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Gauss decomposition with prescribed semisimplepart in chevalley groups II exceptional casesErich 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 (1995): Gauss decomposition with prescribed semisimple part inchevalley groups II exceptional cases, Communications in Algebra, 23:8, 3085-3098

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COMMUNICATIONS IN ALGEBRA, 23(8), 3085-3098 (1995)

GAUSS DECOMPOSITION WITH PRESCRIBED

SEMISIMPLE PART IN CHEVALLEY GROUPS I1

EXCEPTIONAL CASES

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

Let G be a Chevalley group over a field I<. Let B = H U be a Bore1 subgroup of G, where H is the semisimple and U is the unipotent part of B. Let r be a group generated by G and some element a, such that o normalizes the group G. More precisely

a X , ( ~ ) U - ~ = x , (a , t ) /. \

for all root subgroups .Ye of G, where a, E Ii depends only on a . ( I t is possible that there are relations in addition to (I) , but we stipulate G < T.) In [EG] the following theorem was proved under the assumption that G is a classical Chevalley group, i.e. G is of type A,, B,, C,, or Dr.

Theorem 1. Let 7 = ug where g E G, y $! Z(r) For any element h In the group H there is an element T in G such that

where ul E L'-- and u? E U . Here U - is the g o u p generated J3.v all iiegativc root scl bgroups of G.

The main goal of this paper is t o give a proof of this theorem whcn G is an exceptional Chevalley group, i.e. G is of type E g , E7. E R , F4, or G 2 .

- - - -

Research supported in part by NSERC Canada Gran t A7251

Copyrtght 0 1995 by Marcel Dekker, Inc

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ELLEKS A N D GOKDEEV

.is ill !EGI W P apply Theorem 1 to the problein of representatio~l of a siinplr Cl~cvnl le~ g~oiip as ;\ square of a conjugacy rlass. Throl-cm 1 implies tile esistericc o f all elerncnt h in H such that Ch Ch = G , where Ch is the coiijugacy c.l;iss of 11

in G , whenever G is simple and the ground field is sufficiently large. So under these conditions, ti-' E Ch and therefore every element in G is a commutator. This confirms Ore's conjecture in the case of C h e d l e y groups over l a g e enough fields (see [O]) These results are generalizations of those of Brenner's and R. Tho~npson's for special linear groups (see [Br], [TI]. [T2]). It should be noted that Theorem 1 is also a generalization of Sourour's theorem for general linear groups (see [S]). For other references and the l~istory of the question see [AH] and [EG].

2. NOTATION A N D TERMINOLOGY

We shall continue to use the notation of [St,], as we did in [EG], where the symbols h,(t), h,, w,, H, U, Up, and N were defined. Let R denote a root system corresponditlg to G: { a l , . . . .a ,} denotes a simple root system of R. If D l , . . . , f i t E R, then { P I , . . . . 3 k ) denotes the lattice Zpl +. . .+Z@I. Finally, let R+ denote the positive roots of R and let R = R+ \ ( a z , . . . ,or) .

In [EG] we derived results which are true for ?;,cry Clievalley gr-oup \Ve shall ilse these results here. They also exhibit the main steps of tlie proof of Theorern 1. Therefore we shall briefly state them here.

The first step in the proof of Theorem 1 is the beginning of the induction process, nnrnely the proof of Theorem 1 for a C h e d l e y group of rank 1. Lemma 1 in [EG] supplies this part of the proof, it shows that if f E ( X , , S - , ) , f @ h , I,, then for every t E Ii* there is some s E K such that

where q # 0. Multiplying this equality on the right-hand side by x,( -s) we get an vlt~rnent which ic c n n j l ~ ~ n t c t o rr f and has a prrccrihcd ccin~sirnplr ! , a r t

Suppose we have proved our statement for all groups of rank < r . Let GI be the subgroup of G corresponding to the root system generated by i n z . . . . .ct,}, let I+', be the Mtyl group of { a 2 , . . . . a , ) , let W = U IVl ulk CVl Le tlie deco~nposition of IY

w r ELI' into cioul~le cosets with respect to W1, let P be the correzponriin~ pi~ri~bolic sul~group of G and [ ' ( P ) be the unipotent radical of P. Let L 7 - ( P ) = (.Yo 1 (i E -11). Lcsrma 2 in [EG] siioiss that for any root for which ( h p , h,, , . . . h,,,) = 11. t11c1c I > ;in cic111cnt conjugate to y which has the form

If h = h a ( t l ) h , ? ( t ~ ) . . . h,.(t,), here h is from Theorem 1. then our next g o d is to change lig(t) in (2) to hg( t l ) and wb to some element in Lr- , by conjugation of (2) by some element in G. tVe obtain an element in the same cunjugacy class as y which has thc form

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GAUSS DECOMPOSITION. 11 3087

Lemma 4 in [EG] states, if u ha(tl)gl 4 Z ( T l ) , where rl = ( G 1 , u h p ( t , ) ) , the11 we can obtain the desired element by conjugation of ( 3 ) by some element from G I . If u hp(t l )gl E Z ( r l ) , then our goal is to "correct" the factor D ha(tl)gl by conjugation of ( 3 ) so that we get a noncentral element in r l .

Thus if we obtain below an element in the same conjugacy class as y in the form ( 3 ) , we suppose automatically that a ha(t l )gl E Z ( r 1 ) . After correction we get a new factor in the form (3): u ha(tl)gi $ Z ( r 1 ) with g', 6 GI , so we may apply Lemma 4.

By Lemma 2 in [EG] we may assume that

where wk E W \ W1, gl E G I , u E U, t E K* and p is a positive root such that ( h g , h ,,,. . . , h,?) = H . As a rule we shall not distinguish between an element in W and its preimage in iV. Note that a change of the preimage of wt in (4) may result in a change o f t and gl.

.4ny wk E I+' can be factored

W L = wg, ' ' ' wg, , ( 5 )

where J l , . . . ,6, are positive roots, wg,, . . . , wa, are the corresponding reflections in CV and s is the number of eigenvalues of wk which are distinct from 1. Let

If w6 is interpreted as an element in G, then we always mean the element w6 defined in [St], see also [EG]. Recall. if cu and /3 arr roots, then the a-string of roots through B is the sequence

where B + z c u is a root for -1 5 t 5 q but p - ( t + l ) a and 13 t ( q + l)or are not roots.

Lemma A. Assume the root system R is not of type Gz. Suppose that the numbers of factors in (5) is minimal for all possible elements ulk in (4) . Then

Moreover, every element in the 6,-string ofroots through 6, is contained in i S 1

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ELLERS A N D GORDEEV

Proof. Suppose E ( a z , . . . , a r ) for some t . Then

Clearly wg,wb,,, tug, = for j = 1,. . . ,s - I, where 6;+, E R+. Further, w6, E WI because E ( a z , . . . , a r ) . Thus wk in (4) may be rrplaced by w ; = wg, . . . wg,-, log;+, . . . WJ:,

havlng only s - 1 factors. This contradicts the choicc of I V ~ . Thus A C 12.

Consider the pair of positive roots 6,,6j, where i < j . Then

for some 4, $ E (6;, 6j) . Suppose j = i + 1. Then we can replace wa, wa, in the product wg, . . . wg, by w6, w+ or by W*WK? If 4 or q!~ is in ( a Z , . . . ,a,) , then we can reduce the number of reflections in the product above. Thus q , y E 112. If j > i + 1 we can move wg, to w6, using the rules established above. Suppose {6,,6,} is of type A2. Then the 6,-string of roots through 6; consists of 6, and one root of the form 6, 1 6j, but 6, f 6, = 1 4 or 6; f 6, = f$, a contradiction. Suppose {6,,6,) is of type BE If 6, and 6, have different lengths, then the roots in (6,, 6,) are i6,, 1 6 , , f +b and f +. Thus all elements of the 6,-string of roots through 6, arc contained in {6,, 1 4 , i*). Our contention follows. If {6,, 6,) is of type B2 and 6, and 6, have the same length. then both are short roots and the 6,-string of roots through 6, consists of 6, and also of two long roots, say $' and $'. Clearly wg,wa, = W Q ' W ~ , , and so dl,$' E Q. Our contention follows. Finall?: {6,, 6,) is of type G2 only if R is of type Gz. O

Consider two conditions for the set A:

There is a root 6( E A such that (hg,, h,,, . . . , h,.) = H

6, bj is not a root for any pair i, j with 1 < i < j < s.

If the condition ( * ) holds we may assume that = h e .

Lemma B. Suppose ( * ) and ( * * ) hold. Then there vxists an element z E G such that

Proof The condition (**) implies that a, a , = a, a, for every a, E ( X 6 , , X + ) and

E (X6,, X-g,), z # 1 . Since ws, E (X'a,,X-a,) we may assume that = 6,. Further by Lemma 1 in [EG] there are I , , . . . , t , E K such that

where q, # 0 for i = 1, . . . , s.

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GAUSS DECOMPOSITION. I1

Let x = x a . j 0 , ) . . . xa, ( e l ) . Then ( 6 ) implies

Below, when we deal with the cases E6, E7, EB, and F4, we shall assume that the element wk in (4 ) satisfies the conditions of Lemma A . We shall not make this assumption in case G 2 .

We shall now consider the cases E6 and Ei. The enumeration of roots in Tables V and VI in [B] will be referred to as standard enumeration. For E6 and E7 we reverse the enumeration of the roots, i.e. a, -+ a,+] -, . Then {a?, . . . . a,) is of type D5 in case Eg and of type E6 in case E7. In both cases Tables V and V I in [Bj confirm

We shall see now that the conditions (*) and (**) hold. Lemma A implies A c R, hence 6, = al+k2 a2+...+kr a,. Sinceall roots have the same length, ( h 6 ; , h,,, . . . , h O v ) = H for every 6, E A. Thus (*) holds. Further 6, + 6, = 2al + . . . # R. Therefore 6, + 6, is not a root for any pair 6,,6] E A. Suppose 4 = 6, - 6, is a root. Then p @ iR. This contradicts Lemma A. Thus (**) holds.

By Lemma B we may assume

Put : = x - a , ( q l ) . . . x - 6 , ( q a ) . We shall see that there exists a root 4 E R such that

where q # 0, 21 E U - ( P ) , and zz E GI \ Z(G1). Let 6c E A. Then -at + 4 is a root for some 4 E R. Indeed, -6t + E is a root for some E E R, otherwise ( X a l , X - a l } would be a normal subgroup of G. Moreover, e @ -R by (8).

Let

Then 4 E R and -6t + 4 is a root. Also 4 $! A by (**). Since all factors x - ~ ; ( ~ ; ) of z commute with each other, we may order these factors as follows. First we write all factors where -6, + 4 is not a root, then those where -6, + 4 is a negative root and finally those where -6, + 4 is a positive root. For q E I<* let D

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3090 E I L E R S A N D GOKDEEY

Yote, that -6, + $J (a? , . . . . o r ) . Hence ;my cornrlilitator 111 (11) is citller trivial or be!ongs to a root subgronp of G I . The commtit;it,or

is either trivial or beiorlgs to a root subgror~p ci)rri~sporlding to some root from -11. Thus wc may collect all co~nmutators in (11) on the right-hand side and obtain

1v1ie1.e z l 6 L r - ( P ) . Since I < J if -6, + o < 0 anti - A , 4- 0 > 0 for ail pairs 7.1, \vc haw

8

-2 = 1-1 [x-6.(4,). rm( - r i j ] = ~z ~ 3 ,

,=I

where y z E U - n G1 and y3 E U fl GI. Moreover. either y2 or ya does not belong to Z(TI) , because [x-nl(qc). x6(-q)] # 1. Thus we have proved (10).

Since we assume that the factor a hp( t l )g l fiorn (9) belongs to Z ( r l ) we obtain 91 E H and

where v' = x ~ ( ~ ' ) x ~ ( ~ ) u x ~ ( - ~ ) E U. Here n h3( t l )g l z2 @ Z ( r l ) Hence we may assume

= v l a h ~ ( t 1 )(/I 1 1 , (13)

where vl E Ub(P) , "2 E U(P), and cr hs(tl)gl $ Z(T1). Since Theorem 1 is true for r l , where G1 is a group of type Ds, we can apply Lemma 4 of [EG] to (13) and obtain Theorem 1 for the group r, where G is of type E6 Then we can repeat the procedure for Ei

Next we deal with the case E8. Again we reverse the enumeration of roots in Table VII of [B]. Then {az,. . . ,us) is of type Ei . Let

Then R = 121 U R2 U Q3 and all roots in R1 U R2 have the form

We distinguish between certain possibilities for the set A

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GAUSS DECOMPOSITION. I1 309 1

I. Suppose A satisfies (1;) and ( * * ) and A # (0. By Lemma B we may asslirne

Then -6 + Q is a root in ( a z , . . . , a s ) aid 9 = cul + k2a2 + . . . + kaaa . Siucr, il # {(} and since b -( is a root for every 6 E RI U f l z , the condition (**) yields ( 6 A. Thus all roots in A have the form a1 + kz a2 + . . . + ks as. Hence we can use the same arguments as in the cases Es and E7. We consider the element 7 1 = ~ + ( q ) - y x + ( - ~ ) and obtain 7 1 = u l u h o ( t l ) g 1 ~ 2 , where uhg(t1 )gl $ Z ( r l ).

11. Let A = {(I. Here we take 9 = a l . By Lemma 1 in [EG] we may assume Y = x- ( (y )o ha, (t)gl u .

Let II' = €8 + ~ g . Then -( + $ = -s: + ~6 = - a ] . Further

x O , ( s ) ~ - a ~ (7-10 ha, ( t ) = x - a l ( ~ l )a ha, ( t l )% ( P I )

for some s # 0. Also ql # 0. Then

%' = xol(s)71 ~ 0 , (-s) = ~ - $ ( q ' ) ~ - ~ ( q ) x - a ~ ( 9 1 ) ~ hcrl ( t l )gl u l ' , (I8)

where v" E U(P).

Let w = €7 - € 5 . Then -$ + w is not a root, -6 = -( + w E 4, and -a1 + w = D Z . Thus ( 1 8 ) implies

where P , b # 0. Hence a ha,( t )xo,(b) $ Z(i-I).

111. Assume A does not satisfy ( * * ) . LVe introduce the operation of conjugation on the set R1 U Rz.

i f 6 = & s f € k 1 j k 1 6

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3092 ELLERS A N D GORDEEV

Fro111 (19) we gct o + h = < =Eg: +:;. ('0)

Now we shall show that we can choose thr factors o f u ) ~ such tllat .A = {6,6), w11ex.e

L\ t ii 1 2 2 . ( 2 1 )

Let h , , 6, E A \ ( 5 ) such that one of 6, 6, 1s a root Clearly 6, - h, E (nz, , a , ) , which by Lemma A is impossible Thus 6, + 6, = 2 a 1 + ~b a xoot So h, + h , = ( and therefore 6, = 6, by (20)

SupposedE A a n d C # 6 , 6 , < . ~ h e n 6 - d o r 8 - d i s a r o o t , becauseh+6-4 = - 4 is a root for all q+ E R. Further. A = {6, 6, 5 ) is impossible by our choice of A. If = { & , ( I , then we can exchange for 8, because 6 +- 6 = (. This siiows (21). Now we

may assume r = w , a hs(t)gi v (22)

for some 6 E R1 UR2. Note that (/la, h,,, . . . , h,,) = H, because h = a l -t A,? 0 2 i-. . . + !is a,.

Lemma C . Let F be a Chevalley group of type A2 and let { c I , ~ } be a simple root system for F. Then for every h l , h2 t (h,, h8) there are s l , sz E K such that

Proof. Let F = SL3(K) . There is a natural homomorphism F + F which maps transvection groups of 6 into root subgroups of F. Our assertion can be confirmed by a straightforward calculation in SL,(K). 0

Let us return to (22). Since {6,i) is a simple root system of type A z , where 6 + 6 = (, we can apply Lemma C and suppose

Now take 4 from (14) and consider

--6 + @ is a root in (aZ, . . . , ar),

6 + 4 is not a root,

-( + ~3 is a root in -Q

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GAUSS DECOMPOSITION. 11 3093

Kest we deal with the case f;; \Ve rmwse the enumeration ~f roots 111 Table VIII o f [B]. Let

and

Thrn R = 0 1 U $22 U $23. First we show. if s , + ( - I ) d ~ i E A. then

Suppose 6' E A and 6' # 6. Then 6' = p or 6' = u, where p = E I + ( - l ) d + ' ~ k and u = + (-l)d+'~); * E ( 5 E ~ ) , since otherwise 6 - 6' or 6 - 26' is a root in ( a 2 , a s , a4). This contradicts Lemma A. Moreover A cannot contain a root of the form p and a root of the form u , because p - 2v is a root in (a2,cv3, a4). Also the difference of any two distinct roots of the form Y is not a root in (a2, a3,a4). This shows (24).

Now we show, if A il R z = 0, then

where 6 E Q3 and 6 = - 6.

Suppose 6,s' E A and 6.6' f : I . Then 6' = € 1 - 6 , because otherwise 6 - 6' is a root in (az,a3,a4). Hence if \ A / > 1, then A = {6,b} or A = {cl ,6) or A = { ~ ~ , 6 , 8 } . The last case contradicts the choice of A. In the second case we can exchange e l for 6, because 6 + 8 = E I .

Further comparing the numbers ( p . 6) and ( ( I , a l ) for any weight p we get

if 6 E R, 6 # € I , then (ha,ho,,h,,,h,,) = H. (25)

Let us return to the case 6 = E I + E A. Then A satisfies (*) and (**) by (24) and (25). By Lemma B we may assume

Let Q = + ( - l ) a + l c t , where a i s as in (24). The same calculations as before show that the element z,+(r)y ze(-7.) satisfies the condition of Lemma 4. Hence we obtain our result.

Now we consldcr the case A n R2 = fl Suppose A = ( € 1 ) . We may assume

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3094 ELLERS A N D GORDEEV

for some s , q E Ii, q # 0. anti I,' E l i ( P ) . Lct 4 = 4 ( ~ 1 t 62 t 5.1 + ~ . j ) . A simple c;~lcnlntion yields

where v" E U ( P ) and a , q1 # 0 . We apply Lemma 1 of [EG] and get

where b # 0. The formulas (28) and (29) imply

where v"' E U ( P ) and 4 = $ ( E ~ + E Z + .c3 + E.,).

Let 1G = + ( E X + cz - ~3 - ~ 4 ) . Observe -a l = + e:, + ~3 + E ~ ) , - E X , - 4 = - € 2 - € 3 - € 4 ) , - 0 1 1 + = € 7 , - E l + Ij, = + E:, - 6 3 - E 4 ) , - 4 + 21 = - ~ 3 - €4 . These roots belong either to ( a z , as, a 4 ) o r to -Q. Also E~ and -e3 - ~4 are linearly independent. Therefore in

we can collect x , , ( b l ) z - , , - , , ( d l ) on the right-hand side commuting x , , ( b l ' ) with the other roots and obtain

x $ ( i n ) ~ - ~ , ( b ) x - , , ( q " ) x - + ( d ) x d - ) = x i x z , ( 3 1 )

where x l E U - ( P ) and x 2 E G1 \ Z ( G 1 ) . From ( 3 0 ) and ( 3 1 ) we obtain

Now let A = (6 = $ ( & I + ( -1 jac2 + ( - - I ) ~ C ~ $ - ( - I ) ~ E ~ ) } . The formula (25) lmplies (*). so Lrmnia B y1c4ds

Let w = E X t ( - 1 ) " ~ ~ . Then z,(P)y xu(-!) satisfies the condition of Lemma 4 in [EG].

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GAUSS DECOMPOSITION. 11 3095

Finally let A = { A . 6'). b , 6 ' E 123 and h + h' = 1 . By (25) ai~d Lcmrna C lve may assume

; :: I , L ~ , ( ( I ) . T - ~ ( / I ) ~ - ~ , ( C ) C ~ / 1 6 ( t , )gl I ,

- x - ~ ( ~ ) z - ~ , ( ~ ) x - ~ ~ ( c ) u hn(tl)gl U . (37)

where b, c # 0. We may assume 6 = f (€1 + € 2 i ES i € 4 ) and 6' = + ( E , - sz c4). Let w = ~1 - ~ 2 . Then

where X I E U - ( P ) , x2 E GI \ Z(G1) and x3 E U ( P ) . From (32) and (34) we obtain

where xi E U ( P ) .

Finally we shall deal with the case G2. Here we take the standard enumeration of roots, see Table IX of [B]. Then

Further, A = {OI] or A = 1301 + 0 2 ) or A = {01,3cul + ?a2]. In all cases A satisfies ( * ) and (**).

Consider (35). Let n = -01, b = 3rr1 + az. Thcn n + b = 301 + cr?, 2a + b = 0 1 + 012,

3a + b = a2, 3a + 2b = 3al + 2 ~ 2 . Thus { ( I , b) is a simple root system for G2.

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3096 ELLERS AND GORDEEV

wherr 1.2 is a product of elements from root subgroilps of the roots a + b, 2a +b, 3a+2b. and xl = rn,+b(r) where r # 0 (see [St] p.151 (3)ja)) . Thus

where a hg ( t l )gl sl E rl \ Z(T1 ) and x; V' E U ( P )

Consider (36). Then

where s # 0

Consider (37). Then

where x-,,(s)x,,(d) E rl \ Z(T1) and v ' , u l ' E U(P). 0

Let G be a quasisimple group with a split BN-pair. Suppose that for every noncentral conjugacy class C of G and for every h E H there is an element c E C such that c = ul h 212, where ul E 15- and uz E U. If hl and 112 are regular elements in H arid if CI and C2 C G are their corresponding conjugacy classes. then (see the Proposition in Section 5 of [EG])

Ail cienlent h E H is called regidas if the centralizer C'r; (h) C .Y Note that the condition in this definition is weaker than that used in [EG] where \\.e required CG(h) = H. The Proposition in Section 5 of [EG] is nevertheless true, because its proof requires only Cu(h) = 1 and this is still guaranteed since U 0 N = 1.

Now let G be a Chevalley group of rank r over a field 11' and let R be the corresponding root system We put

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GAUSS DECOMPOSITION. 11

'21. + 2 if G is classical. m(R) =

lRf 1 if G is exceptional

T h e o r e m 2. If lK*I 2 m(R) , then there is a conjugacy class C C G such that

Moreover, if G is simple, then C C ' = G .

Theorem 2 is a consequence of (38) and the following lemma.

L e m m a D. If Ih"J 2 m ( R ) , then there exists a regular clement h E H such that Ch = C h - I . where Ch and Ch - are the conjugacy classes of h and h-' , respectivel~v.

Proof. The result is trivial for groups of type A,. For groups of type B,, C, and D, it

follows considering the action of H on the sum of weight spaces C ICE, (or considering , = I

natural representations). For exceptional groups we require JK'/ 2 J R + / . Then regular elements exist. Indeed, it is easy to see that a ( H ) = K* for every a E- Ri. Thus there is an element h E H such that a(h) # 1 for all a E R+ and so h is regular. Moreover. W ( E ; ) , Tti(E8), W(F4) , and II'(G2) contain -1. Hence in these cases C,, = C h - I for every h E H. If G is of type Eg we consider the subgroup H F of H corresponding to the group F = (X*,,,S* ,,.. Y*,,.S+,,). Then a(HF) = I(* for every a E R, so there is a regular element h in H F . The group F is of type D4 and so C,, = Ch-1. because -1 E ?Y(D, ) . 0

Remark 1. The equality C C = G implies C' = C-' and therefore every g E G is a commutator.

Remark 2. The estimates for the number of elements in the field Ii glven here for the exceptional cases may not tx sharp

[.AH] Z Arad and ht Herzog Cds , l'roducls of conpgacy classes rn gTlJUp$. Lec(ure Notes In Math- ernatlcs, no. 1112. Springer \Prlag. i 'cu York, 1985

[B] N. Bourbaki, Groupes el aigkbrea de Lte, IV, V , VI, Hermann, Paris, 1968.

[Br] J . L . Brenner, Couenng theorems for Jnasrgs X, Ars Combinatorla 1 6 (1983), 57-67

[EG] E W Ellers and N Gordeev, Gauss decompos~tron wrth prescribed semtszmple par t rn classrcal Cheualley groups, Comm Aigebra 22 (1994), 5935-5950

[O] 0 Ore, Some r e m a r b on commutators. Proc. Amer. Math. Soc 2 (1951), 307-314

IS] A.R. Sourour, A faclonzalton theorem fo r matrices, Linrar and hlultliinear Algebra 19 (19861, 141-147.

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['l'l] It C 'l'l~ornpqon. L'ornrrrutillors t n [he spectal and general lznctlr groups, '1'r;ms A n ~ e r Math Soc 101 (1961). 16-33

Received: October 1994

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