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Gauss decomposition with prescribed semisimplepart in chevalley groups iii: Finite twisted groupsErich W. Ellers a & Nikolai Gordeev ba Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 1A1,Canadab Department of Mathematics, Russian State Pedagogical University, Moijka 48, St,Petersburg, 191-186, Russia

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To cite this article: Erich W. Ellers & Nikolai Gordeev (1996): Gauss decomposition with prescribed semisimple partin chevalley groups iii: Finite twisted groups, Communications in Algebra, 24:14, 4447-4475

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COMMUNICATIONS IN ALGEBRA, 24(14), 4 4 4 7 4 7 5 (1996)

GAUSS DECOMPOSITION WITH PRESCRIBED SEMISIMPLE PART IN CHEVALLEY GROUPS 111: FINITE TWISTED GROUPS

Erich W. Ellers' Nikolai Gordeev* Department of Mathematics Department of Mathematics

University of Toronto Russian State Pedagogical University Toronto, Ontario M5S 1 A l Moijka 48, St. Petersburg

Canada Russia 191-186

Continuing the investigations of [EG] and [EGII], we shall show that Theorem 1

below is also valid for twisted CheMLley groups over finite fields. Let G be such a group.

Here we consider only groups G 2 Z F / 2 , where z is a universal Chevalley group over

s finite field K, F is an automorphism of z, and Z is a subgroup of G contained in

the center z(@). Suppose B = HU is a Bore1 subgroup of G. Let I? be a group

generated by G and some element o normalizing G in I? and acting on G as diagonal

automorphism.

Theorem 1. Let 7 = og E I', g E G, and 7 $! Z(I'). If h is any fized element in the

group H , then there is an element r E G such that

As in [EG] and [EGII] we shall give here applications of Theorem 1 to problems of

representations of a simple group as a square of one conjugacy class and representations

of elements in G as commutators.

' Research supported in part by NSERC Canada Grant A7251

Copyright O 1996 by Marcel Dekker, Inc

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2. NOTATION A N D TERMINOLOGY

ELLERS AND GORDEEV

In this paper G = gF/z, where 6 is a universal Chevalley group over a finite

field K, F is an automorphism of 6 , and Z 5 Z ( Z F ) . I f Theorem 1 is true for G = G F , then it is also true for any G = Z F / 2 . Thus we shall suppose G = eF.

- - - - - The F-stable subgroups B, W, H, N , U, and 6- of will have the same mean-

ing as in [St, $111. Let B = g F , W = EF, H = f i F , N = 2 F , U = cF, and

U- = 6-F be the corresponding subgroups of G. Then

G = B N B .

Since 6 is universal, H = EF 5 (U, U-) = G (see [St, §Ill).

Below, when we use the Bruhat decomposition of G, we shall identify any element

in N with its image in W under the natural homomorphism N + W, so we shall write

G = BWB. The graph and field automorphisms connected with F will be denoted by p

and 0, respectively. We assume that the field K has characteristic p and lKI = q = pm.

Let k = K e be the subfield of K, containing all @-invariant elements in K. Let R

denote a root system corresponding to the group G, and let { a l , . . . , a , } be a simple

root system for R. If a 6 R, then X, denotes the corresponding root subgroup of G

and xu denotes any element in X,.

Let a E R and { p , p(P), . . . , p'(/j ')} be a p-orbit of roots of 5 corresponding to

a . If this orbit contains more than one element, we put

h a ( t ) = h+4(t)hpta,( te>. . . h#(p,( te') ,

where t E K g . Here p is a long root if P and p ( p ) have different lengths.

If the orbit consists of only one root p , then we put

- - Since G is universal, every element of H is represented uniquely as a product

- h p , ( t l ) h p , ( t z ) . . . h P . ( L ) , where { A , . . . , A } is a simple root system for G and hPi(ti)

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GAUSS DECOMPOSITION. 111 4449

are semisimple elements in (Xp, , X-p,) (see [St, 531). Thus for every subset I c

(1,. . . , n} such that the corresponding Dynkin diagram is connected, the subgroup

(X*p, ( i E I) is also universal. Clearly, the elements in { a ] , . . . ,ar) can be obtained as

porbi ts of PI , . . . , P,. Thus for every subset 3 C { l , . . . , r) such that the corresponding

Dynkin diagram is connected, the subgroup (Xiai 1 j E J) is a group of F-fixed

elements of a universal Chevalley group. Further,

ha = (h,(t) I t E K*) = 5 n (X,,X-,), for all a E R,

- (ha, , . . . , h,.) = H F = H (see [Cl, 13.71).

For CY E R, let w, be the corresponding reflection in W ( R ) . We shall identify

w, with one of its preimages in N n (X,, X-,).

Let Rl be the root subsystem of R, generated by {a2, as,. . . , ar}, R = R+\ R:,

Wl = (w,, I i = 2,. . . ,r), and P = BWIB. Then P is a parabolic subgroup of G and

P = LU(P), where L is the Levi factor of G and U(P) = (X, I a E R) is the unipotent

radical of P. Let GI = (X, I a E RI). Then L = HGI. If P E R is a root satisfying

(ha, ha,, . . . , ha,) = H , then L = hgGl and hence

(Recall that GI = @, where 51 is a universal Chevalley subgroup of and therefore

H n GI = (ha, , . . . ,ha.).)

If (ha, h ,,,. .. , ha.) = H for some p E R, we shall suppose below that the

element h in Theorem 1 is written in the form

For the proof of Theorem 1 for twisted Chevalley groups we proceed in a similar

fashion as we did in the proof of the same theorem for proper Chevalley groups. The

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4450 ELLERS A N D GORDEEV

first step here is to establish an analogue to Lemma 1 in [EG], i.e. we deal with groups

of rank 1.

Lemma 1. Lei G be a iwided Chevalley group of rankl. I f f E G \ B , then for every

h E H there is an element z E U such that

where vl E U-, vl # 1 , and u2 E U.

Proof. Since G is a group of rankl, we have G = B U B w B where w E W. So

f = ulwhluz for some ul ,ua E U and h' E H. Since oUo-' = U, we may assume

Case ZAz(q) . Here we consider only the case G = SU3(q), (the other possibilities are

factor groups of SU3(q) by subgroups contained in the center). We have [I(( = q Z and

U = {x , ( r , s ) I r , s E K, s + se + ?re = O} (see [el, p.2411).

We may assume

where t E K*

From (2) and (3) we obtain

for some t E K*. Since K is a finite field, there is a pair r, s E K such that

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GAUSS DECOMPOSITION. 111

Let

From (4) and (5) we get

where y @ wB, because the upper left-hand corner of the matrix y is not zero. Hence

y = vlhnvz for some v l E U - and v? E U , where vl # 1, because there are some

nonzero elements below the diagonal of y. Since the upper left-hand corner of y is t,

we have

But h" E G and from (3) we get h" = h,(t) = h.

Case 'B2 (see [G, p.1641). Here l K ) = 2'"+'.

Let Q : K -+ K be an automorphism such that 2Qa = 1. We may assume

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ELLERS AND GORDEEV 4452

Then

Further, rZ+e- l t l+e = t'+e for some r E K*. Let

Then x o f = a y , where the entry in the upper left-hand corner of the matrix y is t '+e.

So we obtain the required decomposition.

Remark. Let y = v lhv2 , where vl E U- and vz E U. Then we can write the element

vl in the form

8 V I = z -a (a ) z -p (a ) (a )2-,-2p(a)(b)z-a-p(4(a1+e + be),

where cr is a long root of the simple root system of B2 [St, Lemma 63). Since the entry

yzl of the matrix y is not zero, we have 2-,(a), z+ , ) (ae ) # 1 (see (G, 3.2 (3 .12)] ) .

We shall use this later when we consider 'F4.

Case ' G 2 . Here 8 is an automorphism of K such that 3e3 = 1. We consider the

adjoint representation of G . Let e, be an element of the Chevalley basis corresponding

to the root a. Further, let

uo ( e - ( s a l + ~ a 1 ) ) = e ~ o t + l a t + . . . (see [Cl, p.247)). Further, D

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GAUSS DECOMPOSITION. Ill

and

(see [Cl, p.248]).

We put ul(s) = h(s)uoh(s-I). Using ( 6 ) and (7) we obtain

ul(s)(e-sal-zar) = ~ ~ e ~ a ~ + z a ~ + . . . . With respect to the basis

{ e - ~ ~ ~ - z ~ , , . . . , e - ~ ~ , e - ~ ~ , e ~ ~ , eq,. . . ,e3a,+~az},

the element f has the matrix

Let L be the entry in the left lower corner of (9). Since (K*', - K a 2 ) = K* (see [Cl,

pp.248,249]), there is an element s E K* such that s2e = t or -sat = t . Now it follows

from (6) and (8) that either au(s)o-' or aul(s)a-' satisfies the condition of z. Indeed,

the entry a l l of the matrix a = u(s)f or a = ul(s)f is equal to t-' # 0. Hence the

matrix a belongs to the Gauss cell wBwB = U-HU and the semisimple part h of

the Gauss decomposition a = vl hvz is equal to h(t) (this follows from (7)). Moreover

vl # 1 because f @ B.

Remark 1. Multiplying the right-hand side of avlhvz by z-I we obtain Theorem 1

for groups of rank 1. The corresponding statement for A1 has been proved in [EG]. If

G is of type (Al)" (where n = 2 or 3 (see [St, Lemma 63])), we can obviously use the

result for A , .

Remark 2. We observe that Lemma 2 of [EG] is also true for twisted Chevdley

groups. Indeed, there we only used that u normalizes all root subgroups and also

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4454 ELLERS AND GORDEEV

that a parabolic subgroup cannot contain a noncentral normal subgroup of G. This is

trivially true if G is quasisimple. An inspection of the list of nonsimple groups gives

us the same result. Thus as in [EG] and [EGII], we may assume that for any p in R+

such that H = (ha, ha,, . . . , ha.) we have

where wk 4 Wl, gl E GI, and u E U(P).

If wk = wp for some p E R, we can apply Lemma 1 to (10) and obtain (as in

Lemma 3 of [EG]) a conjugate 71 of 7 in the form

where 2-8 # 0 and v E U(P).

Further, if ohp(tl)gl # Z(r l ) , where rl = (uhg(tl),G1), then we obtain the

assertion of Theorem 1 in the same way as in [EG] and [EGII] (see Lemma 4 in [EG]).

Therefore, whenever we obtain an element 71 conjugate to 7 in the form ( l l ) , we shall

automatically assurneahp(tl)gr E Z ( r l ) and then change the factor uhp(tl)gl E Z ( r l )

into a noncentral element in r 1 .

Case 2A2,(q), r > 1. Here R is of type B,. The roots of the simple root system are

The elements 1, w,, , w,, can be chosen as representatives in the decomposition of W

into double cosets with respect to WI (see (EG]). Then according to (10)

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GAUSS DECOMPOSITION. I11 4455

We may take p = €1, since (h.,, ha,, . . . , ha.) = H. Indeed, we may assume

G = ~ U ~ P + I ( Q ) . Then h,,(t) = d i a g ( t 1 1 , . . . 1 ) and

h,,(t) = diag(t, 1,. . . , 1, t-IT, 1,. . . , 1, T-I). The group (h,, , . . . , ha.) consists of matri-

ces of the form diag(l,sl, sz,. . . , S Z , - ~ , 1). Hence

Now we consider (12). We may assume 7 has the form ( l l ) , i.e.

where x-,, # 1. Let z., E X,, . The commutator relations for 'Az,(~) show that

where z., E X,,, z,,+~, E X e , + r , and z,, # 1 if z,, # 1 (see also [Cl, p.2651). Hence

In (13) we may actually assume

where x-,, # 1. Then

[z-e,,z.,+e,I = 5-',+,,Z.,,

where x., # 1 and therefore 71 = z-,,z-,,+,,oh,,(tl)glz.,v. Again we may apply

Lemma 4 of [EG] to 71. 0

Case 'As(q). Here R is of type Cz. Let {al,crz} be a simple root system, where (1.1

is a long root and a 2 is a short root. As in [EG] the elements 1, w,,, w,,+,, can be

taken as representatives in the decomposition of W into double cosets with respect to

Wl .

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ELLERS AND GORDEEV

Indeed, we assume G = SU4(q) (see [Cl, p.268)). Then it is easy to see that

So for p = t we take e = ( f t ) - I s E 1. Now we may assume

7 = ~ - ~ , ~ h a , ( t ~ ) g l v

In case (15) we use

[x- , , , ~ 0 , + = , 1 = 20, Z P ~ + ~ O Y I

where za,+, , , X U , , Z O , + ~ Q , # 1.

In case (16) we use

[ ~ - ( a ~ + o ~ ) ~ f a l ] = ~ - 0 a z - a i - 2 a a 7

where z , , , 2 -o , , x - o , - ~ o , # 1.

Thus in both cases we obtain a new element.

- 1 ~ 0 , + 0 , 7 ~ , , + a , or z=, 72:;

which has the form uloha( t l ) g l v where uha( t l )g l $ Z ( r 1 ) .

Case 2Az , -1 (q ) , r > 2 . Here R is of type C,. The roots a1 = E I - ez , a2 =

- 63, . . . ,a, = 26, form a simple root system. Using the decomposition of W into

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GAUSS DECOMPOSITION. I11

double cosets (see [EG]) we obtain

Consider (17). We have

[z-alr z2r11 = z~e,zr1+r,t

where z~r, ,z2r, , zr,+., # 1. Thus

~ 2 r 1 7 z G = ~ - a I ~ h a l ( t ~ ) g ~ ~ z t a ~ ' ,

where v' E U(P).

Consider (18). We have

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ELLERS AND GORDEEV

Then

Any commutator of elements corresponding to the roots from ( a z , . . . , a p ) with ones

from the root subgroups of -R is either trivial or has the form

Thus

where Z I E U - ( P ) , 2 2 E GI . Moreover zz = xc,+,,z-,,+,,zz,, # 1 because e.g.

x,,+,, # 1. Thus we have

where .i, E U ( P ) and ah,,(tl)g;'zz E I'1 \ Z ( r 1 ) .

Case 2Dr+r(q). Here R is a root system of type B,. Let a1 = el - E z , a z =

EZ - € 3 , . . . ,ar-1 = - E ~ , a, = er be a fundamental root system and let F =

(X*,,,X*.,). Then F SU,(q) (see [Cl , p.2681) and

Let h' E hash.,. Then h1 = h,,(s)h,,(t) for some s E k g and t E K'. For t l E K* and

tlFl = tis-' put ! = t t;'. A simple calculation shows

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GAUSS DECOMPOSITION. 111

Hence (h., , h e , ) = ( h a , , h , , ) and therefore

Thus we may assume that

Y = x - a , u h a , ( t ~ ) g l v

In order to change the factor uh, , f t l )g l or o h C , ( t l ) g l into a noncentral one in r1 we

consider

where x.,+.,, x,, # 1 and v', v f f E U(P). 0

Case 3D4(q). Here R is a root system of type D,:

Let

Indeed,

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ELLERS AND GORDEEV

Let p = ~1 + ez and v , be a vector in the p-weight space. Then

ha,( t )v , = vr for all t E K',

hai(s)vfi = for all s E k*, (21 )

ha(e)v, = eeeeeJv, for all e E K*.

Since the norm Nxlk : K* -i k* is an epimorphism, for every s E k* there is an e E K g

such that eeele2 = s. Thus (21) implies ( (ha , ha,)l = ( ( h a , , h a , ) ( . This proves (20).

where x-,,, x-p # 1 .

Let (9, $ 1 be a simple root system for Gz. Then

It is easy to calculate that z,, Z+ # 1 implies

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GAUSS DECOMPOSITION. 111 446 1

Here z,,,z-,, # 1, v' E U(P), ul,u" E U - ( P ) .

We conjugate the element 7 in the forms (22), (23), and (24) by z,,+,, # 1,

z,, # 1, and xa,+sa, # 1, respectively. Using (27), (28), and (29) we obtain an

element of the form v l o h a , ( t 1 ) g ~ u ~ or v l a h a ( t l ) g ~ u z , where v l E U - ( P ) , vz E U(P),

9; , g:1 E G I , and ~ h , , ( t ~ ) g : , uha( t l )g; E rl \ Z ( r l ) .

Case 'Ee lq ) . Here R is a root system of type F4. In the notation of [B] , let

{al,. . . ,ab) be a simple root system of Ee. Putting a , = $(al +a,), a2 = f(a3 +a,),

a3 = a4, and a4 = az, we get a simple root system { a l , a z , a s , a , ) for 'Ee (see [ C l ,

p.2221). We can express a ; in the form

(see [B, Table VIII], but note that we have reversed the enumeration).

Let W = U Wlwk Wl be a decomposition of the Weyl group into double w,€W

cosets with respect to W I . Then every representative wk can be decomposed into a

product of reflections

where wsi is a reflection corresponding to a root &. In [EGII, pp.9 and 101 it was shown

that a representative wk of any double coset distinct from W l can be chosen such that

the corresponding set of roots A = (61,. . . ,6.) is one of the following:

1) A = { E ] + ( - l ) a ~ t ) for some a = 0 , l or A = {cl rt e k ) ,

2) A = ( € 1 + ( - - l ) a ~ ~ , f ( € 1 + (-l)O+l€k + ( - I ) ~ E ~ + ( - l ) C ~ l ) ) for some a , b,c =

0,1 a n d k , p , e = 2 , 3 , 4 , k Z p # e ,

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Let A = (6) or A = {6,6') be a set in 1 to 4. Then either 6 f 6' is not a root,

or 6,s' are both short, 6 + 6' = €1, and (6,s') = Az. Moreover in the last case

If A = (6) or A = {6,6') where 6 f 6' is not a root, we can repeat the procedure

of Lemma B in [EGII], replacing Lemma 1 of (EG] by Lemma 1 and using (10). We

obtain an element f conjugate to 7 in the form

where zs, zp # 1

If A = {6,6'), 6 + 6' = € 1 , then using (lo), (30), and Lemma C in (EG] we

obtain an element 71 conjugate to y in the form

where X-6,z-61 # 1.

Now we show

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GAUSS DECOMPOSITION. 111 4463

if B is a short root belonging to R and p # € 1 . Indeed, let ,4 = f (a + 6 ) where a, ii are

conjugated roots from Ed. If P E R, then (up to permutation of a and 2)

6 I

a = a1 + k i a i , 6 = a 6 + C e j a j . i=2 j=5

Suppose ka = 0, then el = 0 and

where h' E (h.,, . . . , h,,). Thus we have (33). Suppose ks # 0. Then ko = 1 and

Since the coefficient of crl is 2, we get p = E I + ... . Finally P = €1, because P is a

short root.

We consider now the cases 2 and 3. Using (33) we may in both cases replace P

in (31) and (32) by 6 E A and t by t l , where 6 is a short root. First assume we are in

case 2 and put rq = el + (-l)C+let and 4 = ( - 1 ) a + l ~ ~ + ( - ~ ) ~ + l ~ t . Then for z, # 1

we get

[x-6' zy] = X+, (2-6, zy] = 1, (34)

where z + # 1 and z + E GI. From (31) and (34) we obtain

Second assume we are in case 3. Let A = (6) and put cp = € 1 + ( - l ) a ~ b ,

II, = +(&I + (--l)'&k + (-l)'+lcp + (--l)C+l~C, and w = ( - I ) * + + ' E ~ + (-l)C+le(. For

z, # 1 we have

(5-6, xyl = xu"+, (35)

where zw,z+ # 1, x, E GI, and z + E U(P) . From (31) and (35) we get

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4464 ELLERS AND GORDEEV

Now let A = { b , 6 ' ) and 6 + 6' = E. Define cp, $J, w as in the preceding case. Put

X = (-1)" ~ k , p = -€I+(- EL, and v = 4 (-€I + ( - 1 ) " s ~ + ( - l ) b + l ~ , + (-l)c+l~().

For x,,, # 1 we get

where Z , , X X , X + , Z ~ , z u # 1 and xw,zx E G I , z, E U-(P) , and z + E U(P). From

(32), (35), and (36) we obtain

where v" = z;v1 E U(P), zxz, E GI \ Z(G1) and xxz, # 1. Hence ahs(tl)glzAx, E

rl \ z ( r l ) .

[x-6,xe] = X - ~ X - ~ , [x-6,zLp] = 1.

F'rom (31) and (37) we obtain

7 2 = z y ~ z y l = ~ - 6 z - ~ o h ~ ( t ) ~ ~ v ~

or

72 = ~ - ~ ~ ~ - 6 z - ~ o h ~ ( t ) g ~ u ~ .

According to (33) we may assume P = 8. Moreover x -e commutes with z-6 and 2-61.

Thus instead of (38) we write

7 2 = ~ - e ~ h e ( t ) ~ - 6 9 1 ~ Dow

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GAUSS DECOMPOSITION. I11 4465

(Note that the elements x - 6 , x - p , q 1 , and v in ( 3 9 ) may be different from those in

(38)')

Lemma 1 yields

~ e z - e u h e ( t ) = z - e u h e ( t l ) z e

for suitable y e , z e E X e and z-e X - e . From ( 3 7 ) , ( 3 9 ) , and ( 4 0 ) we get

Let

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ELLERS AND GORDEEV

From (42) we obtain for z, # 1:

where y = ZX,ZX, E GI \ Z(G1) and z = z + E U(P) , or

y = z ~ , z ~ , z ~ , E GI \ Z(G1) and r E U(P).

F'rom (41), (43), (44), and (45) we get

where vl E U-(P), vl E U(P) , and ohe( t l )g l E rl \ Z(r1) .

Let A = {el}. We may assume /3 = a1 in (31). Put cp = f (c1 + + e3 + E l ) .

Then

72 = zy7izy1 = ~ - ~ , z - ~ , u h ~ ~ ( t ) g l ~ ' = ~ - ~ ~ u h ~ ~ ( t ) z - ~ ~ ~ ~ v ~ . (46)

Lemma 1 and (46) yield for some suitable y,, E X,, that

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GAUSS DECOMPOSITION. 111

Let x = z - ~ , z - , , z - , . Then we get from (48) for z + # 1:

where y = z - . , z ~ ~ - , , z ~ z - ~ E U - ( P ) and x.,z -,,-,, E GI \ Z ( G I ) .

From (47) and (49) we get

Case 2F4(q). Let { € ; , e i f cj (i < j ) , f(cl zt € 2 & € 3 & e 4 ) } be the set of positive roots

of F4 (see [B, Table VIII]). In 'F4 this set of roots is distributed into 8 disjoint subsets

(see [R, (3.2) p.4061):

Every set Ei corresponds to a new root in 'F4. It is easy to see that {E2, E8) is a

simple root system for 'F4 (here we identify the sets Ei and the roots in the system

lF4) . We put

a l = E s and a z = E z .

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4468 ELLERS AND GORDEEV

Since the corresponding Weyl group is a dihedral group, we may assume that

the nontrivial representative wk of the double coset W1wkWl is a reflection. Thus

according to (10) we may assume

where 6,p are positive roots of 'F4 and p satisfies (h,, ha,) = H .

Every set Ei consists of roots a , p(a) , a + p(a) , a + 2p(a) or a , p(a) , where a

is a long root and p(a) is a short root. Therefore

Let i # 2 and a, p(a) E Ei, where a is a long root and p(a) is a short root. Then the

coefficient of €1 for a or for p(a) is nontrivial and also the coefficient of a* is nontrivial

for a or for p(a). Moreover, the coefficient of E I or €1 is zero either for a or for p(a).

Hence

h ~ , ( t ) v = tmv , (51)

where u is a nonzero vector from the weight space V",, or V,, and m = f 1 or f 2 0 .

Further

hg,(t)v = u (52)

because the coefficients of EI and €2 are zero for all roots of Ez. Since char K = 2 the

map t -i tm is a bijection of K'. Therefore (51) and (52) imply

for every i # 2. Hence in (50) we may assume P = 6.

Applying Lemma 1 we can obtain from (50) an element 71 conjugate to 7 such

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GAUSS DECOMPOSITION. 111 4469

Suppose that for every P = Ei, i # 2, there is an element y = y(P) E G such

that

[ x - ~ , Y ] = zlZZz3,

Then 72 = yyly-I = x - ~ z 1 o h ~ ( t l ) g l z 2 z ~ v , where o h p ( t l ) g l z ~ E rl \ Z ( ~ I ) ,

z3 E U ( P ) . Thus we only need to prove the existence of an element y = y(P) satisfying

the condition (54) .

Let

Let s E K' and

Then y E G (see [R, p.407]). Now we show that y satisfies the condition (54) .

Let p = Ei = { a , p ( a ) ) or { a , P ( Q ) , Q + p(a) , a. + S p ( a ) ) , where a is a long

root and p(a) is a short root. Using the definition of Ei and (55) , (56) one can

check that the roots a, p(a), 4 6 ' are linearly independent. The group L generated

by z - , , z -A , ) , xs, 2 6 , is nilpotent, because every element from the set

has a unique decomposition provided by the basis - a , - p ( a ) , 6,6'. The elements 2 - 8 , y

are in L by definition. Now consider (2 -8 , Y ] mod [ L [ L L ] ] . We obtain a product of

commutators of elements from root subgroups X- , , X-,,(,) or X-,, X - p ( p ) , X-a-P(a)

with elements from root subgroups Xs, Xs, or Xs, Xst, Xs+s,. According to the Remark

to Lemma 1 we may assume that the element x-8 from (53), or X - ~ , ) ( P ) in the case P =

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4470 ELLERS AND GORDEEV

El has a nontrivial factor z-,(e) in its decomposition into elements of root subgroups

X-,, X-,,(,I or X- , ,X-~ , ) ,X-a-~o) , X-a-ZP(o). In this case [x+, y] mod [L[LL]]

contains a factor [x-,(P), zs(s)]; this follows from (57), or [~-~,)( t ' ) , zs(s)] in the case

= El. Hence [z+, y] also contains the factor [x-,(e), zs(s)] or [z-~,)(!), Z&(S)] in

the case p = El. Moreover, this factor will appear in any order of the decomposition

of [ x - ~ , y]. From the definition of Ei and (55), (56) we get

1 # [ ~ - o ( e ) , ~ s ( ~ ) l E (Xicl+t,,X*el,Xk.,)

or (58)

[~-1~(e),3ra-r1(~)1 = z-e1(a)~-s2-s1(e),a # 0.

According to (57) the element y is also an element from a root subgroup of G. Therefore

the commutator [ x + , y] can be written as a product of elements of root subgroups of

G. We can collect the elements from U-(P) on the left-hand side. We denote this

product by zl. The elements from G will be collected in the middle, their product

will be denoted by 22. Finally we collect the elements from U(P) on the right-hand

side and denote their product 2s. Since every element in the group L has a unique

decomposition into elements of root subgroups of FI and since (58) holds, we have

Z 2 # 1 . 0

We shall apply Theorem I to confirm a conjecture of J. Thompson (see [AH],

[Le3]) for simple groups of Lie type over sufficiently large fields. J. Thompson conjec-

tured that every finite nonabelian simple group G contains a conjugacy class C such

that C2 = G. As a consequence we obtain a solution to Ore's commutator problem

which asserts that every element in a finite nonabelian simple group is a commutator

(see [O]). The authors learned recently that 0. Bonten also solved the Ore problem for

simple groups of Lie type over sufficiently large fields, using character theory (see [Bo]).

It is known that J. Thompson's conjecture holds for alternating groups (see [BL]), for

Suzuki groups (see [ACM]), for PSL,(K) (see [Le3]), for PS,,-(K) if charK # 2 and

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GAUSS DECOMPOSITION. 111 447 1

-1 E KO2 (see [Gow]), and for simple nonabelian groups of order less than 10" (see

[K]). The solution to the Ore problem is also only known for the groups mentioned

above.

In [EG] and [EGII] we proved that if G is a Chevalley group over a field K which

contains sufficiently many elements, then there is a conjugacy class C C G such that

CZ > G \ Z(G). Moreover, if G is simple (so Z(G) = I), then there is some C c G

such that C2 = G. The field K contains sufficiently many elements if we can find a

real regular semisimple element h 6 H (recall that h is regular if Co(h) C N and h is

real if h is conjugate to its inverse h-' in G). If we can find a real regular semisimple

element h E H, then its conjugacy class C satisfies the condition pointed out above.

For proper Chevalley groups this is an easy consequence of Theorem 1. Moreover, if G

is any group with BN-pair containing a real regular element h E H and if Theorem 1

hoids for G, then the conjugacy class C of h satisfies the condition C Z > G \ Z(G)

and C2 = G if G is simple (see [EG, Proposition]). Here we have proved Theorem 1

for finite twisted groups, so we are now able to confirm Thompson's conjecture for

all finite simple groups of Lie type under the condition that the basic field contains

sufficiently many elements. In [EG] and [EGII] we have shown the existence of a real

regular element in a Chevalley group G over a field K if IK'I > JR+( , where R is the

root system of G. If G is classical, the existence of such elements follows from the

inequality 1K.I 2 2r + 2 where r is the rank of G. Here we prove an analogous result

for twisted groups. We put

( r +4)2 if G is of type 'Az,(q) or 'Az,-l(q)

(2r + 3)2 if G is of type 'D,+l(q)

63 if G is of type D4(q)

252 if G is of type 'Es(q)

3 if G is of type 'Bz(q)

5 if G is of type ' G z ( ~ )

9 if G is of type 2F4(q).

Lemma 2. If q = (Kf 2 m(G), then there ezists a real regular ~emi~imple element in

G.

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4472 ELLERS AND GORDEEV

Proof. Case2An. WemayassumeG=SUn+~(K) . IfIK1 2 ( r + 4 ) 2 , t h e n I k l > r + 4

and lk \ {O, 1, -1)l 2 r + 1. We can find elements s l , 32,. . . , s, E k \ {0,1, -1) such

that si # s,, syl for i # j, where m = e if r = 28 and m = L + 1 if r = 2L + 1. Let

t l , t 2 , . . . , t ( be the preimages of s l , . . . , s t with respect to the norm map N : K* k'

a n d l e t t , = t f i f m = e a n d t , = s ~ i f m = e + l . Weput

The eigenvalues of ho are distinct by the choice of ti. Obviously, ho and h;' are

conjugate in SU,+l(K).

Case 'Dr+l. Let F = (X,(t) ( a E R, t E k). Then F is a proper Chevalley

group of type B, over k. If IkJ 2 2r + 3, then there is a regular semisimple element

h E F which is automatically real because -1 E W(Br). Using the structure of the

root subgroups in G one can check that h is also regular in G. Thus if IKI = q = Jk12 2

(2r + 3)', then there is a real regular element in G.

Case 'D4. Let q = p"., p # 2. Then (k( = pA 2 6 and there is some

a E k* such that a4 # 1. Let t be a preimage of a in K with respect to the norm map

N : K' -r k'. Consider the element

of the group D4(q), where z, y E KO. We put z = -1, y = t. A simple calculation

shows that h(-1,t) is regular in D4(q). Obviously, h(-1,t) is F-invariant and hence

belongs to G. Since -1 E W(D4) and therefore -1 E W('D4), we have that any

element in H is real. Let q = 2". Since q > 6 we have q 2 8. Let s be a generator of

the group k*. A simple calculation shows that h(s,s4) is a regular element in G.

Case 'Ee(q). Let F = (X,(t) I a E R, t E k). Then F is a proper Chevalley

group of type F 4 . Let H F = H il F. The action of H on a root subgroup X, of G

gives a character X, : H -, K*, namely hX,h-' = X,(x,(h)a) (recall that in case

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GAUSS DECOMPOSITION. 111 4473

'Es(q) we have Xa(a) = ~ ~ ( a ) z ~ ( ~ ) ( a ) or zS(a) , where /3 is a root of Es, a E K or

a E k ) . One can check that x,(HF) = k*. Hence Ikerx, n HFI = l ~ ~ l / l k * l . If

1k1 2 25, then I ker X , n HF( 5 HF/24. Let M = U (kerx. n HF). Since (R+( = 24 aER+

we have ]MI < 241 ker X , n HFI 5 IHFI. Thus HF \ M # 0. Hence there is an element

h E HF \ M. Since h 4 ker X , n HF for every (Y E R+, the element h is regular. Since

h E F and -1 E W ( F 4 ) , the element h is real.

Case 'Bz (q) . Let z ( t , u ) be the general element in U and let h(s ) be the

general element in H. Then h(s )z ( t , u)h(s)-' = z ( ~ ~ - ~ ' t , sZeu) (see [St, $11, proof of

Theorem 361). If s # 1, then sZe, sz-le # 1. Since q 2 3 we may take h = h(s ) where

s # 1. Note that all semisimple elements in " z ( ~ ) are real because -1 E W ( B 2 ) .

Case 'Gz(q). Let z ( t ,u ,u) be the general element in U and let h(s ) be the

general element in H. Then h(s)z( t , u, v)h(s)-' = z (+Set, s-'+'~ u , sv ) (see [St,

$11, proof of Theorem 361). Further, q = 31a+', O = 3". Since q 2 5 we have a > 1.

Thus IK'J = 3'"+' - 1 > 2 . 3" - 1. Let s be a generator of the cyclic group K'.

Then sZ-3e,s-'+3e # 1. Indeed, if s2-3e = 1, then s2e-3e' - - s 2e-1 - - 1 and hence

IK*I = 2 0 - 1 = 2.3"-1. This is impossible. If s-'+~' = 1, then s - ~ + ' ~ ' = s-'+' = 1

and hence s = se. But s is a generator of K*. Thus h(s ) is regular. Since -1 f W ( G 2 ) ,

all semisimple elements in H are real.

Case 'F4(q). Consider the elements ak( t ) defined by Ree (see [R, p.4071).

Every a k ( t ) contains exactly one factor of type za, ( s ) , where s = t , te or te+' for

t E K and pk is a root of F4 of type E ; & e j ( i < j ) . If hak(i)h-' = f f k ( t ) for some

h E H , then hzab(s)h-' = zPb(s ) . Let B = { E ; Zt ~j I i < j } . For /3 E B we define the

character X B : H -+ K* by hza(6)h-' = za (xB(h)6) (here ~ ~ ( 6 ) is an element in 5, a

group of type F4). It is easy to verify that im x p = K * and that ker X B , = ker xp, if

Hence (MI = I U kerxP( < 81 kerxB( = 8(H(/(K9I < (HI if (K'I 2 8. Thus if 1K1 2 8, B E E

there is an element h E H \ M . We have hzg(s)h-' # za ( s ) for all P E B, s E K ,

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4474 ELLERS AND GORDEEV

s # 0. Hence hak( t )h- ' # a k ( t ) for all ak and t E K , t # 0. Since every element

in U can be written uniquely as a product of elements a k ( t k ) for tr, E K , we obtain

h u h - ] # u for every u E U, u # 1. This implies that h is regular. Since -1 E W(F4) ,

the element h is real.

Now we add to the definition of m ( G ) cases when G is a proper Chevalley group,

namely 2r f 3 if G is classical,

m ( G ) = if G is exceptional.

Using results of [EG], [EGII], and the preceding considerations we obtain

T h e o r e m 2. Let G be a finite simple group of Lie type over a field K . If 1K1 2 m ( G ) ,

then there is a semisimple conjugacy class C of G such that CZ = G .

Remark. Theorem 1 and the approach to the J. Thompson conjecture based on this

theorem is a generalization of a corresponding theorem and approach of Sourour (SJ

who proved similar results for SL,(K).

Remark. We are sure that the estimates for m ( G ) can be improved

[ACM] '2. Arad, D. Chillag, and G. Moran, C r o r p * with D , m a l l e o r e n n g n.rnbcr, Chapter 4 in [AH].

2. Arad and M. Herzog, Eds., P r o d r c t . of c o n j r g e c y clr, .rr tn gro.p,, Lecture Notes in Mathematics, no. 1112, Springer Verlag, New York, 1885.

N. Bourbaki, Gro.pc, e l olg ibrcn de Lnr, IV. V, V I , Hermann, Paris, 1968

0. Bonten, ~ ~ b c r K o m m r t e t o r e n in e n d l i c h e n e t a l a c h e n C r r p p r n , Aschener Beitrige zur Math- ematik, Bd. 7 , Verlag der Augustinus-Buchhandlung, Aschen, 1993.

J.L. Brenner and R.J. Lit, Appl ic . t ion of pnrf i l ion fhcor , t o gr0.p.: C o v e r i n g the a l termat ing

g r o u p b y p r o d r c l r of c o n j r 9 a c y c l a , ~ r r , Number Theory (J.M. De Koninck and C. Levesque, eds.), Walter de Gruyter, Berlin, New York, 1989, pp. 65-71.

J.L. Brenner, C o v e r i n g t h c e r r m r l o r fina.1,. X, Ars Combinstoria le (1983), 57-67.

R.W. Carter, S i m p l e group, of Lnr trpc, John Wiley k Sons, London, 1989.

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GAUSS DECOMPOSITION. I11

[CZ] , F i n i t e groupr of Lie t v p r , John Wiley & Sons, Chichester, 1993.

[EG] E.W. Ellers and N. Gordeev, C a u r r d e c o m p o r i t i s n wi th prrrcribcd . rmrr implr part in cla,,ical

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[EGII] , C e n r , d r c o m p o r t t i o n v i t h prcrcr ibcd . crn i r implc pert In C h c v a l l r g g r o r p r 11: e r c r p -

t i o n a l c a r e * , Comm. Algebra 23 (1995), no. 8, 3085-3098.

D. Gorenstein, F i n i t e r t m p l r g r e r p r : A n inlrod.ction to t h e i r c l e r , i f i c o t i o n , Plenum Press, New York, 1982.

R. Gow, Comm. to tor . tn the t y r n p l c c t i c g r o u p , Arch. Math. (Basel) 50 (1988), 204-209.

S. Karni, C o v r r i n o n r m b r r t of g r o u p . of s m a l l o r d e r and s p o r a d i c g r o u p r , Chapter 2 in [AH].

A. Lev, P r o d r c t r of c y c l i c conj.gacl c l a r r r r i n the p r o r p r PSL,(F) , Linear Algebra Appl. 179 (1993), 59-83.

, P ~ o d . c t s of cyc l ic , ,mi Iar i tp c l a r i c , i n t h e g r e r p r GL,(F), Linear Algebra Appl. 202 (1994), 235-266.

, P r o d a c t , of c o n j r g e c y clo,rr. In t h e g r o s p r PSL,(F) , Ph.D. thesis, Tel-Aviv Univer- sity. 1994.

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0. Ore, S o m e r e m a r k . o n c o m m r t o t o r r , Proc. Amer. Math. Soc. 2 (1951), 307-314.

R. Ree, A fami ly o j s i m p l e g r o u p , a , roc ia i rd wi th t h e r i m p l o Lie algebra oJ t ype (F t ) , Amer. J . Math. 83 (1961), 401-420.

A.R. Sourour, A f a c t o r i z a t i o n t h ~ e r c m for m o t r i c e r , Linear and Multilinear Algebra I 9 (1986), 141-147.

R. Steinberg, L r c t r r r s o n C h r v a l l e y grasp., Yale University, 1967.

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Received: December 1995

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