n t r a d i c t i o n . This

(2.6) Proof o f Co ro l l a r y C.

Let Xo = id : B ~ B c P

-1 and Xi = a i : B - , B ~ P f o r i = 1,2.

Then, by Theorem 1, there ex i s t s a pa i r j • k E {o ,1 ,1 }

x jZ(N) ~ X~I(M). As

- I U. xi (N) =N ~ SM = Xo(M)

and also -1 a. I(M) ×o (N) = N $ M I = × i f o r i = 1,2

by hypothes is , we have { j , k } = {1 ,2 } . But then

a I a 2 N % M a2 or N ~ M a l

and the c o r o l l a r y holds.

s a t i s f y i n g

§ 3


Notat ion and proper t ies of groups w i th weak BN-pair o f rank 2.

19

Since t h i s paper depends very much on the c l a s s i f i c a t i o n of weak BN-pairs of rank 2

by Delgado-Stel lmacher in [4] we repeat the d e f i n i t i o n f o r our purpose:

We say the group G has a weak BN-pair of rank 2 w i th respect to PI,P2 i f

i t s a t i s f i e s :

( I ) G : <P1,P2>, where the Pi are rank 1 parabo l i c groups in char .p . (p is

the char of the weak BN-pai r ! )

(2) B = P I n P2 is a Borel-subgroup of P1 and P2"

(3) Let S E Sy lp(B) . Then S G = 1.

(4) Let ~ i : oP ' (P i ) and M i = Op(Pi) . Then C~.(Mi) ~ M i f o r i = 1 and 2. I

B = ( B n ~ l ) ( B n ~ 2 ) . (5)

This d e f i n i t i o n is s l i g h t l y d i f f e r e n t from the d e f i n i t i o n in [4] s ince De lgado-Ste l l -

macher do not demand (5 ) , wh i l e instead of (3) they demand the s t r i c t e r B G = 1.

But i f H is a p'-complement to S in B then i t is obvious w i th (3) and (5) tha t

B G = H n Z(G) n G'. Hence a group w i th weak BN-pair in our sense is a pe r fec t cen-

t r a l extens ion of a group w i th weak BN-pair o f rank 2 in the sense of De lgado-Ste l l -

macher by a p ' -group' contained in B. Since we have to consider FF-modules fo r weak

BN-pairs such extensions have to be admit ted. ( I . e . one has to admit SL3(q) instead

of PSL3(q)! ).

We of ten re fe r w i t hou t reference to the c l a s s i f i c a t i o n of weak BN-pairs of rank 2.

(See Theorem A on page 100 of ~4] . ) So the reader should make h imse l f f a m i l i a r w i th

the s t r uc tu re of ~ i ' i = 1,2. In case tha t G is l o c a l l y or parabo l ic isomorphic

to a L ie - type group, t h i s s t ruc tu re is described in [ 18 . ( 3 . 2 ) ] as fa r as i t is ne-

cessary in t h i s paper. I f ~ i /M i ~ L2(2 ) f o r i = 1,2 the reader should use tab le 1

in Goldschmidt 's paper [5 ] .

20 F, G. TIMMESFELD

I f G is of type 2F4(2 ), 2F4(2 )' or F 3 the de f i n i t i on on page 99 and 100 of [4]

suf f ices. In case G is parabolic isomorphic to J2' then i t is well-known that

P1 ~ (D8*Q8)A5 and P2 is the product of a special group U 2 of order 26 with

Z(U2) = @(U2) ~ ~2 × ~2 with z 3x ~3"

A fa i l u re of fac to r i za t ion (short F F)-module for a group G with weak BN-pair

of rank 2 of char p is a nont r i v ia l GF(p)G-module V sat is fy ing:

IV:Cv(A) I z IAI

for some nonident i ty elementary abelian p-subgroup A of S. Such subgroups w i l l

be cal led offending subgroups. For the convenience of the reader we state the clas-

s i f i ca t i on of such modules in [3] , which is our second main reference. To make this

statement shorter we use the fo l lowing vague notat ion: I f G is loca l l y isomorphic

to a rank 2 Lie-type group L in char p, then we cal l a GF(p)G-module V a loca l l y

natural module, i f P1 and P2 act on V as the parabolic subgroups of L on the

natural GF(p)L-module. Of course i t is not at a l l c lear that such a natural module

is uniquely determined. (Up to equivalence of representat ions.) (A natural G2(q),

q = 2n-module V is the GF(2)G2(q)-module obtained from the embedding G2(q)

Sp(6,q).)

Let for the next two lemmata G be a group with weak BN-pair of rank 2 with

respect to PI,P2; B ~ PI n P2' Mi = 0p(P i ) ' ~i = oP'(Pi) and S E Sylp(B). I f

G is loca l ly isomorphic to . . . . . . we wr i te often G is Z . i . , L e t V = <Cv(S)G> be

an FF-module for G, V = V/Cv(G ) and G = G/CG(V ).

(~.1) Proposit ion. One of the fo l lowing holds:

( i ) G ~ SL3(q) and V is a natural G-module.

(2) G ~ A 6 and V is a natural Sp(4,2)'-module.

(3) G is Z . i . to SL3(2), F~J = 26~

(4)

(5)

(6)

(7)

(8) G is / . i . to

(9) G is Z . i . to

In any case Pi m

dual module of

(3), (4) or (6),

Proof, To apply the resul ts of [3] to

form also a weak BN-pair of rank 2 for G.

below B n CG(V ) = 1, since V = <Cv(S)G>

ON AMALGAMATION OF RANK 1 PARABOLIC GROUPS 21

is / . i . to SL3(q), bVl : q6, l-Cv(S)l : q2, CV(Mi) = [~,Mi ] : [~ ,~ i ]

is the d i rec t sum of two natural SL2(q)-modules and [V,Mj] = Cv(S ) for

{ i , j } = {1 ,2} .

~ Sp(4,q) and V is a natural G-module.

is Z . i . to Sp(4,2), IVI = 2 6 •

is Z . i . to U4(q) or U5(q ) and V is l . a natural G-module.

~- (6 ,q) , q = 2 m and # is £. a natural ~-(6,q)-module.

G2(q) , q = 2 m, V is l . a natural G-module.

NG(Cv(S)) for i = i or 2 and e i ther V is selfdual or the

is again an FF-module. Moreover in a l l cases, except possibly

is an i r reduc ib le module for G.

and G we need to show that PI,P2

But th is is obvious, since by ('3.3)

is a nont r i v ia l G-module.

Now al l statements of (3.1) , except that G : SL3(q) in (1), G ~ Sp(4,q) in

(5) and G~A 6 in (2), are contained in [3]. I f q = 2 in ( i ) and (5) resp. G / . i .

to L3(2 ) in (2) th is is an easy computation. (In L4(2 ) ~ A8! ) So to prove (3.1)

i t remains to show G : SL3(q) resp. Sp(4,q) i f G is Z . i . to one of these groups,

q > 2 and # is Z.a n~tural G-module. Since this w i l l j us t be a part of a fo r th -

coming paper of A.Chermak we only sketch the proof.

Let G be the free amalgamated product of P1,P2 over B. Then there ex is t

homomorphisms G £ G ~ G.

Hence ~o is a representat ion of G on ~ and we may assume without loss G = G.

Let F = F(G,P1,P2) be the coset graph of PI,P2. Then F is a tree. Define a

map X from F in the set of a l l subspaces of V by

22 F . G . TIMMESFELD

X: o ~ ~ia = C'](Ma) for m E F.

Choose by [~] a Weyl-group W = W(H) and an apartment T : T(H,W) for which the

uniqueness and exchange condition of [4] are sat isf ied. (Exists in our cases by [4]!

Let ~ be the equivalence relat ion on ? defined by T for which F/~ is a

projective plane respectively a generalized quadrangle. (See [4 , (3 .6) ] . )

We need to show:

(*) I f 0 ~ ~ in ? then X(o) = X(~).

Suppose (*) is shown already. Then X defines a map ~:?/~ ~ ]~=~(V) = {~EaEF}.

Let * be the incidence relat ion on V defined by ~ * #B i f and only i f ~ _

VB or ~{~ c ~ and cal l points the elements of p~ with I-#~I = q and l ines the

elements of ~ with I ~ I = q2. Since obviously I~I ~- 2 (q2+q+1) = IF/~l resp.

2 (q3+q2+q+1) = I?/~I , ~ is a b i ject ion. Moreover, ~ respects incidence, since

X maps neighbours in F on incident pairs in ~ . Claim

~-1 (**) X also respects incidence.

I f (**) also holds then ~ is by [4 , (3 .6 ) , (3 .7 ) ] a classical project ive plane or

a generalized quadrangle of type q,q) over GF(q). Hence G/K ~ PSL3(q) resp.

PSP(4,q) where

i t follows that

PSP(4,q). Hence

elements of

is the kernel of the action of G on V. Since obviously SNK=

PiK/K, i = 1,2 s a pair of parabolic subgroups of PSL3(q) resp

G/K ~ PSL3(q) resp. PSP(4,q). Since G = 0P'(G) and since the

act as diagonal automorphisms on V the resul t fol lows.

I t remains to prove (**) and ( . ) . We f i r s t show (** ) .

is a point in ~ , Suppose that (~o,Oi,02,m3) is an arc in F such that Voo

a l ine and # * # . Then % 03 V = ~ • # = #

° 1 a o 0 2 0 3 '

oN AMALOAMATIONOFRANKIPARABOLIC GROUPS 23

a contradict ion since ~ is b i j ec t i ve . This shows (**) in any case, since d(P,L)~

for each point P and l ine L of F/~.

Now to the proof of (~). By de f i n i t i on of ~ we may without loss assume that

a,~ E T and d(a,B) = 6 resp. d(a,B) = 8 in F. Suppose f i r s t that X(a) is a point.

I f d(a,~) = 6 le t ~ = s o, B = a 6 and (ao,a I . . . . . m6) the arc in T from a to B.

Then obviously JX(~I)I q2 = = IX(~3)I, X(~I) = X(ao) e X(a2) and X(~3) =

X(a2) e X(m4). Since X(al) # X(a3) we have X(a4) n X(~l) = i by the action of H.

This shows that : ' \ 2 ' \ 4 ' V Vao

Since T is H- invar iant , there exists a K < H which centra l izes ~ and a 4

acts f ixed po in t - f ree l y on Val. Since Va6 is also H- invar iant , th is implies

As • and since there exists also a subgroup of H cent ra l - V~ 6 V~ I • Va 6 Va 2

iz ing V- 2 and acting f ixed po in t - f ree l y on ~ , th is shows that X(~) = X(B)

in th is case.

Now a l ine in ~ is in any case the sum of two points of distance 1. Hence th is

shows (~) in case G Z . i . to SL3(q) Moreover, we also only need to show (~) in

case G Z . i . to Sp(4,q) i f X(a), X(B) are points in ~ . For th is le t again

(s o . . . . . a8) be the arc in T jo in ing ~ = a o and B = ~8" The action of H on

shows again that

v = ° \2 ° \4 ° \ 4 : L2° \4 \ 6 ' v 8 (Detai ls are l e f t to the reader!)

Now by the action of Pro4 on # there ex is ts a subgroup K of H.. with [#,K] =

-Va2 e V~o _and C~(K)= V~o e-Va4. Hence -Va8 ~V~o e-V 4. I f now q > 4 then

V o and Va4 are nonequivalent GF(p)H-modules and so are the only nont r i v ia l

GF(p)H-submodules of ~V~o • Va4, which shows X(a) = X(B) in th is case.

24 F.G. TIMMESFELD

So only the cases q = 3 or 4 remain to be t reated.

Assume f i r s t q = 4. Let w E W(H), which induces a r e f l e c t i o n on a 4 on T.

Then w 2 = i and w ~ Z(P'a2)# in Pa4 as one e a s i l y shows. But then l [ # ,w ] l = q

and so ~ e #B ~ C~(w) which shows X(B) = x(a) w = x (a ) .

In case q = 3 one shows that the r e f l ec t i ons w on a 4 is the product of two

GF(3)- t ransvect ions on #. Since w 2 is the invers ion on V e # th is shows a2 ~6

again that w cen t ra l i zes ~ m VB' which again impl ies X(a) = ×(B). This

shows (*) in the f i n a l cases and thus proves (3 .1) .

(3.2) Coro l la ry .

and choose notat ion so that

(1) A ~ M. fo r i = I or 2 1

(2)

(3)

(4)

(5)

(6)

Suppose A ~ S is a q u a d r a t i c a l l y act ing of fending subgroup

PI ~ NG(Cv(S))" Then the fo l low ing holds:

and i f A ~ M I then IAI = IV: Cv(A) l .

IAI = IV :Cv(A )

Cs(C~(A)) : A,

C~(A) = IV,A] ,

A ~ M 2 except

I f A ~ M 2 and

except in case ( i ) or (5) of (3.1).

except in case ( I ) , (5) or (7), i f G~ U5(q)

except i f (1) or (5) of (3.1) holds and IAI = q = [I~,AI].

n case (1) or (5) of (3.1).

IAI > q, then C~(M2) ~ [~,A]. Moreover, i f case ( i ) - (5)

of (3.1) holds then C~(M2) = [~,A] = IV,M2].

(7) I f (3) , (4) or (6) of (3.1) holds, then C~(A) i f P2- invar ian t , but is not

an i r r educ ib l e ~2-module.

(8) In case (8) or (9) of (3.1) ex i s ts a g E G wi th [#,A] n C#(M~) = o.

(9) ~i ~ NG(A) f~ r i = i or 2, except in case (1) or (5) of (3 .1) .

(3.3) Lemma. Let G = <P1,P2 > be a group with weak BN-pai6 of rank 2 wi th

respect to P1,P2 and S = Op(B), where B = P1 n P2" Suppose N n S • 1 fo r


some normal subgroup N of G. Then the f o l l o w i n g holds:

(1) G = N.S

( 2 ) G = N or G is of type ~6' G2(2) ' Aut(Ml2) °r2F4 (2)

25

and I G : N I ~ 2.

Proof. Let I , x E S n N and Ui = Op(Pi ) ' ~ i = oP ' (P i ) " Then

G x n S ~ U i f o r i = 1 or 2,

as S G = Io Hence ~ i = U i (NNP i ) s N.S. So, as G = <PI,P2 >, to prove (1) we

may assume tha t N n S ~ Uj f o r { i , j } = {1 ,2 } . But then [U i ,NNP i ] ~ U i NN ~ Uj

and thus U i n Uj < Pi" Now an inspec t ion of the cases occur ing in the Delgado-

Stel lmacher- theorem shows tha t t h i s is not poss ib le .

So (1) holds and thus P i= (P inN)U i fo r i = 1 and 2. Hence i f G , N, then

U i > U i n N f o r i = 1,2 and thus

( . ) [ P i , U i ] ~ (U iNN)U~ < U i f o r i = I and 2.

I f now G is parabo l i c isomorphic to a L ie - t ype group, then (2) is a consequence

of [ 9 . , ( 3 . 3 ) ] . I f G is parabo l i c isomorphic to MI2 or J2 ( * ) cannot hold. I f

f i n a l l y G is of type 2F4(2 ) ' or F 3 then the desc r i p t i on on page 99 and i00 of

[4] shows tha t ( . ) also does not hold.

(3.4) Lemma. Let G be a group w i th weak BN-pair of rank 2 w i th respect to

PI,P2 and 1 • x E U 1 = Op(Pl) . Then there ex i s t s a g £ G such tha t G =

<X,Plg>.

Proof. I t su f f i ces to show G = <xh,p~> fo r some h,g E G. Suppose t h i s is

not the case. As B = P1 n P2 is maximal in P2 we obta in T = x G N S ~ U 2 =

= 0p(P2) and thus T ~ U I . Now there ex i s t s a y E T n U I such tha t y ¢ U 1

o o <Ba,y~> fo r some B E PI" f o r some o £ P2" As y E T ~ S ° we have PI

Set L = <U2,Y~> ~ P?. Then P~ U?L and, since U I ~ N(U2) , i t fo l lows tha t ~a

L g P I .

26 V.O. TIMMESFELD

Let R = S ~ N L a U 2. I f R > U2, then P2 = <B,R> s <Pl,y6>

holds. Hence R = U 2 is a p-Sylow-subgroup of L and thus

U 1 n U 2 = U 1 n L ~ P I '

which is as in (3.3) by the Delgado-Stel lmacher-theorem impossib le.

and (3.4)

(3.5) Lemma. Assume the hypothesis o f (3 .4) and in add i t i on tha t G is l o c a l -

l y isomorphic to Sp(4,q) . Suppose G acts on the natura l Sp(4,q)-module V

such tha t G = G/C(V) ~ Sp(4,q) and PI is the s t a b i l i z e r of a maximal t o t a l l y

i s o t r o p i c subspace. Then we may choose g E G w i th G = <x,P~> such tha t V =

= [V,U I ] ~ [V ,U~] .

Proof. Use the no ta t ion of (3 .4 ) . Then V 1 = [V,U 1] is the t o t a l l y i s o t r o p i c

subspace s t a b i l i z e d by PI" Fur ther , obv ious ly P2 is the s t a b i l i z e r in G of a

po in t P in V I and i f Z = Z(~2) then IV,z ] = P fo r each z E Z #,

Pick h E P2 such tha t

h h pg g E P1 such tha t V I = P m and

and thus x E P~. But as [x ,V] s V 1,

h h x £ U 2 fl P1 but x ~ U 1. I f

Since P~ : <U~,P~NP~> t h i s

Conjugat ing in P1 we may assume x c U 1N U 2. h h p±

V 1 , V 1. then V 1 + V I = and we f i nd a

V = V I m V . I f x E Z, then x E U I ~ P2

x

h So we may assume x ¢ Z. As U 1 n U 1 = Z,

x £ U~, then as x E U 1 we would obta in

cv,x] = Ev ,x] n v 1 = o ,

which is impossib le. Thus ~ : <x,U~> ~ <x,P~>.

also impl ies G = <x,P~>. This proves (3 .5 ) .

The proof of the next two lemmata is t r i v i a l and so we leave i t to the reader.

(3.6) Lemma. Suppose G is a group w i th weak BN-pair of rank 2 w i th respect to

PI,P2 which is l o c a l l y isomorphic to L3(q). Let U i = Op(Pi) and Z = U 1N U 2,

Then f o r i = 1,2 there ex i s t s a g C G w i th Z ~ Pgi, but Z ~ U i . ~g


(3.7) Lemma. Suppose G is a group with G = oP'(G), G/C i = SL3(q), where

q = pn, for i = 1,2 and ICi/C i n C j l t q - i for { i , j } = {1,2}. Then C I = C 2.

The next lemma is part of a more general resu l t , but we only need i t in th is

special case.

(3.8) Lemma. Suppose G is a group with weak BN-pair of rank 2, which is

l oca l l y isomorphic to L3(2 ). Assume G/N ~ L3(2), where N is an elementary

abelian normal 2-subgroup. Then e i ther N = I or INt = 28 and N is the

Steinberg-module (= adjo int ) for G/N.

i

Proof. Let G be the free amalgamated product of two z4 s over the common

2-Sylow-subgroup. Then there ex is t homomorphisms:

~ G I G/N = L3(2 )

such that to maps the ~4 s isomorphicly on a pair of parabolic subgroups of

L3(2 ) with common Borel-subgroup. Further K = kernel to is f ree. Now there are

three p o s s i b i l i t i e s to accomplish the proof:

(1

(2

(3

Use a resu l t of Solomon and T i t s , which shows that

module (as ~-module!) for G, i f

dins 4.

Show that the s p l i t extension of

BN-pair of rank 2 of type L3(2 )

1 4.3 of [11] to show

Hn-l(~) is the Steinberg

G is a rank n Lie-type group with bu i l -

L3(2 ) by i t s Steinberg-module has a weak

(which is t r i v i a l ! ) and use exercise 1 of

rank K = 21- 2.7+ 1 = 8.

Show by elementary, but tiresome, computations that an extension of L3(2 )

by i t s natural module has no weak BN-pair of rank 2 of type L3(2 ).

(3.9) Lemma. Suppose G is a group with weak BN-pair of rank 2 with respect

to P1,P2 and G/C ~ L3(2), where Pi ~ C = 1 for i = 1,2. Suppose G admits

a f a i t h f u l F2G-module V, sa t i s fy ing :

28 F.m.T~MMESFELD

(1) V = <zG>, where E 2 = Z ~ Cv(PI) .

(2) There ex i s t s a hyperplane H of V w i th

i = 1,2.

Then C = 1 and

Hence [V,C] ~ U and

[H,U 1] = Z, where U i = 0p (P i ) ,

V is a natura l SL3(2)-module.

Proof. Let V be an F2G-module of minimal dimension fo r which the hypothesis

of (3.9) ho lds, but the conc lus ion is f a l se . Then one has:

(a) EVl ~ 2 7, #

since G is generated by U I and two i n v o l u t i o n s conjugage to U I .

(b) Cv(G ) = o,

since the hypothesis of (3.9) holds f o r ~ = V/Cv(G) and ~ = G/Cc(~ ). Hence i f

Cv(G ) • o then by m i n i m a l i t y of V, ~ is a natura l SL3(2)-module and C = CC(~)

is a d i r e c t sum of dual modules, a con t r ad i c t i on to (3 .8 ) .

(c) V is i r r e d u c i b l e .

Suppose U < V is an i r r e d u c i b l e F2G-submodule. Then, as Z ~ U by (1) , U ~ H.

Hence U n H is a hyperplane of U cen t ra l i zed by U 1 and V = U+H. I t f o l -

lows by (3.1) tha t U is a na tura l SL3(2)-module and V/U ~ U*- the dual module.

[U,C] = o, which shows tha t C is an elementary abe l ian

2-group.

Suppose C is the a d j o i n t module f o r G/C. Then [Z,C] ~ C/Cc(Z) as Pl-mOdule

But as Cc(Z ) = Cc(U+Z) and C is i r r e d u c i b l e of order 2 8 as G/C-module, we

have [Z,C] = U. This is a c o n t r a d i c t i o n to the s t r uc tu re of C as G/C-module,

since in C there ex i s t s no Pl-SUbmodule C o w i th C/Co ~ ~2"

Hence (3.8) shows C = I . Now by ( i ) i t is obvious tha t V is indecomposable.

But on the other hand i t is easy to see by (2) t ha t there ex i s t s a P l - i n v a r i a n t

complement Ho/Z to ( H N U ) + Z / Z in H/Z, a con t rad i c t i on to GaschUtz's-theorem

since V = U+H o and H o n U = o.

ON AMALGAMATION OF RANKIPARABOLIC GROUPS 29

Now as V is i r reduc ib le and f a i t h fu l we have 02(C ) = 1. Since IVl z 27

order arguments imply C S Z(G). On the other hand, since U i s G' fo r i = 1,2,

i t fol lows that G = G'. Hence G is an odd perfect central extension of L3(2 ),

which implies C = 1. As IVl s 27 i t is obvious that th is proves (3.9) .

(3.10) Lemma. Let G be a group with weak BN-pair of rank 2 with respect to

P1,P2; U i = Op(Pi), B = PIN P1 and S E Sylp(B). Suppose there exists an

elementary abelian normal p-subgroup V I of PI sa t is fy ing VIU 2 = S. Then one

of the fo l lowing holds:

(a) G is loca l l y isomorphic to L3(q), Sp(4,q) or U4(q) and e i ther V I = U I or

JUI: Vlf = 2 and G loca l l y isomorphic to Sp(4,2).

(b) G is parabol ic isomorphic to Aut(Ml2 ) and JVII = 16.

Proof. I f G is loca l ly isomorphic to a Lie-type group this is a consequence

of [18 , (3 .2 ) ] . I f Pi/Ui ~ L2(2 ) fo r i = 1,2, (3.10) fol lows from table i of [5] .

I t is c lear that G is not parabolic isomorphic to J l" (See descr ipt ion in the

int roduct ion to § 2!) So the cases G of type 2F4(2 ), 2F4(2 )' or F 3 remain to

be treated.

Suppose f i r s t G is of type 2F4(2 ) or 2F4(2)' Use the notat ion of page 99

of [4] . Then i t is easy to see that B3 ~ QI and A 2 ~ Q2 (since B6/B4 ~ 2 4 × 24! )

As [B3,A 2] ~ A 1 = Z(S), the structure of P2 implies IB3A2f ~ 26 . This shows

that also B4 ~ QI'

holds in this case.

Assume f i n a l l y G of type F 3

[ Q l , t , t ] * 1 for each 3-element t E S-QI.

Delgado-Stellmacher's notation and V 1 ~ A 2.

[QznQ2'Q1 ] $ A 1 and so A 2 ~ Q~ ~ S' m Q2'

case.

since otherwise [B4,Q1] = B 3, which impl'ies that (3.10) also

and use the notation of page 100 of [4]. Then

Hence our P1 is also the P1 in

Now A 1S B 3. Hence

which proves (3.10) in this f ina l

30 F. G. TIMMESFELD

G be a group with weak BN-pair of rank 2 with

respect to PI,P2; U i = Op(Pi), B = P I n P2' ~i = oP'(Pi ) and S E Sylp(B).

Suppose V is an FpG-module sa t i s f y ing :

( . ) There ex is ts a B- invar ian t 0 m Z ~ Cv(S ) such that

fo r i = 1 and 2.

(3.11) Proposi t ion. Let

Pi ~ NG(Z)

I f G is l . i to L3(q) suppose fu r ther that Z is a non t r i v ia l i r reduc ib le B-

module. ( l . e . q > 2!) Then [V,Ui,U i ] • o for i = 1 and 2.

Proof. Let V be an FpG-module of minimal dimension sa t i s f y ing the hypothesis

but not the conclusion of (3.11). Then Cv(G ) = o, V = <zG> and by (3.3) Cs(V )= I

Assume wi thout loss [V,U1,U I ] = o. Then ¢(UI) _~ Cs(V ) = 1 and thus G is l . i .

to L3(q), Sp(4,q) or U4(q).

Now 7 2 = <UI,U~>, g E ~2" Let W ° = [V,UI ] , W = <C W (s)PI> and A = U~ N U 2. 0

Then UIA = S and ~1 = MI<-A'AX> for some x E "PI" Let PI = ~1/U1" Then, as - PI

[W,A,A] = o and W = <Cw(S ) >, [ 8 ] and [19 , (2 .3 ) ] imply:

(+) W = Cw(PI) m [W,P 1] and [W,P 1] is the d i rec t sum of natural

P1 ~ SL2(q) or SL2(q2)-modules.

<wPI> contains a nonsp l i t extension of a natural by a t r i v i a l

w E Cw(S ). Thus CW(~I) n [W,A] , o. But on the other hand:

,U1,A] < [V,U I ] n [V,U~] ~ CV(~2),

Cv(G ) = o and G = <PI,P2>.

Namely suppose that

Pl-mOdule for some,

[W,A] ~ [V

a cont rad ic t ion to

Suppose f i r s t that G is not of type L3(q). Then T = U I n U~ = Z(~2) = Z(S)

and [V,T] ~ CV(~2). Now C[W,~I](S ) ~ [Wo,A] ~ [V,U 1] N [V,U~] ~ CV(~2). Hence

i f IV,T] $~[W,PI], then Cw(P1) n CV(~2) # o, a cont rad ic t ion as before. Because

of U 1 = <TPI> we obtain W = W ° = [W,~ I ] and CW(~I) = o. This impl ies Cw(S ) =

= [W,A] = Cv(P2) n W.

ONAMALGAMATIONOFRANKIPARABOLICGROUPS 31

Let U = [V,P 2] and claim Cu(S ) ~ Cu(~2). We have U = [V,U 1] + [V,U~].

Pick u = u I + u 2 E Cu(S ) where u I E [V ,U I ] , u 2 E [V,U~]. Then u 2 E Cu(MI) n

n CU(M~) = Cu(~2) and thus u I E Cw(S ) ~ CV(~2), which proves our c laim.

Now, as U + Cv(S ) = U + Cv(P2), th is is a con t rad ic t i on to cond i t ion ( . ) . So

G is 1 . i . to L3(q). Then again [V,~ 2] + Cv(S ) = [V,~ 2] + Cv(P2). Since

C[V,U~](UI) ~ CV(~2) we obtain Cv(S ) ~ [V,U 1] + CV(~2) = W o + Cv(P2). Hence

Cv(S ) = C W (S) + CV(~2) and (+) impl ies Cv(S ) ~ CV(~I) + CV(~2), since o

C[W,~I](S ) = [WYl ,A ] ~ [V,U I ] n [V,U~]. Now the ext ra cond i t ion shows Z ~ CV(~i)

f o r i = 1 or 2, since B = ( B n ~ 1 ) ( B n ~ 2 ) , a con t rad ic t i on to (~).

In case G is 1 . i . to L3(2 ) we need the fo l l ow ing subs t i t u t i on fo r (3 .11) :

(3.12) Lemma. Suppose G is a group wi th weak BN-pair of rank 2 w i th respect

to P1,P2, which is / . i . to L3(2 ). Let U i = 02(Pi) and S = P1 n P2" Suppose

the GF(2)G-module V s a t i s f i e s : P.

(~) IVi l = 4 fo r i = 1,2; where V i = <z i> and z E Cv(S ).

Then [V,Ui,U i ] • o fo r i = l a n d 2.

Proof. Suppose that V s a t i s f i e s the hypothesis of (3.12) but [V,U2,U 2] = o.

Then

(~) [Vi,U j ] = <z> fo r i • j ~ 2 and

[V 1+V2 ,s ] = 1 fo r s E U 1 n U 2.

P1 Let W 1 = <V 2 >. Then by (~) W 1 m V 2 + V 1 and thus e i t h e r

(~) IWll = 16

or (B) IWII =25 .

As U1 # PI U1 # = s a l l elements of induce in case (a) GF(2)- t ransvect ions on

W 1, Hence they induce a l l t ransvet ions to the same po in t or to the same hyperplane,

a con t rad i c t i on to [WI,U 1] = V 1 = CwI(UI).

32 F . G . TIMMESFELD

Therefore (~) holds. Suppose IWI: CwI(U1)I = 2. Then V 2 <= CwI(U1), s ince

otherwise [Wl,S] = o, a c o n t r a d i c t i o n to the d e f i n i t i o n of W 1. Hence by the #

same reason as above a l l elements of U 1 don ' t induce GF(2) - t ransvect ions onW I .

But then [Wl,S] = U 1, a con t r ad i c t i on to [WI,S,U 2] = o.

(3.13) Lemma. Let y = SL2(q) ' q = pn, V the natura l GF(q)Y-module and

A = Aut (GF(q) ) . Let V be the module V considered as GF(p)Y-module. Then the

f o l l o w i n g holds:

(a) VeF~ ~ ~ (Ve FVo) as GF(p)Y-module. ~EA q

(b) Let X E HOmFpy(VeFV,T) where T is a trivial P

GF(p)Y-module. Then Ix(V~FV)I ~ q. P

(c) Suppose q > 2 and X E HOmFpy(VeF~,V). Then X = o.

Proof. W = Fqm (VmFV) is the d i r e c t sum of n P

as GF(p)Y-module. On the other hand

W ~ ( F q ~ V ) ~ F ( F Q ~ V ) ~ ( e V o ) ~ ( ~ #~) q oEA zEA

Ù,TEA TEA oEA

copies of Vo FV considered P

Now m (V ~ Vo)T ~ m (V m Vo)p as GF(p)Y-module f o r each pa i r ~,p E A. Fur ther oEA oEA

i t is wel l -known tha t , i f o * 1, V ~ Vo is an i r r e d u c i b l e GF(q)Y-module and

V ~ Vo is not equ iva len t to V ~ VT even as GF(p)Y-module fo r o * T.

(S te inbe rgs - tenso r product- theorem!) Since V ~ V is the ad j o i n t module, which is

the d i r e c t sum of a t r i v i a l and a 3-dimensional i r r e d u c i b l e i f p is odd and is

indecomposable i f q > 2 is even, (a) is now an easy consequence of the K r u l l -

~emak-Schmidt-theorem.

(b) is now obvious i f q > 2. I f q = 2, then

W = V~F~ = Cw(Y') ~ [W,Y']

where [W,Y'] ~ V and Y acts n o n t r i v i a l on Cw(Y' ) as an easy mat r ix computation


shows. Hence (b) also holds in th i s case.

Since V ~ W is not the d i r e c t sum of copies of V considered as GF(p)Y-

module, (c) is also obvious.

The proof of the next lemma is t r i v i a l and thus omit ted.

33

(3.14) Lemma. Let Y = <A,B> = SL2(2n); where A,B E Syl2(Y ). Suppose V is

is a GF(2)Y-module s a t i s f y i n g V = Cv(A ) m Cv(B ) and 1[V,x] I ~ 2 n fo r x E A ~.

Then V is the natural GF(2)Y-module.

(;3.15) Lemma. Let G be a group wi th weak BN-pair of rank 2 wi th respect to

PI,P2 which is l . i . to G2(3n). Set ~i = 03 ' (P i ) ' B = P1N P2 and S E Syl3(B ).

Suppose o is a 3'-automorphism of G normal iz ing PI and P2 wi th [BN~I ,O] ~ S.

Then ~ induces an inner automorphism according to some element of B on G.

Proof.p. Use the desc r ip t i on of the Pi of [ 1 8 , ( 3 . 2 ) ] . Let U.I = Op(Pi ) ' Z = Z(S),

T i = <Z l> = Z(Ui ) and Z i = Z(~ i ) = ~(Ui) f o r i = 1,2. Then TIT 2 = U 1 n U 2.

Suppose that UI/Z I is the d i r e c t sum of two natural ~l-mOdules. Then U 2 acts

quad ra t i ca l l y on U1/Z I and so [UI /Z I ,U 2] = CUI/ZI(U2) = T2/Z 1. Hence

[UI,U 2] = T 2. But. as ~ i ~ 2 we would have

T 1 = [U1,U 2] = T2, impossible.

So UI/Z 1 is indecomposable. Let H be a ~ - i n v a r i a n t complement to S in B.

By hypothesis o induces no f i e l d automorphism on ~ I /U I ~ SL2(3n). Since

PI/U1 ~ GL2(3 n) acts n a t u r a l l y on UI/T I there ex i s t s a T E H such that

[Ul,O%] s T I . Hence [PI,OT] S U I and [UI,~%] = I , since UI/Z I is indecompos-

able and o% is a p ' -e lement . This impl ies [Pl,O~] = i and then also [ P 2 , ~ ] = I .

which proves the lemma.

34 ~.GTIMMESrELD

§ 4 G i ~ N(Z) fo r some i E I in the constrained case.

We assume in th i s sect ion the fo l low ing :

(4.1) Hypothesis.

(1) G is generate by three rank 1 parabol ic groups Po,P1,P2 in char.p wi th

common Borel-subgroup B.

(2) B G = I and F(1) is a t r i a n g l e , where I : { o , i , 2 } .

Introduce the fo l low ing f u r t he r notat ion:

S E Sylp(B) , H a f i xed p'-complement to S in B,

Gi : oP ' (G i ) ' ~ i

and K i = BG. 1

Then obvious ly

= oP' (P i ) f o r i E I ,

K i

G i = <Pj,Pk > fo r { i , j , k } = I ,

Mi = Op(Pi) ' Qi = SGi= n Sg' g E G i

Let Z = ~ I (Z (S ) ) ,

Now the f i n a l hypothesis:

= Q i (HnK i ) and QiQj ~ M k fo r I = { i , j , k } .

G i ~ P. Z i = <Z >, F i = Cz. (Gi ) , C i = C~ (Z i ) and T i = <Z l>.

1 1

(3) Cs(Qi) ~ Qi fo r i E I . ( l . e . we are in the constrained case!)

The f i r s t consequence we draw from (4.1) i s :

(4.2) The fo l low ing holds:

( i ) Gi /Qi is a group wi th weak BN~pair of rank 2 with respect to ~ j ( B n ~ k ) / Q i

and ~k (Bn~ j /Q i fo r each i E I and { i , j , k } = I . ( ( I ) does not need

Hypothesis (3 ) ! )

(2) Suppose Z i * Z. Then Cs(Zi) = Qi" ( I t is obvious that Cs(Ti) = M i i f

T i , Z ! )

(3) Without loss Z = T o .


Proof.

( I ) As Mj rl M k .~ Pj and ~ Pk' since F(I) is a t r i ang le i t follows that :

C~ (Mp/Q~) _c M£ fo r £ = j , k . - ~ ~ |

35

This impl ies 1).

Assume Z i , Z. Obviously Qi ~ Cs(Zi)" I f now Qi < Cs(Zi) then (1) and

(3. 3 ) imply G i = CG (Zi)B and thus Z i = Z, a con t rad i c t i on . This proves (2) . 1

Suppose Z * T i fo r each i E I . Then, as r (1) is connected Propos i t ion

(3.1) of [19] appl ied to the group Gi/Q i impl ies J(S) ~ G, a con t rad ic t i on

to Hypothesis (4 .1 ) (2 ) .

We assume now fo r the res t of th is sect ion by way of con t rad ic t i on that Z ~ G i

f o r each i £ I . Then we have:

(4.3) The fo l l ow ing holds:

(1) Z • T i f o r i = 1,2

(2) J(S) < qo

(3) r o = 1.

Proof. ( i ) is obvious. (2) fo l lows again by apply ing Propos i t ion (3.1) of

[19] to Go/Q o. To prove (3) l e t G= <~i I i E I > . Then G = GH and, as G =

<Go,Po >, (4 .2 ) (3 ) impl ies F ° < ~. Now (3) is a consequence of S G = 1.

Now the f i r s t serious reduct ion:

(4.4) Suppose ~ (S) 0 M o ~ Qi

s a t i s f y i n g :

(a) A $ Qi"

(b) (AnQ i )Z i E ~ (S ) ,

f o r i = 1 or 2. Then there exists no A c a(S)

36 F.G. TIMMESFELD

Proof. Suppose (4.4) is fa lse for i = I and le t AEC~(S) with A ~ QI'

but B = (AnQ1)Z1E C~(S). Then B ~ J(S) ~ Qo and thus

(A gnQ1)z l = B g ~ Qo for each g E G 1.

G 1 Let V 1 = ZI<Z ° >. Then, as Z o ~ Z(J(S)) and O~(S) n Q1 * 6 we have V 1

Z(J(QI) ), Since B g E ~(S) i t fo l lows that V 1 = ZlCvI(A ). G I

Let R I = <A >. Then [VI,R I ] ~ Z I and by (3.1) and (3.3) e i the r GI =

CIR I or p = 2, IGI : CIRll = 2 and ~I/Q1 is ] o c a l l y isomorphic to Sp(4,2).

Moreover, as C~6(S) n M o ~ QI' again (3.1) implies that ZI/F I is an i r reduc ib le

~l-mOdule and M 2 ~ ClJ(S ) ~ CIQ o. Hence

( , ) [Zo,R 1riG o ] ~ Z o n Z I = T 2

by the act ion of P2/Q1 on Z I . Further T2/F I is a natural SL2(q)-module for

~2/M2. As ~2 ~ M2(P2nRI) (4.3)(3) and (*) therefore imply that Z o has an

i r reduc ib le ~o-submodule Z' sa t i s f y ing Z o = Z~Z. 0

Let - be the natural homomorphism from ~o on ~o/Co. As M~ = [M2,P 2nR I ] ~ M 2 n R l ,

( . ) implies that M2' acts quadra t ica l l y on Z o and so is elementary abel ian.

Further, as G O = <P1,P2nRI> we have Z ° = <(TIT2)PI> by ( . ) . This shows that

there is no ~1- invar ian t subgroup in MI n M2" As M~ is elementary abelian and

IM2: M~I ~ 2 now (4.2)(1) and the Delgado-Stellmacher-theorem [4 ] show that:

~i/Qo ~ q2.SL2(q) for i = 1,2.

- # Now a l l elements of M i are conjugate in ~i for i = 1,2. Hence ( . ) impl ies:

[Zo,X] ~ CT2(X ) = Z for each x E M 1 - M 2

since MI ~ P2 n R 1. As [Zo,M I ] is P l - i nva r ian t we obtain [Zo,M 1] ~ CTI(~I) .

Since F o = i and T2/F 1 is a natural SL2(q)-module for ~2/M2, we obtain

ICTI(~I) I ~ q. I t now eas i l y fol lows that IZ~I ~ q3 and thus Z o = TIT 2,

cont rad ic t ion to MI n M2 * 1, This proves (4.4) .


We can st rengthen (4.4) to

37

(4.5) For i = 1 and 2 ex i s t s no A E (~C(S) s a t i s f y i n g

(a a # Qi '

(b (An Qi)Zi E ~ (S) .

Proof. Suppose (4.5) is f a l se fo r i = I . Then (~.4) impl ies 6~(S) n M ° =

~(S) n QI * 6" I f (4.5) is also fa lse fo r i = 2, then

J(Q1) = J(Mo) = J(Q2)

a con t rad i c t i on to (4 .1 ) (2 ) . Hence (4.5) holds fo r i = 2 and thus ( 3 . 2 ) impl ies

that ~2/C2 is l o c a l l y isomorphic to SL3(q) or Sp(4,q) and Z2/F 2 is a natu-

ral module. But the act ion of these groups on t h e i r natura l modules and (4 .3 ) (2 )

imply: G%(S) n M o ~ S ) n M o n M 1 ~ Q2"

Hence again J(Q1) = J(Mo) = J(Q2), a con t rad ic t i on .

As a consequence of (4.5) we obta in:

(4,6) The fo l l ow ing holds:

(a) ~ i / C i is isomorphic to SL3(q) or Sp(4,q) and Z i /F i is a natural

module fo r i = 1,2.

(b) ~ (S) n M ° = 6.

(c) Z ° = Z~Z Z' an i r r e d u c i b l e Go-module O

(d) Qo is elementary abel ian and Zo $ Qi fo r i = I or 2.

Proof. (a) fo l lows d i r e c t l y from (~.5) and ( 3 . 2 ) . Suppose ~(S) n M o , 6.

Then (4.5) and (a) imply J(Mo) ~ Qi ' i = 1,2 whence by the same reason J(Q1) =

J(Q2) ~ G, a con t rad i c t i on .

To prove (c) l e t Z' be an i r r e d u c i b l e Go-submodule of Z o Then Z' is O " O

n o n t r i v i a l , as F o = i . I f Zo/Z ~ is a t r i v i a l To-module (c) holds. Thus assume

i , Zo/Z o L is n o n t r i v i a l

38 ~.OT~MM~SFELD

Pick AE ~(S) Then by (a) and (b) IA: AnMol s q and A n M o $ QI'

Hence there ex i s t s a conjugate B ~ M o of A n M o wi th B $ M 2. As

IZo:CZo(B) I $ q lB : CB(Zo)I, ( 3 . 1 ) now impl ies that Z'o and L are FF-modules

fo r Go/Q o. Since T1/F 2 and T2/F 1 are both natural SL2(q)-modules fo r ~ i / M i ,

I i = 1,2, L cannot cota in a quo t ien t which is equ iva lent to Z o as ~o/Qo-module.

Hence i t is easy to see that ( 3 . 1 ) impl ies that ~o/Qo is l o c a l l y isomorphic to

SL3(q) and the only n o n t r i v i a l composit ion fac to r of L is the dual module of

Z' O "

NOW the above impl ies C = (BnQo)Zo E C~(S). Hence by (b) Z o # M o and thus

QoZi = Mj fo r { i , j } = {1 ,2 } . This in turn impl ies by (a) tha t Gi/Qi is l oca l -

ly isomorphic to Sp(4,q) f o r i = 1,2 and so, as [Z1,Z 2] s Z I n Z, i t fo l lows

eas i l y tha t [T i ,Z i ] = i , i = 1,2. Now

[ZI ,Zo,Z 2] ~ [T2,Z 2] = i = [T1,Z I ] ~ [Z2 ,Zo,Z I ] .

Hence by the three-subgroup-lemma [Z I 'Z2 ] ~ Qo' a con t rad ic t i on to [MI,M 2] $ Qo'

This proves (c ) .

Suppose Zo ~ QI n Q2" Then by [ 19 , (2 ,1 ) ] J(Qi ) ~ Qo' i = 1,2 and also

J(Mo) ~ Qo" Hence arguing as in (4 .4) wi th AE ~(Mo) we obtain J(Q1 ) = J(Mo) =

J(Q2), a con t rad i c t i on . This proves the second par t of (d) . To prove the f i r s t

par t assume Z o $ 'QI . Then by (a) QiZo = M2, since Z o is no FF-module. Hence

[Zi ,Qo] = [Z I ,Z o] ~ Z I n Z o : T 2 G

and thus R o = <Zl°> cen t ra l i zes Qo/Zo. As F o = i (c) impl ies {(Qo) = I or

Z' ~ {(Qo) and we may assume the l a t t e r . Then oP(Ro) cen t ra l i zes Qo/{(Qo) o

and so cen t ra l i zes Qo' which is obviously impossible. This proves (4 .6 ) .

(4.7) The fo l l ow ing holds:

(a) E i ther IZi = q and F 1

IF~l = q fo r i = 1,2.

2 = F 2 = 1 or IZl = q and Z = FIF 2 wi th

Proof.

Zo # ql" Zo $ q2"

ONAMALGAMATIONOFRANKIPARABOLIC GROUPS 39

(a) is obvious by (4 .6 ) (a ) . To prove (b) we may by (4 .6) (d) assume

Then by (4.6)(a) ZoQ I = QoQI = M2, since Z o is no FF-module. Thus

(4.8) One of the fo l lowing holds:

(1) q = 2 n, ~o/Qo is l oca l l y isomorphic to Sp(4,q) and ~ i /Qi is l oca l l y

isomorphic to SL3(q) for i = 1,2.

(2) q = 3 n, Go/Qo is l oca l l y isomorphic to G2(q) and Gi/Qi is l oca l l y iso-

morphic to SL3(q) for i = 1,2.

Morover, in any case [ZI ,Z 2] = 1.

Proof. Suppose f i r s t p is odd. Then, as QoQi = Mj for { i , j } = {1 ,2} ,

(4 .6)(d) impl ies that Mj/Q i is elementary abel ian for { i , j } = {1,2} . Hence

~ i /Qi is l oca l l y isomorphic to SL3(q) for i = 1,2. This implies by (4 .7) (b)

ZiQo < Pj/Qo and IZiQo/Qol = q for { i , j } = {1 ,2} . Since ~ i /Mi ~ SL2(q) for

i = 1,2 ( 4 . 2 ) ( i ) and the Delgado-Stellmacher-theorem imply tha t . (2 ) holds.

So assume p = 2. I f Gi/Qi is of type SL3(q) for i = 1,2 then, arguing

as above, (1) holds. So assume that ~ I /QI is l oca l l y isomorphic to Sp(4,q).

Then iZiQo/Qo i = q2 and is a natural module for ~2/M2. I f now G2/Q2 is

also l oca l l y isomorphic to Sp(4,q), then iZ2Qo/Qo i = q2 and the Delgado-Stel

macher-theorem implies that ~o/Qo is l oca l l y isomorphic to SL3(q), A 6 or ~6"

But then [ZI 'Z2] ~ Qo' which is arguing as in (4.6) , impossible.

So ~2/Q2 is by (4.6)(a) l oca l l y isomorphic to SL3(q ) and tZ2Qo/Qoi = q.

Hence Z2Qo/Q o ~ Z(S/Qo) and so by the Delgado-Stellmacher-theorem, Z2Q ° < ZIQ °

But then T 1 = Z[Z2,Z o] ~ Z[Z1,Z o] = T 2,

which is obviously impossible.

40 r.m. TIMMESFELD

Now in any case IZiQo/Qol = q fo r i = 1,2, and thus

{ i , j } = {1 ,2 } . Since ZIQ2 ~ Po' th i s proves (4 .8 ) .

Zi ~ QjQo ~ C(Ti) fo r

We are now in the pos i t i on to obta in the f i na l con t rad i c t i on . By (4.8)

ZiQo/Q o play fo r i = 1 and 2 the r o l l of a long and a short root subgroup of

Go/Qo . Hence i t is easy to see tha t

S = Qo<Z~, Z~> fo r some elements g,h E G O .

As JQo: QonQi I = q2 th is impl ies IQo: Zi ~ q4. Since IZl = q or q2 depending

i f ITIT2r = q3 or q4 we obta in by (4.8) in any case

Qo n QI = TIT2 = Qo n Q2

and thus Z o = T I T 2, a con t rad i c t i on . This shows:

(4.9) I f Hypothesis (4.1) holds, then Z ~ G~ fo r some i E I.


§ 5 The main reduct ion.

41

In th is sect ion we assume;

(5.1) Hypothesis.

( i ) G is the f ree amalgamated product of the 3 rank i parabol ic groups

Po,PI,P2 in char.p over the common Borel-subgroup B.

(2) B G = i and r (1) is a t r i a n g l e , where I = {o ,1 ,2 } .

Use the notat ion introduced in § 4. As in § 4 we assume:

(3) Cs(Qi ) ~ Qi fo r i E I .

and, u t i l i z i n g the main r esu l t of § 4

(4) Z ~ G 2.

As a f i r s t consequence we obta in:


( I ) (4 .2 ) (1 ) holds.

(2) G o n G 1 = P2 and

CZ(~2) = i . (3)

(4) Cs(Zi) = Qi fo r

Proof.

(P2) G = 1.

i = o,1.

(1) and (4) are ( 4 . 2 ) ( I ) and (2). (3) is a d i r ec t consequence of

(5 .1 ) (4 ) . The f i r s t par t of (2) fo l lows from [6 , ( 7 . 9 ) ] . ( I t is a well-known

property of f ree amalgamated products ! ) . To prove the second par t l e t R = (P2) G.

I f R s B, then R = I by (5 .1 ) (2 ) . So assume R $ B. Then the s t ruc ture of

the rank I parabol ic group P2 impl ies P2 = RoB. As C~2(M2/Qi) ~ M 2 fo r

i = o,1 since 2 is connected to o and 1 in r ( 1 ) , we have R n M 2 ~ Qi fo r

i = o,1. Hence (3. 3) impl ies G i = RB fo r i = o,1, a con t rad ic t ion to G o , G 1

42 F, G. TIMMESFELD

We int roduce the fo l l ow ing no ta t ion :

Let F = r(Go,G1) be the coset graph of Go,G 1 in G. Then G

an edge- t ran t i ve automorphism group of £ wi th edge s t a b i l i z e r

s t a b i l i z e r s G O and G I . For each ver tex a of F l e t

• A(a) : : {B E G } ( a , B ) is an edge} ( I ~ ( ~ ) I = ~ !)

I f a = Gig, i = o or I l e t

Ga' ~ ' Qa' Ka and Z a

be the corresponding conjugates of Gi ' ~ i ' Qi ' Ki resp. Z i and C a = C~ (Za),

Ga = Ga/Ca" Let d(a,B) be the usual d is tance metr ic on F and

d := Min{d(a,B) la ,B E F and Z a # QB }.

of ve r t i ces of F wi th d(~,u) = d and Z~ ~ Qu w i l l be ca l led

One has:

a pa i r (~,~)

a c r i t i c a l pa i r .

is by (5 .2 ) (2 )

P2 and ver tex

(5.3)

( i ) K a is the kernel of the act ion of G a on &(a) .

0 ' (2) K a is p-closed with Qa = p(Ka)"

(3) 1 5 d < ~.

(4) CQaQ (Za) = Qa fo r each 6 E A(a) .

Proof. To prove. (1) we may assume a = G i . Then K a= (P2)Gi i s by ( 3 . 3 )

and ( 5 . 2 ) ( i ) the kernel of the act ion of G i on A(a). (2) is obvious and (4) is

(5 .1 ) (3 ) . Since G acts f a i t h f u l l y on F we have Z a ~ K B fo r some B E £.

As Za ~ Qa th is proves (3) .

Let now (~o,~d) be a c r i t i c a l pa i r and ~ = (~o,~1 . . . . . ~d_1,~d) an arc frol

&o to kd" Then


( i ) L ] n [ZXo,ZXd ~ ZXo ZX d"

(2)

(3)


Without loss l~d I ~ tZ~ :Cz~ (ZXd)I (Considering ^ : G x ~ G~o/C~Xo(ZXo ) O O O

Either Z~ is an i r reducible FF-module for GX or IZXo I = q6, i~Xd i = q2 0 0

= IZXo: CZx (ZXd)I and [ZXo'ZXd] = ZXo n ZXd = CZx (ZXd) has order q4. O O

!)

Proof.

(5 .3)(4) .

As ZX g ~ and ~ g GX (1) is a consequence of o QXd-I - GXd Zld - QxI o

Now i t is clear that e i ther (2) or the symmetric statement with the

ro l l s of o and d interchanged, holds. (3) is now (3. 1 ).

Now we choose notat ion so that Xo = Go' X1 = GI" (Exchanging o and 1 i f nec-

essary!) Since the main object ive of th is section is to show that in almost a l l

cases d = I , we assume d ~ 2 by way of contradict ion for the rest of th is

section. To s impl i fy notat ion we drop the index ~ for the various subgroups

corresponding to vert ices of the arc ~ . Set

V = Za<ZBI B e A(G)> ~ QG (as d ~ 2!)

for a E r. Next we show:

(5.5) The fol lowing holds:

(1) QoQ1 = M 2 or IM2:QoQ11 = 2 and ~o is l . i . to s 6.

(2) ~i * CGi(Z4)N~i(ZiZ j ) for { i , j ) = {o , I } . Especial ly

C r and each ~ E ~(m).

Proof. Since Z ~ P2' (.5.4)(3) and (3. 2 ) imply P2

QIQ2 ~ QI<Zd > = M 3

or Go ~ ~6'~6 and IM2:QoQzl = 2. Suppose Gi

Gi = CGi(Zi)NGi(QinQj) as Qi n Qj = cQi (z iz j ) ,

[Qi'Qj ] ~ Qi n Qj. Since by ( I ) and ( 3 . ~ ) :

IGi : C~i(Zi)<QjGi> I z 2

this implies ING'(Qil n Q j ) : c ~ i ( Q i / Q i n Q j ) l ~ 2. A contradict ion since

V > Z Zg

= C~i(Zi )N~i(ZiZj ) . Then

Now Qj ~- NG i~ (QinQj) and

for each

44 F.G. TIMMESFELD

P2 ~ NGi(QiAQj) ' but oP(P2) acts by (1) n o n t r i v i a l l y on QiQj/Qj and thus

also on Qi/Qi n Qj. This proves the f i r s t part of (2). The second part is a

d i rec t consequence of the f i r s t .

(5.6) The fo l lowing holds:

(1) Z o is l oca l l y a natural Go-module. ( l . e . case (3 ) , (4 ) and (6) of (3.1) do not hold!)

(2) Z o is not l oca l l y the natural U4(q) or Us(q)-module.

Proof. F i r s t suppose by (5.4) and (3.1) that Go is # . i . to U4(q) or U5(q ) and

Z is the natural G -module. Then o o

lZ d : ZdnQol = q4 = iZo: ZonQdl = t[Zo,Zd] I

and [Zo,Z d] is the natural ~2/M2 = SL2(q2)-module. Hence by (3.1) Gd is £ . i . to

U4iq), U5(q) or Sp(4,q 2) and Z d is the natural Gd-mOdule. Now in any case there

ex is ts a subgroup U of index q3 in Z d such that IZo: C z (U)I = q2. Now the o

act ion of Gd on Z d implies IZd:CZd(CZo(U))I = q 2 a cont rad ic t ion to (3.2)(9)

This proves (2).

Suppose (1) is fa lse. Then by (5.4)(3) Z d is also an FF-module for Gd"

As iZoQd/Qd I = q2 ~2/M 2 ~ SL2(q) and [Zdl ~ q6 a l l by (5 .4) (3) , ( 3 . 1 )

shows that Z d is also not natura l . Especial ly

[Zb'Zd] : CZo (Zd) = CZd(Z°) = z ° n z I = Z dnzd_ I .

Pick now by (3, 4 ) ~ E &(Xd) with G d = <Zo,GdnG >. I f d ~ 1(2) then

Z I ~ Z d in G. I f d - o(2) and Z $ Q1 then Z I ~ Z~ in G and Z I is also an

FF-module for 61 . Hence, i f Z # QI' then Z I is in any case an FF-module for

GI which is not i r reduc ib le , since P2 does not act i r reduc ib le on Z 1 n Z 2.

Hence Z o n Z 1 = Czl(Z ) = [Z~,Z ] = Z d n Z

as above. But then Zd_ I N Z d = Z d n Z , a cont rad ic t ion to the choice of ~.

ONAMALGAMATIONOFRANK1PARABOLIC GROUPS

QI and [Zv,Zo] = [Zd,Z o] ~ Z d. But then Thus Z _

t r ad i c t i on to (5 .5) (2) .

G d ~ N(ZdZ ), a con-

45

(5.7) G ° ~ SL3(q) or Sp(4,q) and

l[Zo,Zd]J = IZd :Z dnQol = IZo :Z onQd I = q.

Proof. Suppose (5.7) is fa lse. Then by (3.2) and (5.6) T = <ZP2> ~ [Zo,Zd].

Pick by (3.4) ~ E A(~d) with G d = <Zo,G dnG >. Then (3.2) implies as in

(5.6) that Z $ QI"

Claim IZd: Z dnQo I = IZo: Z onQd I. I f not then by (3.1) and (3.2) Go ~ SL3(q)

or Sp(4,q) and IZo :Z onQd I = q resp. q2. Hence d , I (2) and e i the r [ZI ,Z u] =

Z d n Z or I[Z1,Z ]I = q = IZ :Z nQ1 I. Now the f i r s t case contradicts the

choice of ~. Thus by (3 .1 ) , (3 .2 ) G1 ~ SL3(q) or Sp(4,q) and Z 1 is the natural G1-

module. As Z 1 ~ Z d we obtain G1 ~ Sp(4,q) ~ Gd" Now we may by (3.5) choose

with the addi t iona l property that Z d = (Z dNZd_I)(Z dnZ ). As Z o cent ra l -

izes (Z dnZd_ l ) [Z1 ,Z ] we obtain IZd: Z dnQo I = q ~ IZo: Z onQd i , which

proves by (5.4) our claim.

Hence Z d is also an FF-module for Gd and, as Z $ QI' Z I and Z are in

any case i r reduc ib le FF-modules for G1 resp. G . As above

Hence J[ZI 'Z~] I = q' G1 : SL3(q) or Sp(4,q) and

and the same holds for ~.

Now obviously Gd is not l . i . to SL3(q ). I f Gd is Z . i

as Z o cent ra l izes (Z dnZd_ l ) [Z1 ,Z ] , we obtain choosing

IZd: Zd Qo = q' a cont rad ic t ion. Thus by (3~I) and (5.6)

~-(6,2 n) and [Zo,Z d] = CZd(Zo).

Choose now by (3.2)(8) p E A(~d) with [Zo,Z d ] n Zp = 1.

ZpnZ dnC(zo) = i . As Zp centra l izes Z d

by the choice of p, Z o $ C(G dnGp), where

C<Zo,~ dnGp> = CN~d(ZdZp), a cont rad ic t ion to (5 .5) (2) . This proves (5.7) .

Z d Z ~ [ Z I , Z ] .

Z 1 is the natural Gl-mOdule

to Sp(4,q) or A 6 then,

u ~s before that

Gd is Z . i . to G2(2n),

Then [Z1,Z p]

we obtain [Zo,Z p] = [Zo,Zd]. Now

C : C~d(Zd). Hence ~d =

46 F. G, TIMMESFELD

(5.8) Suppose V ° z Qd_lQd . Then Go ~ SL3(q) ~ G1

natural SL3(q)-modules.

and Zo,Z 1 are

Proof. Suppose false. By (5.7) there exists a p E ~(~o) such that

Zp N [Zo,Z d] = i and Z o = (Z oNZ )(Z oNZI). Then Go = Co<GoNG ,Zd> , where

C O = C~o(Zo) and so by (5.5)(2) [Zd,Z ~]=Z d N Zd_ I. Suppose Gd ~ Sp(4,q)

and let A = Z d N Qo" Then

[Zd,Z p] = [Zo,Zd][A,Z u] z ZoZ u,

since otherwise [A,Z ] ~ [Zo,Z d] n Z = I, which is impossible. But this is

again a contradiction to (5.5)(2).

Thus Gd ~ SL3(q)" Suppose Go ~ Sp(4,q). Then d ~ 1(2), IZIZo/Z ° = q

and for ~o = Vo/Zo we have:

[~o,Qo ] = 1, l [#o,Zd] l = q = l # o : C v (Zd) l . 0

Hence by ( 3 . 1 ) Go/C~o(~O) ~ Sp(4,q) and T 0 is an extension of "the other"

( i . e . nonequivalent to Zo] ) 4-dimensional GF(q)Sp(4,q)-module by a t r i v i a l module.

As oP' !c~ (To)) ~ C ° th is is a cont rad ic t ion to I [#o,Zd] l = q. 0

SO Go ~ Gd ~ SL3(q)" Next suppose that GI ~ Sp(4,q) and Z I is the natural

Gl-mOdule. Then [Vo,Qo] ~ Z o.

(Exists by (3.6)!) Then, as

Z d N Zd_ 1 = [Z6,Z d] ~ Z 6 and

so as above i~ol = q3. Hence

(,) v o

And so

Pick 6 E ~(~o) with Z d ~ G O n G 6 but Z d ~ QoQ6

G6 ~ Sp(4,q) and Z 6 is the natural module, R = N

n G 8 _ ~ Z 6 = <R G° >. This implies T ° = [Vo,G o] and

= ZoZIZ X for a l l X E A(~o) with Z onZ I ,Zo 'NZX.

This implies Qo

So i t only remains to be shown that GI ~ Sp(4,q) and Z I

module. Let 8 E A(8o) be as before. Then [Zs,Z d] = [Zo,Z d]

[Vo,QonQ1 ] s Z oNf IZ~ , where ~ runs over a l l ver t ices in

A(~o) sa t i s f y ing ( . ) .

N QI = CQo(Vo) < Go' a cont rad ic t ion to (the proof of) (5 .5 ) (2 ) .

is the natural GI-

or [Z6,Z d] = R.


In the f i r s t case ( 3 . 2 ) implies G6 ~ Sp(4,q) or SL3(q) and Z 6

module. Since we assume (.5.8) is false the second case holds.

Let L be generated by three conjugates of Z d sat is fy ing

the natural

Go = C O • L and

S ~ Qo L. Then we may pick a

i [Vo,L] l ~ q6 ~ iVo: CV (L)I 0

W o = <Z~> S [Vo,L] . Z 1 and

Since Z6 ~ Wo

i t follows that

47

6 with the above properties in ~ . Then

and R ~ [Vo,L] N Z 6. I t immediately follows that

I [Vo,L]I = q6. Moreover, Wo = [Wo 'L] m C~ (L). o

by the above, and since Z6 is a nontr iv ia l module for L N G 5

IZ 1 n [Wo,L]I = q2 and IWo:Zl l = q. Hence

C~ (L) = ~ Z6 6 c xL ' 1" 0

Suppose ~ = C~o(L ) • 1. Then, as Qo s N(#), also ~ : C#(Qo) • 1. Now

[E,Q o] ~ Z o n ~ ZS, 6 c ~ and so [E,Q o] = 1. As IE: CE(Zd)I = q th is im-

pl ies E = Z ox CE(QoL), a contradict ion to Z ~ Z o and S ~ Qo L.

So C = i , Wo = [Wo 'L] has order q3 and IZII = q4. Now Qo acts as

offending subgroup on Z I and i t is by (3. I ) obvious that GI = Sp(4,q) and

Z 1 is the natural Gl-mOdule.

(5.9) Either d ~ I or Go = SL3(q) ~ G1 and Zo,Z 1 are natural SL3(q)-

~odules.

Proof. Assume false. Then we f ind by (5.7), (5.8) without loss a c r i t i ca l pair

~o,~d with Gd = Sp(4,q), l[Zo,Zd]1 = q and V o $ Qd_lQd , V d $ QoQ1 , Let A

Z d n Qo and pick ~ C ~(Xo) with Z o = (Z oNZX)(z onZ1) and [Zo,Z d] n Z~ = I

and t o = Co<G oNGX,Zd> where C o = C~ (Zo). Suppose Z~ ~ Qd_IQd. Then by 0

(5.5)(2) 1 , [Z~,A] ~ Z~ n (Z dNZd_l) n Zo,

since by (~.7) Z~ is the natural module for G~ ~ SL3(q ) or Sp(4,q). This im-

pl ies

and thus

Z d n Zd_ I = [Zo,Zd][Z~,A] ~ Z °

[Z~,Z d ] s Z o, a contradict ion to (5.5)(2) .

48 F.G. TIMMESFELD

Hence Z~ $ Qd_lQd for a l l such ~ E &(~o). We obtain 1 ~ R~ = [Z~,Zd_l]

Z~ n Z ° n Zd_ 1 ~ C z (Zd). Suppose Go ~Sp(4 'q ) and claim o

(*) CZo(Z d) = [Zo,Z d ]~R A, where the product is taken over a l l ~ E &(~o)

with Z~ n Z 1N Z o = i . I f not, then I[Zo,Z d]nRAI = q2 and is a to ta l l y iso-

tropic plane of Zo, considered as 4-dimensional symplectic space. Hence we find

a ~ E ~(~o) with

Z o n Z n [Zo,Zd]nR~ = 1 = Z ° n Z n ZI,

a contradict ion. So our claim holds and thus Czo(Z d) ~ Zd_ 1. Hence C Z (Zd) = Zd_ I ^ O

i f Gd-i ~ SL3(q) or IZd_l: C Z (Zd)I = q i f Gd_ I ~ Sp(4,q) and so in any case O

I[Vo,Zd_1]I ~q. But this is obviously a contradict ion to ( * ) .

This shows Go ~ Gd-1 ~ SL3(q) and d > 2. Moreover, as

= C z (Z d) = [Vo,Zd_ I ] = Zd_ I n Zd_ 2. o

Pick now Xd+1 c ~(~o) with (Z dnZd+l ) (Zdnzd_ l ) = Z d. I f

(~l,~d+1) is a c r i t i ca l pair and vle obtain as above

Z 2 n Z 3 = [Z2,Vd+I] = Z d n Zd+ I.

But this is impossible since d > 2 and [Zd,Zo] ~ I .

R~ # I , Z o n Z 1 =

Zd+l $ QI' then

Hence Zd+l ~ Q1 for al l such ~d+l E ~ (~d)

[Zo,Zd+ 1] ~ Z o n Z 1 ~ Zd_lZ d.

and

G d n Gd_ I Since QdQd_l = Qd<Zo > we obtain [ZdZd+l,QdQd_ 1] ~ ZdZd_ 1 for al l such

~d+l" Let H d = Qd<Qd_1,Qd+l >. Then Gd = CdHd by a well-known property of sym-

plect ic groups. By symmetry H d ~ N(ZdZd_lZd+l). (We may for example choose ~d+l

such that ~d_1,~d+l are interchanged by an involut ion in Hd! ) But as

iZd_lZd+l/Zd I ~ q2, Hd/H d n C d = Sp(4,q) cannot act non t r i v ia l l y on Zd_lZd+l/Zd,

while H d n C d ~ C(Z dnZd_l ) . Hence H d s NGd(ZdZd_I), a contradiction to (5.5)(2)

(5.10)

Vo s Qd- l '

Proof .

ONAMALGAMATIONOFRANKIPARABOLIC GROUPS 49

Ei ther d S 2 or there ex i s t s a c r i t i c a l pa i r (~o,2d) wi th

Suppose (5.10) is fa lse . F i r s t we show:

( i ) There ex i s t s a c r i t i c a l pa i r (~o,~d) wi th

Z ° n Z I ~ Z I n Z 2 or Z d n Zd_ I ~ Zd_ I N Zd_ 2.

Assume (1) is also fa lse . Since we assume that (5.10) is f a l se , there ex i s t s a

E A(20) such tha t (p,%d_l) is a c r i t i c a l pa i r . Since ( i ) is fa lse fo r a l l

c r i t i c a l pai rs and since V ~ Qd-2 we obtain as above:

Z d N Zd_ 1 = Zd_ 1 n Zd_ 2 = . . , = Z I n Z 2 = Z I n Z o.

Pick now ~ E A(~O) wi th Z o = (ZonZ~) [Zo ,Zd ] . I f Z~ < Qd-1 then [Z~,Z d]

Z d N Zd_ 1 _~ Zo, a con t rad ic t i on to Go = Co<GonG~'Zd>' Co = C~ (Zo) and O

(~ .5 ) (2 ) . Hence (k,Xd_l) is a c r i t i c a l pa i r s a t i s f y i n g ZxNZ o • Z o N Z 1.

This proves (1) .

Next we strengthen (1) to:

(2) There ex i s t s a c r i t i c a l pa i r (Xo,kd) wi th

Z o n Z I • Z I n Z 2 and Z d n Zd_ I • Zd_ I n Zd_ 2,

50 F.G. TIMMESFELD

Suppose (2) is false. Choose by ( I ) a c r i t i ca l pair

Z 1 n Z 2. Since (5.10) is false

with Gd_ 1 E n(6d_l)

and so

Zx $ Qd-1

: > . Cd_I<Z~,Gd_I n G

Z n Zd_ 1 , Zd_ 1 n Zd_ 2.

(Xo,Xd) with Z o n Z 1 •

for some X E ~(Xo). Pick

Then [Z~,Zd_l](Zd_ l n Z ) = Zd_ 1

= [Zo,Z 3] ~ Z 3,

: C3<Zo,~ 3nGp> and (5 .5) (2) . But th is implies

a cont rad ic t ion since we assume (5.10) is fa lse.

Hence Z ~ Qo" As d > 2, [Z~,Z ] ~ C Zo (Zd) = z° n Z I . I f [Z~,Z ] =

[Zo,Z d] ~ Zd_ I we get a cont rad ic t ion to (5 .5) (2) . So

Z o n Z 1 = [Zo,Zd][Z~,Z ~] ~ Vd_ I ,

I f now d > 3, then [Vd,Vd_ I ] = 1 and thus, as Z I= (Z onZ1)(Z INZ2) , V d~Q1

a cont rad ic t ion to the assumption that (5.10) is fa lse.

Thus d = 3, Z 1 n Z 2 = Z 2nZ 3 and, since (2) is fa lse,

(*) [Z ,Zp] = 1 for a l l ~ E n(~2), p E ~(~3) with

Z n Z 2 • Z 2 n Z 3 • Z 3 n Zp.

Let V2, I = <Z J~ E &(X2), Z n Z 2 , z 2nZI>z 2. Then Z~ ~ N(V2,1) and thus

Z o n Z I = [Zo,Z3][Z~,Z ~] ~ V2,1.

Hence Z I < V2, I and by (*) Zp ~ QI for a l l p E &(X3) with Zp

Pick such a p with Z 3 = [Zo,Z3](Z 3nZp) , Then [Zo,Z p ]=Z o N Z1,

otherwise [Zo',Z p]

a cont rad ic t ion to ~3

and so V3 ~ QI'

n Z 3 , Z 2 n Z 3 .

since

Z I < V 3

Now pick a c r i t i c a l pai r (Xo,~d) sa t i s f y ing (2). Let ~ E A(~o) with

(Z~nZo)[Zo,Z d] = Z o. I f Z~ s Qd-i then [Z~,Z d] = Z d n Zd_ 1 ~ V o. As

Zd_ 1 = (Z dnZd_ l ) (Zd_ lnZd_2) by (5.9) th is implies Vo ~ Qd-l ' a cont rad ic t ion.

So Z~ $ Qd-1 and

1 . [Z~, Zd_ 1] ~ C Z (Zd) n Z~ = Z o n Z 1 n Z~. o

ON AMALGAMATION OF RANK l PARABOLIC GROUPS

This implies Z ° N Z 1 = [Zo,Zd][Zx,Zd_ 1] ~ Zd_ 1 and so Vd ~ QI as before.

This f ina l cont rad ic t ion proves (5.10).

51

. Then C = (5.11) Suppose (Zo,~d) is a c r i t i c a l pair with Vo - Qd-1 o

C~ (Vo/Zo) and Vo/Z ° is a natural ~ -module dual to Z o, o o

Proof. By (5 .5) (2) we have [Vo,Z d] = Z d n Zd_ 1. As [Vo,Q o] ~ Z o by (5.9)

T o = Vo/Z o is an FF-module for Go/Qo with

(*) I E~o,Zd]l : q ~ I~o: C V (Zd) l . o

Let C o = C~o(~O). Then C o normalizes a l l Z ° N Z~, ~ E A(Xo), since Z o n Z X =

[ZoZx,Qo]. Hence C o can only induce scalar automorphisms on Z o and

IC°/C ° N C o I l q - l . Now (3. i ) and (*) imply ~o/C ° ~ SL3(q) and

T o = Cx U; C = C~o(~O) , U = [Vo,G o] the natural SL3(q)-module.

This shows that C o s tab i l i zes a l l ZX, ~ E A(Xo) since i t s tab i l i zes a l l

Z o n ZX and thus can also only induce scalar automorphisms on Vo' Now (3. 7 )

= C° ~o o) , which proves the f i r s t part of (~.11). implies C O as = oP'(G

As P2 s tab i l i zes Z1 i t is obvious that U is the dual module of Z o.

Suppose q > 2. Then P2 N ~o contains an element h of order q-1 act ing f i xed-

po in t - f r ee l y on ZlC n U. Hence ZIC n U = [Z l ,h ] = Z1, thus # = i and (5.11)

holds.

So we may assume q = 2 and Go ~ SL3(2)" Further I~I = 2, since IZl l = 2.

As [Vo,Z d] = Z d n Zd_ 1 th is implies

(+) Z d n Zd_ 1 ~ Z 1.

We w i l l lead (+) to a cont rad ic t ion . For th is assume f i r s t d > 3 and pick by

(3.4) X_I E A(~o) such that Z_I n [Zo,Zd] = 1 and Go : <~onG- l 'Zd >. Then

by (5.5) Z d ~ N(ZoZ_I). Hence [V_I,Zd_ 2] ~ CZ_I(Zd) n Zd_ 2 S Z_INZ onZ 1nZd_ 2

52 F.G. TIMMESFELD

I f now Zd_ 2 N Zd_ 1 = Z d N Zd_ 1 then, as d > 3 and Z d n Zd_ 1 s Vo, e i ther

V-I ~ Qd-2 or Zd_ 3 N Zd_ 2 = Zd_ 2 q Zd_ I = Z d A Zd_ I . Now in the second case

[V_I,Zd_ 2] ~ (Z dnZd_ l ) N (Z_INZ1) so that by (+) also V_I g Qd-2"

So assume Zd_ 2 n Zd_ I . Zd_ 1 n Z d, Then Zd_2Zd_ I = Zd_2(Zd_ I nZd ) and thus

V_I ~ N(Zd_2Zd_I), whence V_I ~ N(Zd_ 2nZd_ l ) . Hence [V_I,Zd_ 2] ~ Z_I n Z 1 h

Zd_ I in th is case as we l l . We have shown

(a) V-1 s Qd-2 or

(b) Z ° n Z 1 S Zd_ 1 (by (+))

in any case. Assume f i r s t that (a) holds. Then

[V_I,Zd_ 1] s Z_I n C(Zd) n Zd_ 1 = Z_ InZ lnZd_ 1

As V_I cent ra l izes Z d n Zd_ 1 th i s implies [V_I,Zd_ 1] s (Z l n Z l ) n (Z dnzd_1)

_ ~ N(UV_I ) and so by (+) V-I ~ Qd-l" Hence IV l ,Zd] = [Vo,Z d] ~ U and thus t o _ G o

and [Ho,V_I] ~ U for H o = <Z d > . Now U # V_I, since otherwise G ~ N(V_I).

Hence IU/UNV_ll a 4 and [V_I,02(HoNG_I )] ~ Z o n Z_I by (5.5)(2) a contra-

d ic t ion .

Next suppose (b) holds. Claim Vd ~ QI in th is case. Let ~ E ~(~d) with

[Zo,Z d ] n Z~ = I . Then, as [Z1,Z ~] ~ Z I n Z d n Zd_l, Z~ ~ QI" Hence V d

ZdZd_I<Z~I~ E ~(Zd), Z~n [Zo,Z d] = I> i f our claim is fa lse. Especial ly Vd/Z d

is not the sum of a natural and a t r i v i a l SL3(2)-module.

Now IVd: V d n Q11 ~ 2 and [Vd,Z 1] ~ [Zo,Z d] by (+). I f V d n Q1 s ZdQo

then Vd/Z d is an FF-module for Gd and Z o induces a GF(2)-t.ransvection on Vd/Z d,

a cont rad ic t ion to (3.1) and the above. So (V dnQl)Qo = QIQo . Since Z o $ V d we

obtain IV dnQ l , vo ] = (Z o N z l ) ( z dnzd~ l ) ~ Zd_ I , As C~d(Vd/Zd) ~ C d, (3.9)

applied to the act ion of Gd/C~ (Vd/Zd) on Vd/Z d again supplies a cont rad ic t ion .


This proves the claim Vd ~ QI" Now pick Xd+ 1 E A(Xd) w i th [Zo,Z d] n

Gd ~ Zd+ 1 = i and = <Zo,G dNGd+l >. As Z o N Z I ~ V d and e i t h e r ZIZ 2 =

(Z 6 A Z I ) Z 2 or Z o n Z I = Z 1 n Z2, we obta in as before

[Z2,Vd+ 1] s (Z I n Z 2 ) n CZd+I(Zo) g Z 1 n (Z dnzd_ I n Z d + 1 ) ,

and so Vd+1 s Q2 by (+). Since also

[ZI,Vd+ I ] ~ Z 1 n (Z dnZd_ 1 n Zd+l) .

we obta in [Zo,Vd+ 1] ~ U d = [Vd,Gd]. But then we get a con t rad i c t i on as before

wi th the r o l l s of X_l and Xd+ 1 interchanged.

So the case d = 3 remains to be t rea ted . By (+) Z 1 n Z 2 , Z 2 N Z 3. Hence

Z 2 = (Z 1AZ2) ( z 2nZ3) s V o and e i t h e r V 3 n QI g Z3Qo or [v 3 n Q l , v o ] = Z 2.

Now as before e i t h e r (3.1) or (3.9) imply that V3/Z 3 is the d i r e c t sum of a

natural and t r i v i a l SL3(2)-module and thus V 3 g O 1 . But then we have symmetry

between Xo and ~3" Hence Z o N Z I • Z I N Z 2, s ince Xl ~ X3 and so otherwise

ZoZ1 = ZlZ2 ~ Q3' This impl ies ZIZ 2 = V ° n V 3.

Now, as ZoZIZ 2 = CZ 1 = CZ 2, we have [02(GonGI ) ,Z lZ 2] ~ Z ° n Z 1. Since

Z I = (Z o A Z I ) ( z I o z 2 ) , th is impl ies G1 = CI NGI(ZIZ2) ' a con t rad ic t i on to

(5 .5 ) (2 ) . This f i n a l l y proves (5 .11) .

(5.12) Suppose (~o,~d) is a c r i t i c a l pa i r wi th Vo ~ Qd-l" Then e i t h e r

d s 2 or Vd ~ QI"

Proof. Suppose d > 2 and Vd $ QI" Pick ~d+l E ~(~d) w i th Zd+l # QI"

Since CZoZI(Zd) = Z 1 (5.11) impl ies

Z d n Zd_ 1 = [Vo,Z d] ~ ZoZ 1 n C(Vd) = Z I n Z 2.

I f now d = 3 then, as V2/Z 2 is the natural G2-module, th is impl ies ZIZ 2 =

Z2Z 3, a con t rad i c t i on . Thus d > 3 and, as Vd ~ QI ' we have ZIZ 2 ~ Z2Z 3

and thus Hence Z 2 = (Z 2 n z 3 ) ( Z dNzd_ l ) and, as d > 2, Z 1 n Z 2 • Z 2 n Z 3 .

54

Vd+1 ~ Q2"

and thus

F.O. TIMMESFELD

NOW by (5.11) Vd+i/Zd+ 1 is the natural SL3(q)-module dual to Zd+ 1

[V2,Vd+ I ] ~ Z 2 n ZdZd+ 1 ~ CZdZd+l(ZiZo) ~ Z d n Zd_ I = Z 1 n Z 2.

Hence V 2 ~ ZIQd+ I . But as V2/Z 2 is also a natural SL3(q)-module we f ind a

p E ~(~2) with Zp n Z 2 . Z I n Z 2 and Zp ~ Qd+l" Hence

Zp n Z 2 ~ [Zp,Vd+l] ~ [V2,Vd+l] ~ Z 1 n Z 2

and so l [Zp,Vd+l ] l s q, a cont rad ic t ion to Zp ~ Qd+l"

We now can prove the main resu l t of th is sect ion:

(5.13) One of the fo l lowing holds:

(a) d ~ 1 or

(b) d = 2, Go ~ G1 ~ SL3(q) and Zo,Z I are natural SL3(q)-modules.

Proof. Suppose (a) does not hold. Then by (5.9) the second part of (b) holds

and, to prove (5.13), we may assume d > 2. Then by (5.10 - (5.12) there ex is ts

a c r i t i c a l pa i r (~o,~d) wi th Vo ~ Qd-1' Vd ~ QI such that Vo/Z ° and Vd/Z d

are natural SL3(q)-modules. Hence

(*) ZI = CVo (vd) = [V°'Vd] = Cvd(V°) = Zd-z

Especial ly d > 3. Choose now by ( 3 . 4 ) ~d+l E A(~d) with ~d = <Zo'GdnGd+l >.

Assume f i r s t Vd+1 s QI" Then

[Zo,Vd+ I ] = [Zo,V d ] s Z I s V d

and thus Gd ~ N(VdVd+I)" Further, i f H d = <ZoGd>, then Gd = CdHd and

[VdVd+I,H d] ~ V d. Now e i the r V d n Vd+ 1 = ZdZd+ I or V d ~ Vd+ 1. Since

Vd+i/Vd N Vd+ 1 : VdVd+I/V d as H d n Gd÷l-mOdule, we obtain in the f i r s t case

[Vd+l,0 (Hd) n Gd+ 1] ~ Z d n Zd+ I s Zd+ I , which implies Gd+ I = Cd+iC~d+l(Vd+i/Zd+l)


a contradiction to (5.5)(2). In the second case G = <Gd,Gd+l > ~ N(Vd+l), again

a contradiction.

Thus Vd+1 $ QI" By (*) and d > 3 this implies Vd+1 $ Q2 and Z I N Z 2 =

Z 2 n Z 3. Since V2/Z 2 is a natural module for G2 we obtain ZIZ 2 = Z2Z 3 and,

as Z 2 n ZIZ 3 = Cz2(Vd+I) = Z 2 N Z3, f ina l ly Z I = Z 3.

Pick now ~d+2 E A(Xd+1) with Zd+2 # Q2" Then (~2,~d+2) is a cr i t ica l pair

I f Vd+2 ~ Q3 then V2 s Qd+1 and so by (*)

Zd_ 1 = Z 1 = Z 3 = [V2,Vd+ 2] = Zd+ 1,

a contradiction to the choice of ~d+l" Hence Vd+2 ~ Q3 and thus d = 4, as

Z 3 = Zd_ I. Pick Ld+3 E A(Ld+2) with Zd+3 $ Q3 and Zd+ 1N Zd+ 2 • Zd+ 2 N Zd+ 3.

Then the last argument in the proof of (6.10) implies Vd+3 ~ Q4 or V3 ~ Qd+2"

Hence by (5.12) Vd+3 ~ Q4 and V3 ~ Qd÷2 and thus by (5.11)

Z 4 = [V3,Vd+ 3] = Zd+ 2,

a contradiction to Zd+2 $ Q2"

56 F.G. TIMMESFELD

§ 6 The contradict ion in the constrained case.

We assume in this section that Hypothesis (5.1) holds. We also use the notation

introduced in § 4 and § 5. By (5~13) ei ther d = 1 or d = 2 and Go ~ GI ~ SL3(q)

and Zo,Z I are natural modules. We wi l l show that both cases are impossible.

Since the th i rd group G 2 = <Po,PI> wi l l be used in th is section again, we

denote the groups corresponding to the vertex ~2 of F(Go,GI) by G~2, Z~2 and

and so on. We f i r s t show:

(~.1) Suppose d = 1. Then without loss IZo:Z onQ11 ~ I Z I : Z lnQo I and one

of the fol lowing holds:

( i ) GI ~ SL3(q) and Z 1 is the natural module.

(2) G1 ~ Sp(4,q) or A 6 and Z 1 is the natural module.

In any case ei ther QiZo = M 2 or EM2: QIZo I = 2 and GI ~ ~6'

Proof. I t is clear that e i ther IZo :Z oNQ1 [ ~ I Z I : Z INQo I • I or the sym-

metric statement holds. So by renumbering i f necessary we may assume the f i r s t .

Hence Z 1 is an FF-module for ~I/Q1 and so one of the cases of (3.1) holds.

As ZoQ 1 ~ P2 and T 2 ~ Z(~2), ~I/Q1 is not Z . i . t o G2(q) or ~-(6,q) .

The same argument as in {5.6) shows that ~I/QI is not l . i . to U4(q) or U5(q ).

I f ~I /QI is loca l ly isomorphic to SL3(q) and IZl i = q6, then I Z I : Z lnQo I =

2 q4. q = IZo: Z onQl ] and IZ onZ l l = Hence ~o/Qo is also loca l ly isomorphic

to SL3(q ) and so ZIQo/Q o is the natural ~2/M2 ~ SL2(q)-module, a contradict iol

since ~2 acts t r i v i a l l y on ZI/Z ° n Z I.

ON AMALGAMATION OF RANK1PARABOLIC GROUPS 57

The same argument also shows that ~1/Q1 is not l oca l l y isomorphic to Sp(4,q)

and iZ i I = 26 . Hence ( 3 . 1 ) and ( 3 . 2 ) imply (6.1) .

G 2 (6.2 Let V 2 = <T 2 >. Then the fo l lowing holds:

(1) V2 ~ Q2 and [V2,Q2] = Z

(2) ~i $ C(T2/Z) for i = o and 1

(3) V2/Z is not an FF-module or the dual of an FF-module for G2/Q2 .

( I .e . There ex is ts no quadra t i ca l l y act ing elementary abel ian subgroup A ~ S,

= S/Q 2 with I [V2/Z,A]I ~ IAI . )

Proof. By (6.1) IT21 = q2 and [T2,S] = Z. Hence Cs(Q2) ~ Q2 and ( 3 . 3 )

imply V2 ~ Q2"

(2) is obvious by (6.1) . Hence ( 3 . 1 ) implies that V2/Z is not an FF-module

for ~2/Q2 and so also not a dual of an FF-module, since the l i s t of modules of

(3. i ) is closed under dua l i t y . This proves (3).

Assume V 2 ~ Z(Q2). Then by ( 3 . 3 ) Cs(V2) = Q2 = Cs(V2/Z)" I f d = I then

by (6.1) Zo $ Q2' whence V2 $ Qo" Hence by (3)

1 * IZo: Z oNQ21 < IV2: V 2nQo I,

a cont rad ic t ion to ( 3 . 2 ) . So d = 2

Now, as [Zo,V 1] ,' i , Vl $ Q2"

(3) implies IV1:V I n Q21 < q2.

and, since Zi ~ V2' Q2 ~ Qi for i = o,1.

Since ~(Vl) = Z1 ~ Q2 and [V1,V 2] { [VI,Q1] ~ Z1,

But th is is by (3.10) a cont rad ic t ion to S = MIV 1.

So V 2 $ Z(Q2), whence T 2 $ Z(Q2) and by (6.1) [T2,Q 2] = Z. This proves ( I ) .

(6.3) Suppose d = 1. Then one of the fo l lowing holds:

( I ) Qo is elementary abel ian.

(2) Ei ther Q2Qo = M I or q = p = 2, IMI: QoQ21 = 2 and

l oca l l y isomorphic to ~6"

~2/Q2 i s

58 F.G. TIMMESFELD

G O Proof. (6.1) impl ies [Qo,Zl ] = [Zo,Z I ] z Z o n Z I . Let R o = <Z 1 >. Then

[Qo,Ro] ~ Z o. Now by (6.1) IZl = q and B acts t r a n s i t i v e l y on Z #. Hence

~(Qo) or Z o n @(Qo) : 1. Z o is an i r r educ ib l e Go-module and thus e i t he r Z o _

In the f i r s t case R o cen t ra l i zes Qo/@(Qo) and so oP(Ro) cen t ra l i zes Qo'

which is impossible. In the second case @(Qo) = 1, which proves ( I ) .

Now by (6.2) and (6.1) e i t he r QIQ2 = M o or IMo: QIQ21 = 2 and ~1/Q1 is

l o c a l l y isomorphic to z6' since [Q2,T2] = Z. Hence e i t he r S = ZoQIQ 2 or

lS:ZoQiQ21 = 2. Computing in ~2/Q2 (3.10) shows tha t in the f i r s t case e i t he r

(2) holds or IZo: Z onQ21 = 16 and ~2/Q2 is parabol ic isomorphic to Aut(M12 ).

But in the l a t t e r case ~ i /Mi ~ z 3 fo r a l l i and thus IV2: V 20Qo I ~ 16, a

con t rad ic t ion to (6 .2 ) (3 ) .

In the second case

also holds fo r Go/Qo

~ i / M i = ~3 f o r i = o ,1

holds in th is case as

G1 = Z6 and IZo: Z onQl I = 4 . I Z I : Z InQo I. Hence (6.1)

and thus also @(Q1) = I and ]MI:QoQ21 ~ 2. Since

i t is now obvious (by Goldschmidt's theorem!) that (2)

we l l .

(6.4) d * 1.

Proof. Suppose fa l se . Then by (6.3) e i t he r q = p = 2 and ~2/Q2 is l o c a l l y

isomorphic to Z 6 or M 1 = Q2Qo acts quad ra t i ca l l y on V2/Z. Hence by (6.2)

and apply ing (3.11) to the act ion of ~2/Q2 on V2/Z, we obtain in any case tha t

~ i /Mi ~ Z 3 fo r i = o , I . Thus ~i/Q2 ~ z 4 or z 4X~2 fo r i = o,1.

Pi Let U i = <T 2 > fo r i = o , I . Then i t su f f i ces by (3.12) to show IUi /ZI = 4

for i = o,1 to obtain a con t rad ic t ion in th is case as we l l . Since th is holds fo r

i = o by (6.1) assume IUl l = 16.

By the above ]QoQ2/Q21 ~ 4. Hence (6 .2) (3) and (6 .3) (1) imply IV 2 :V2n Qol~ 8

Since FUll = 16 and IZl = 2, Z o is by (3. 2 ) no FF-module and also not

ONAMALOAMATIONOFRANKIPAP,~kBOLIC OROUPS 59

the dual of an FF-module for GolQo . Hence (6.1) implies G1 ~ L3(2) and

JZ21 = 8. We obtain

ZIQ ° ~ V2Q ° ~ Q2Qo ~ MI, JMI:QoQ2r = 2

and [Q2'V2 ] ~ Qo"

Hence i t is easy to see that Go/Qo is not parabolic isomorphic to G2(2):,

G2(2), M12 or Aut(M12 ). Hence

QoV2 = QoQ2 = M 1

and Go/Qo is parabolic isomorphic to E6"

Now ~2/Qo is generated by Z 1 and three fur ther conjugates of Z I in G o

which are not contained in M 2. Since ~2/M2 acts non t r i v i a l l y on Qo/Qo N QI'

these conjugates al l induce GF(2)-transvections on Qo n QI" Hence IQ onQl I ~ 8

since CQo(~?)_ = 1. But then

tQo:Qonz l l ~ 8 = IQiQo/Qol

and Z o is an FF-module. This contradict ion proves (6.4).

(6.5) I f d = 2 then V' = Z o. O

Proof. Since the proof is nearly the same as the proof of [12,(4.10)] we only

sketch i t .

Assume V'o = I . Then by (5.11) iVof = q6 Co = C~o(Vo/Zo) and Vo/Z ° is

the dual SL3(q)-module of Z o. Since ~o ~ ~2 this implies

and QoQ I = QoV~2 VoQ~2 : QIQ~2

Let C i = CQxi(V~.);1 i = o , I . Then easi ly C o = Vo(C°nC 2), C 2,= V~2(C°NC2).

This implies @(C O ) = @(C °nC 2) = @(C2). Hence, as Z o $ ~(C2), we obtain

~(C °) : ~(C 2) : 1.

Now ~(QO) ~ C O" Let Qo = Qo/c°" Then the three-subgroup-lemma implies

[Qo,Co] = 1. Since QoQ1/Q2 is the natural module for ~2/M2 we have C~o(~O)= 1

60 F.G. TIMMESFELD

As [qo nQl 'z~2 ] ~ Zl ~ Vo' Q1 centra l izes Oo n QI" Hence Qo is an FF- ^

module for G o ~ SL3(q). Since Qo n QI = CQo(Q1) (3. I ) now implies that Qo

is a natural Go-module dual to Z o.

As Z o # V~2 we have IQo: C°(C 2nQo)I = q2. Hence I[X,Qo]i s q2 for

x E C o n C 2 and so [Qo,Vo<X>] ~ Zo, since th is commutator is Go- invar iant . But

as Z o and 5 o are not equivalent Go-modules, th is eas i ly implies that C o = VoZ(Qo)

and thus, since S/Q o is generated by two conjugates of Z~2, C o = V o-

We obtain iQol = q9 and Vo/Z ° and 5 o are dual to Z o as Go-modules.

Suppose V2 ~ Qo" Then V2 ~ QI' since V 2 ~ QoQI n Q2QI . Hence V 2 < V o and

thus IV21 = q5, since Zo<Z~I> _~ V 2. As M 1 = Q2Qo by (6.2)(1) th is implies

iV2/Z : CV2/z(MI)I ~ q2 a cont rad ic t ion to (6.2)(3)

So V2 $ Qo and the act ion of t o on Qo implies iV2i ~ qg. As ISI = q12

th is implies V2 = Q2 has order q9 and ~2/Q is also l oca l l y isomorphic to

SL3(q). But th is is a cont rad ic t ion to LVo: V onV21 = q.

d , 2 . (6.6)

Proof. Suppose d = 2. Then by (6.5) V' = Z o. Hence 0

QoVI = VoQ 1 = M 2.

By (6.2) Q2 $ Qi for i = o,1 and so S = Q2QoVl = Q2QIVo . Hence G2/Q2

(3.10) l oca l l y isomorphic to SL3(q) or to Sp(4,2n). Moreover, e i ther Q2Vo =

MI,Q2VI = M o or iM i :Q2Vj l ~ 2 for { i , j } = {o,1} and ~2/Q2 is Z . i . to

Sp(4,2).

Suppose V2 ~ Qi for i = o or I . Then [V2,V i ] ~ Z i and V2/Z is the dual

of an FF-module, since iZ i /Z l ~ q2, a cont rad ic t ion to (6 .2) (3) . So V2Q i =

M~ for { i , j } = { o , i } .

is by

O N A M A L O A M A T I O N OF RANK1PARABOLIC GROUPS 61

Suppose next that Vi/Zi = ~i is an FF-module for ~i/Qi for i I . o or G.

Then, as P2 ~ N(ZoZ1) and ~i = <Z:Z1 1>, ~j acts by ( 3 . 1 ) t r iv ia l ly on

Vi/ [V i ,V 2] for { i , j } = {o,1} . But this is impossible since ~. acts non- J

t r i v i a l l y on ViQ2/Q2. The same argument also shows that ~i is not the dual

of an FF-module.

Since [V iNQ2,V2] s Z s Z i , this immediately implies that ~2/Q2 is local -

ly isomorphic to Sp(4,2 n) and Q2Vi = Mj for i , j = o,1. To obtain now the

f ina l contradict ion we use a representation argument.

Let Go = Go/Qo • Then P2 = <V2'V2 g>' g E P2 with g2 £ Qo' and M2 =

(V2NM2) m (~2gNM2) = ~I" Now ~ o / V o ~ l is a natural P2/M2-module and

[ V o W ' M 2 ] g ~I" Since ~2 acts quadrat ical ly on ~o we have

(*) ~o = (VonQz)(VonQ2)(VonQ2 g) and [VonQ2,V2gnM2 ] g C~ (M2). o

Hence M2 acts also quadrat ica l ly on ~o"

Suppose [Vo,M2]/Z 1 is a t r i v i a l P2-module. Then (3.13) implies l[~o,M2]l sq2,

a contradict ion to the above.

- #1 #2 ~ ~ - "~2) 'Zx 2 ] ~1 Assume without loss Z~2 g n . Then [(V oNQ1)(V on - ~ and

thus i [~o,~] l s q2 for each involut ion ~ E S. Let Vo = Vo/[Vo'M2] n C~ (P2).

Then ( . ) implies o ^ - -

[Vo,M 2] = [V V 2NM 2] ~ [Vo,V2gNM2 ] O'

^ - - __ __

Since I[Vo,M2,~]I s q for ~ E S-M 2 by the above, (3.14) implies that [Vo,M2] ^ ~

is a natural module for P2/M2. Since M2 and Vo/V ° n QI are also natural mo-

dules, (3.13) implies that q = 2. But in this case there is exactly one non-

t r i v i a l homomorphism X : VeF2V ~ V, V the natural SL2(2)-module. Moreover, th is

homomorphism sat is f ies X(V®W) ~ o i f v ~ o # w. This is a contradict ion to

IV oNQ2,V2NM 2] = o. This f i n a l l y proves (5.7).

62

§ 7 P rQof o f Theorem 2

F, G. TIMMESFELD

Suppose in th is section that G is a group of minimal rank ( i .e . minimal n= I I I )

sat is fy ing the hypothesis, but not the conclusion of Theorem 2. Then obviously

n = I I I = 3 and r(1) is a t r iangle. Use the notation of § ~ and § 5, ( i .e .

I = { o , i , 2 } ) . Then, by § 4 - § 6 Cs(Qi) # Qi for some i E I . Choose enumer-

ation so that Cs(Qo) # Qo" Then by ( 3 . 3 ) e i ther

(+) Go : Qo * C~o(Qo) or I~ o: Qo*CG o~ (Qo)i ~ 2

and ~o/Qo is local ly defined over GF(2).

By (4.2) the groups ~i /Qi have a weak BN-pair of rank 2 for each i E I .

F i rs t we show:

(7.1) Suppose Cs(Qi) # Qi for i E I . Then Gi/Qi is not local ly isomor-

phic to L3(q), Sp(4,2 m) or G2(3m).

Proof. Suppose false, i .e . suppose that Gi/Qi is local ly isomorphic to one

of these groups. Claim:

( . ) Cs(Mj) # Mj for al l j E I .

Suppose the claim is false for j . Then in the f i r s t two cases QiQj = M k or

IMk: QiQji ~ 2, since otherwise [QiQj,0P(~k )] = I , a contradict ion to Cs(Mj)

Mj. But then [Mk,0P(~k)] ~ Qj, since TGi : Q i * C~ (Qi) l ~ 2, a contradict ion 1

to the structure of ~k/Qj. ((4.2)(1) holds!)

So ~i /Qi is local ly isomorphic to G2(q), q = 3 m.

as above IQiQj/Qi i = q3 and iMk: QiQj I = q2 Since

(a) iQ iQ j :Q j l = q and ~j/Qj is of type Sp(4,q)

or (b) Qi ~ Qj and ~j/Qj is of type L3(q ).

Now by the same argument

[Qi,oP(Pk )] = I we obtain

ON AMALGAMATION OF RANK1PARABOLIC GROUPS

by the Delgado-Stel lmacher-theorem. By the s t ruc tu re of the parabol ics of

G2(3 m) we have ~(Qj) ~ Qi and so

i = #(Qj) < <Gj,Gi> = G.

Now again the s t ruc tu re of the parabol ics of G2(q) impl ies tha t Qj is not

weakly closed in S. Hence Qj is an FF-module fo r Gj /Qj . Since M k = Q j . R k

fo r R k = [Mk,OP(~k)] and since [Qj 'Rk] ~ Qi N Qj, [Q iNQj 'oP(~k ) ] = I case

(a) is by (3.3) impossib le. As CQj(~j) n Qi = I and I Q j : Q i I = q3 we obta in

by [ 8 ] and (3.1) in case (b) :

Qj : CQj(~j) m Qj, Qj : [Q j ,Gj ] a natura l SL3(q)-module

i cQj ~j and .( ) I = q = I Q i i , Qi < Qj"

63

We obta in that Gj/Qj is of type G2(3 m) fo r a l l j E I . Since Pk n Gi

acts f i xed p o i n t - f r e e l y on QjQi /Qi , we obtain Qi ×Qj - { M'k fo r a l l k E I and

{ i , j } = l - { k } . Hence e i t h e r Qi = 1 fo r a l l i E I or Z = Qo ×QI ×Q2 has or-

der q3. In the f i r s t case we have S = Mill j fo r each pa i r i • j and by

[ 1 2 , ( 3 . 2 ) ( 6 ) ] Z(M2) ~ Z(Mo)Z(MI) ~ M o. As Z(Mo) , Z(M2) * Z(M1) we obta in

Z(Mo)Z(MI) = Z(MI)Z(M2), a con t r ad i c t i on , s ince S/M o does not cen t ra l i ze

Z(Mo)Z(MI)/~(Mo).

Now Pk n /M k induces the GL2(q) on Qj. Hence there is a t E B n Pi i nve r -

t i ng Qi" On the other hand by (2.15) Gi<t> = ~ iC~ i< t> (~ i /Q i ) and thus ~ i< t> = N

= GiC~i<t>(Gi ) , s ince Qi - ~ M'k n Z ( ~ i ) , a con t rad i c t i on to t $ C(Qi). This

proves our claim ( * ) .

We obtain CS(Qj) $ Qj f o r a l l j £ I and so Qi n Qj = 1 fo r a l l pai rs

i * j . Now in the f i r s t two cases there is on ly one noncentral Pk ch ie f fac to r

in M k and th i s is a natura l SL2(q)-module fo r each k E I . Hence the Delgado-

Stel lmacher-theorem shows tha t Gj/Qj is also of type L3(q) or Sp(4,q) fo r each

j E I . Thus ~(Mk) ~ Qi n Qj = 1 fo r k E I , { i , j } = l - { k } . But the s t ruc tu re

of B/Q i , i E I , shows that there are only two s e l f c e n t r a l i z i n g elementary abe-

l i an normal subgroups of index 2 in B.

64 F.G. TIMM~SFELD

In the second case we obtain for example Qo ×Q1 = M~, IZ(M2) I = q4 and

M2/Z(M2) is a natural SL2(q)-module for ~2/M2, a contradict ion to (3.13).

This proves (7.1).

(7.2) G i ~ N(Z) for some i E I.

Proof. We show: For each i E I exists a k E l - i such that Pk ~ N(Z).

I t is obvious that th is proves (7.2).

So assume i C I with Pj ~ N(Z) for each j E l - i . Then (7.1) implies

Cs(Qi) ~ Qi' since by the Delgado-Stellmacher-theorem for al l groups, except

(Z), where G. = those l is ted in (7.1), there exists a k £ l - i with ~k ~ NG i i

Gi/Q i . By ( 3 . 3 ) Qi = Cs(Zi) and by [19,(2.1) ] J(S) s Qi"

Suppose without loss i = i . Then M2/QoQI is a nontr iv ia l P2-module, since

otherwise [M2'OP(~2)] ~ QI" Further, since P2 $ N(Z), T2Qo/Q o is a nontr i -

v ial ~2-module. I f QoT2 = QoZI , then [M2,0P(~2)] central izes Zl, a contra-

dict ion to [M2'oP(~2)] $ QI" So we have a normal series

(*) QoT2 < QoZl ~ QoJ(S) ~ QoQ1 < M 2.

Since J(S) is weakly closed in S with respect to G o , i t is easy to see that

( , ) and the Delgado-Stellmacher-theorem show that ~o/Qo can only be of type

2F4(q) or 2F4(2 ) ' or of type F 3. In the f i r s t case ~2/Q2 is also of type 2F4(q)

or 2F4(2 )' Further Cs(Q2) ~ Q2' since such a group has no FF-module. Hence

easi ly e i ther Qo = Q2 = i or IQol = q = IQ21' Qo ×Q2 = z, since QonQ2 ~

G = <Go,G2 >. As Z I is no FF-module for G I , (~) and the structure of the para-

bolics of 2F4(q ) imply IZ1/Qj I = q5 for j = o,2. Hence I Q l : Z l l ~ q and so

~(Mi)QI/QI is a nontr iv ia l ~i-module and- IMi/~(Mi)Qlr = q3 resp. 22 . I t is now

clear that such a group G1/QI with weak BN-pair of rank 2 does not ex is t .


Assume f i n a l y that Go/Qo is of type F 3. Then by the descr ip t ion of the

parabol ics of a group of type F 3 on page 100 of [ 4 ] and

( , ) IQoZl/Qol = 35. Now QoQ1 = QoZl , as QI = Cs(Zl).

By the act ion of ~2 on QoZ1/Qo we have Z 1 ~ Qo(ZIO0P(~2) ), Hence Z 1 =

(Z 1 n Qo)(Z 1 n 0P(~2) ).

Suppose Qo $ QI" Thus, as M2/QoQ 1 is elementary abelian of order 34

and QoQI/QI is central in ~2/Q1 , we obtain by the Delgado-Stellmacher-theo-

rem Qo ~ QI0P(~2 )" Since [Qo,0P(~2 )] = i by (+) we obtain Qo ~ Cs(ZI) = Q1

Thus Qo ~ Q1 and M2/Q1 is elementary abelian of order 34 . But by the

Delgado-Stellmacher-theorem there ex is ts no such group ~I /QI with weak BN-

pa i r of rank 2. This proves (7.2) .

(7°3) Suppose Cs(Qj) ~ Qj for j E I . Then Gj # N(Z).

Proof. Suppose Gj ~ N(Z) and CS(Qj ) _ ~ Qj. Then Qo = 1, since otherwise

Z n Qo * I . Let I = { o , j , k } . Then Pk ~ N(Z) and ~k acts n o n t r i v i a l l y on

Mk/Q j and Qj/Z. Now the s t ruc ture of the parabol ics of the groups with weak

BN-pair of rank 2 shows that G o is of type 2F4(q ) resp. 2F4(2 ) ' or F 3. In

the f i r s t case ~k/M k ~ Sz(q) but IMk/Qj I ~ q5, which is impossible since no

such group Gj/Qj ex is ts .

Hence T o is of type F 3 and ISl = 310 . Suppose Z(Qj) > Z. Then by

[ 4,p.100] ~k acts n o n t r i v i a l l y on Z(Qj)/Z and thus IZ(Qj)I ~ 34 . Now by

( 3 . 3 ) Qj = Cs(Z(Qj) ) and thus IZ(Qj)[ = 34 and IQjl = 36 by the same quo-

ta t ion . But then ~j /Qj cannot act n o n t r i v i a l l y on Qj /Z(Qj) , a cont rad ic t ion

since ~k/Mk acts na tu ra l l y on Qj/Z(Qj) .

So z = Z(Qj) and thus the coimage F k of Z(Mk/Z ) is not in Z(Qj) but in

Qj. Since F k n Z(Qj) = Z th is implies M k = CMk(Fk)Qj , IFkl = 33"

66 F, G. TIMMESFELD

Now easi ly CMk(Fk) n Qj > F k, since otherwise M k would act quadrat ical ly on

Qj/Z. Now the descript ion of the structure of M k on page i00 of [ 4 ] shows

that Mk/Q j is the extension of a t r i v i a l by a natural ~k/Mk-mOdule, since

[Mk,~ k] ~ Qj. But th is is impossible by the structure of the parabolics of the

groups with weak BN-pair of rank 2 over GF(3).

In view of (7.2), (7.3) we may from now on assume that G o ~ N(Z). Next we

show:

(7.4) Cs(Qj) $ Qj for some j E I-o.

Proof. Suppose (6.4) is false. Then for { i , j } = {1,2) ,

t r i v i a l l y on Mi/QoQ j and QoQj/Qo. Further, since O°(~i)

on Zj, ZjQo/Qo is a nont r iv ia l oP(~i)-module and

(,)

Hence (7.1) and

oP(~i) acts non-

acts non t r i v i a l l y

i < ZjQo/Qo ~ QjQo/Qo < Mi/Q o.

an inspection of the parabolics of the groups with a weak BN-

pair of rank 2 implies that ~o/Qo is of type 2F4(q) resp. 2F4(2)' or F 3.

Now in the f i r s t case ~ i /Mi ~ Sz(q) or ~j /Mj ~ Sz(q) and thus one of the

groups Gj/Qj or ~ i /Qi must also be of type 2F4(q). But this is impossible

since by (*) IMi/QoQj I ~ q5 and QoQj/Qj is central in oP(~i). (Or the same

holds with the ro l l s of i and j reversed!)

So ~o/Qo is of type F 3. Since J(S) $ Qi n Qj one of Z i or zj is an FF-

module for the respective group and we may assume that Zj is. Since Cz.(Gj) = 1 J

(3. 1) holds for Zj and Gj/Qj. As Zj n Qo ~ Cz.(OP(~i)) ' Gj/Qj is not of 3~ 6 38. type U4(3 ) or U5(3 ), since otherwise IZjQo/Qol _ 3 resp.

Suppose Gj/Qj is loca l ly isomorphic to SL3(3 ). Then in any case [Zj,M i ] ~ Z

Hence by [ 4 ,p . i00 ] IZjQo/Qol = 32 = IMi/Qjl and so Zj is a natural SL3(3 )-

module. By the same reference there exists a P i - invar ian t subgroup A with

ON AMALGAMATION OFRANKIPARABOLIC GROUPS 67

ZjQ o < A < Qj such that A/ZjQ o is an i r reducible PSL2(3)-module for 33

and Qj/A is the natural SL2(3)-module for ~ i /Mi . Hence easi ly Zj ~ #(Qj)

ZjQ o. Now Qo N 03(~o ) g Z(S) by (+). As Z = ~I(Z(S)) e i ther Z = Qo n

03(Go) or Z g ~ I (z(s) ) and so Z j g ~I (z(Qj ) ) . Now the last poss ib i l i t y is

out by the structure of Qj/Qo described above. This shows that [A,M i n03 (~ i )]

q(Qj). But then Qj/¢(Qj) is an FF-module for ~j /Qj containing a section which

is an i r reducib le PSL2(3)-module for ~ i /M i , a contradict ion to ( 3 . 1 ) .

So f i na l l y we are l e f t with the case Gj/Qj of type Sp(4,3) and IZjQo/Qol = 33

But then [Zj,Mi]Qo/Qo ~ Z(Mi/Qo ) and is a natural SL2(3)-module for ~ i /M i , a

contradict ion since by [4 ,p.100] ~i/Qo contains no such section.

We now come to the f inal contradict ion:

By (7.3), (7.4) we may assume G o ~ N(Z) and Cs(QI) # QI" Then QI = 1, since

otherwise i . Z n Q1 ~ G. Since ~2 = Qo*C~2(Qo) or I~2: Qo*C~2(Qo)l ~ 2, the

structure of the parabolics of the groups with weak BN-pair of rank 2 shows that

e i ther Qo = Z or ~1 is local ly isomorphic to G2(2 )' or G2(2 ) and Qo ~ #4

or Q8" We need to show that both cases are impossible.

Suppose f i r s t GI is loca l ly G2(2 )' Then S ~ ~4 ~ ~2' Qo ~ ~4 and Z =

Qo A S' ~ #2" Sihce ~2/Qo ~ ~4' ~o/Qo is loca l ly L3(2 ). Hence ~ i /Z is a

s p l i t extension of ~4 by ~2 for i = 1,2. This shows that S/Z is the pro-

duct of two elementary abelian groups of order 8 and thus ~I (s /z) ~ #2' As

S ~ #4 ~ 2 this is impossible.

Next suppose ~1 is loca l ly G2(2 ). I f

ca l ly L3(2 ). Now easi ly ~2 = Qo*C~2(Qo)"

Qo ~ Q8 then agaln Go/Qo

(Since S/Qo ~ (~o/Qo)'!)

is a contradict ion to the structure of the parabolics of G9(2 ).

is lo-

But th is

68 r.G. TIMMESFELD

Hence Qo = ~4 in this case too. But then Cs(Qo) ~ ~4 ~ 2 and we get a con-

t rad ic t ion , applying the same argument as above to C~o(Qo).

So we have Qo = Z. I f ~(M2) z Z then we get by the structure of the para-

bolics belonging to the groups with weak BN-pair of rank 2 a contradict ion to

(7,1). Thus 71 is of type 2F4(q) resp. 2F4(2 )' or F 3. In the f i r s t case

~2/M2 ~ Sz(q), a contradict ion to IM2/ZI = q9 (resp. 2 8 in case of 2F4(2)'! )

and the Delgado-Stellmacher-theorem. In the second case IS/ZI = 39 and ~2/M2

SL2(3), again a contradict ion since there is no such group Go/Z with weak

BN-pair of rank 2. This f inal contradict ion proves Theorem 2.


[1] Chermak, A.:

[2] Chermak, A.:

[3] Delgado, A.:

[4]

[5]

[6]

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Large t r iangular Amalgams whose Rank 1 Kernels are not a l l d is t inc t . Com. in Algebra 14, 667-706 (1986).

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Endliche Gruppen I. Springer-Verlag (1967).

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S3-complexes and the al ternat ing groups of degree 7.

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Serre, J.P.: Trees. Springer-Verlag (1980).

Stellmacher, B., Amalgams with rank 2 groups of Lie-type. Timmesfeld, F.: To appear.

[13] Suzuki, M.: On a class of doubly t rans i t i ve groups I. Ann. of Math. (75), 105-145, (1962).

[14] Timmesfeld, F.: Ti ts geometries and parabolic systems in f i n i t e l y generated groups I , I I . MZ.184, (377-396), (449-487), (1983).

[15] Timmesfeld, F.: On the "Non-Existence" of f l ag - t rans i t i ve t r iangle geometries. Mittei lungen aus dem Math.Seminar Gie~en. Coxeter Festschr i f t p.19-44, (1984).

?0

[16] Timmesfeld, F.:




F, G. TIMMESFELD

Tits geometries and parabolic systems of rank 3 J. of Alg.Vol.96, 442-478, (1985).

Locally f i n i te classical Tits chamber systems of large order. To appear in Inv.Math.

Classical local ly f i n i te Tits chamber systems of rank 3. To appear.

Simultaneous pushing up. To appear.

Author's address:

F. Timmesfeld, Math. Ins t i tu t der Justus

Liebig-Universit~t, Arndtstrasse 2, 6300 Gissen, B.R.D.

Documents

On amalgamation of rank 1 parabolic groups