Gauss decomposition with prescribed semisimple part in chevalley groups iii: Finite twisted groups
Transcript of Gauss decomposition with prescribed semisimple part in chevalley groups iii: Finite twisted groups
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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
Available online: 27 Jun 2007
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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ELLERS AND GORDEEV
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
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Received: December 1995
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