Siberian Mathematical Journal, VoL 40, No. 2, 1999

GENERALIZED CONGRUENCE SUBGROUPS OF CHEVALLEY GROUPSt)

S. G. Kolesnikov and V. M. Levchuk UDC 519.44/45

I n t roduc t i on

Congruence subgroups of linear groups of order n modulo an ideal of a basic ring K are exploited for a long time. These have been generalized and successfully developed further from the sixties in connection with various questions of constructing and describing subgroups. The respective general- ization is the so-called congruence subgroup modulo a carpet [1; 2, 16.1.2; 3] or a net [4, 5] of ideals ~r of order n, i.e., a collection of ideals ~rij such that ~ri,~r~j C_ crij, i,j , m = 1, 2 , . . . , n ("carpet" or "net" subgroups).

Initially, Yu. I. Merzly~ov [1] used carpets (the term was introduced later) with quasiregular ideals ~rii. The corresponding carpet subgroup (in the group GL(n, K) it consists of the matrices e + a, a/j E ~r/j, where e is the identity matrix) is always generated by transvections and diagonal matrices [1; Lemma 1]. In [1] an effective method, based on arithmetical functions, was proposed for constructing nonincreasing chains of carpets such that the mutual commutant of the kth and ruth carpet subgroups for "nonsingular" pairs k, rn coincides with the (k + m)th carpet subgroup of the group SL(n, K). The main theorem of [1] and some of its applications were translated to the symplectic groups [6] and, in connection with Problem 6.34 in [7], to the Chevalley groups [8-10]. In this way, the low central series has been constructed for a Sylow p-subgroup of the Chevalley group over the ring g ~ , where p > 2 for the Lie types B,,, U,,, and F4 and p > 3 for the Lie type G2 (see also [11-13]). One of the problems we discuss in the present article is construction of the derived series of such a subgroup for the classical types Bn, Un, and D,,. As opposed to the case in [1], now its terms need not even be centrals. Note that the proof of Yu. I. Merzly~ov's theorem was extended by Kh. Roloff [14] to recurrent description for the terms of the derived and low central series of an arbitrary carpet subgroup of the group GL(n, K) in [1] for 2K = K.

Z. I. Borevich [5] also considered "diagonal-free nets" whose analogs for the Chevalley group associated with a root system ~ are nets of type �9 [15,16] and elementary carpets of type �9 [7, Problem 7.9.8; 17]. As an extension of the latter, we below introduce a special radical carpet ~r :

U q~* --. Id K; the subgroups S(~) and G(~,) of the Chevalley group are analogs of the carpet subgroups of [1]. The notion of a complete carpet of order n was earlier developed for the cases in which all diagonal elements of the carpet coindde with a fixed ideal (see [10,16,18,19], etc.), whereas the general ease leads to cumbersome conditions as was remarked in [16]. In the present article we reveal a close connection between the commutation of carpet subgroups and the introduced commutation of carpets. Under the assumption [~r, r] C ~ fl r , the commutator, as well as ~ and r , turns out to be a special radical carpet, and if 2K = K and, for the Lie type Gz, also 3K = K then [S(o),S(r)] = [G(~),G(r)] = S([~,r]) (Theorem 1; also see Theorem 2). This readily yields recurrent relations for the chains of the respective special radical carpets cr i and pi(~r) (i = 1, 2, 3 , . . . ) of the low central series and the derived series of an arbitrary group F, S(~) _ G(~) (Theorem 3). The main results of [14, 20] are obtained from here as consequences in a uniform manner. Henceforth by an arithmetical carpet we mean a carpet with elements ~rs = jl(s) defined by means of a mapping f : �9 O ~* --, N U {0} (a {)-function) and a fixed quasiregular ideal J . Commutation of carpets

t) The research was supported by the Russian Foundation for Basic Research (Grant 96-01-00409) and a grant of the Krasnoyarsk Regional Science Foundation.

Krasnoyarsk. Translated from Sibirskif Ma~ematieheskif Zhurnal, Vol. 40, No. 2, pp. 336-351, March-April, 1999. Original article submitted November 3, 1997.

0037-4466/99/4002--0291 $22.00 C) 1999 Kluwer Academic/Plenum Publishers 291

induces commutation of C-functions: [a(f),a(g)] = a([f,g]) (see w 2). Therefore, some commutator C-functions f(i) are defined such that pi(a([)) = a((f(i))), etc. (see w 2). Theorem 4 (for 2p(C)K = It') reduces the question of constructing the low central series and the derived series of the carpet subgroup defined by an arithmetical carpet a(f) to the question of constructing the sequences of C-functions f(i) and f(i).

The subgroup C(K, J) . U~(K) for K = Z f , and J = (p) is a Sylow p-subgroup of the Chevalley group C(K). In all cases, it is defined by an arithmetical carpet a ( f ) for which the ~-function f = fz and all C-functions fi+l = [fi, f] of the low central series were factually indicated explicitly (in analytic form) in [1,6,101 (also see the case of C = A1 in [21]). In w we explicitly find the commutator C- functions f(0 and therewith the derived series for the group C(K, J) . UC(K) when 2K = K, as well as (alongside the solvability length) for the Sylow p-subgroup of the Chevalley group C(Zf~), p > 2, of the classical types q~.

w 1. Mutual Commutants of Carpets and Carpet Subgroups

Henceforth we denote by K a unital associative commutative ring. We denote by C the reduced root system of an arbitrary simple complex Lie algebra, denote by H(~) the system of simple roots, and put pC ) = max{Cr, r)/Cs,~) I r,~ e C} (= 1,2 or 3) when C is of rank > 1, and p(A1) -- 2. We denote by C* the root system dual to C. Thus, r*(q) = (hr, q) (r, q E r is the Cartan number and hr = 2r/(r,r) is the co-root (see, for instance, [22, p. 322; 23, 3.6.1].

Considering the Chevalley group C(K), we usually distinguish the root elements z,(u) and the di- agonal elements h(x), h,(t) (r E C), where X ranges over the K-characters of the root lattice and t ranges over the multiplicative group K t of invertible elements of K [23, 24]. The commutator Chevalley formula for r, s, r + s E C shows that

= 1 ] , (1)

where the product is taken over all possible roots ir +js E C, i >_ 1, j >_ 1, in accord with their order. The constants c/i, rs are indicated explicitly, for instance, in [23, 5.2.2]; moreover, 1 < Ic j,, l _< p(c).

In line with [7, Problem 7.281, a collection of ideals {at [ r E r of the basic ring is called an elementary carpet of type �9 if

cij,tsa~ais C_ air+is (r,s, ir + js E C, i >__ 1,j > 1). (2)

Its extension a = {at [ r E q~ U C*} is called a special radical carpet if at . is a quasiregular ideal coin- riding with a_ , . and if a t a - t C at. for all r E q~. Clearly, an elementary carpet admits an extension to a special radical carpet if and only if all ideals ata-r are quasiregular (we obtain an extension by setting, for example, at . = a , a_ , ) ; also see Theorem 2 in [10]. By Lemrna 13 of [17], for p(q,)!K = K the notion of an elementary carpet coincides with the notion of a net of type �9 in [15,18,19]. Here the carpet condition (2) takes the simpler shape a, a8 C_ a,+8 (r, s, r + s E C).

With a special radical carpet a of type q~, we assodate the elementary carpet subgroup of the Chevalley group of type C:

E(a) = gr(x,(a,) I r e C)

and the subgroup sca)= gr(x, Ca,), h,(i + a ; ) l r e c).

By definition, the carpet subgroup G(a) of the extended Chevalley group of type C over K [23, 8.4.7] is generated by the subgroup E(a) and the diagonal elements h(x) for the various K-characters X of the root lattice such that x(q) equals 1 modulo the sum of the ideals at* (r E q~.,r*(q) # 0).

Given arbitrary systems of ideals as and rs (s E C U C*), we define the system of ideals

= c u c * }

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as follows:

r ,q-rE@ rE@,r*(q)#O

~q. = a_qrq + r_q~q (q q @).

By analogy with the partial ordering of nets of order n considered in Z. I. Borevich's articles, we put a _C r if as C_ r~ for all s E r U @*. Monotone chains and intersections a 13 r are defined naturally.

The following theorem reveals a close connection between the introduced "commutation" of carpets with the commutation of the corresponding carpet subgroups.

T h e o r e m 1. Let a and r be special tad/ca/carpets of type @ over a ring K which satisfy the condition [o, rl C_ o N r. Then [a, r] too is a special radica/carpet; moreover,

[s(~), s(,)] c_ [C(o), c(~-)l c__ s([~, d).

Furthermore, i f 2K = K and, for (I, of type G2, also 3K = K then

[s(o), s(,)] = s@, d).

L e m m a 1. The condition [~, r] C a f'l r for special radica/ carpets a and r is equivalent to the fulfillment of the following three conditions:

(1) trrrs C trr+sQrr+s i f r , s , r + s q (I); (2)~,r~.+~q~,.C_oqn~q,~,eE~, ,-*(e) # o; (3) crqr_q C aq. N rq. for every q E @.

P a o o r . This lemma is obvious. The system of ideals t~ = [~r, r] of Theorem 1 satisfies the con- ditions (1)-(3) of Lemma 1. In particular, for every root q E @ the ideal r lies in aq. t3 rq. and is consequently quasiregular. The fact that ~ is a carpet is verified as follows:

Wq~_q C_ qqr_q C_ Wq. = w_q., WqWs C_ Cqrs C_ ~q+s (q, s, q + s E (~).

Thus, t~ is a special radical carpet and S([a, r]) C S(cr) 13 S(r) . We now prove the inclusion [S(a),S(r)] C_ S([a, r]). For q,s ,q + s E @, the definition of w implies

the inclusion aqrs C wq+~. If we also have 2q + s E @ then

Analogously, iy aqr; C_t~iq+js (q,s, i q + j s E @, i > 1 , j ~ 1).

The commutator Chevalley formula yields the inclusions

[=q(~d,=,(,,)] c s@)

for q, s, q + s E q~. These inclusions are also valid for s = - q E q~, as the following relations show:

[Xr(t), x-r(u)] -- zr (c - l t2u)hr (c ) z_r ( -c - l tu2) , c - 1 - tu; (3)

~2r_q c_ ~q(~qr_q) c_ ~q(~q. nTr c_ ~q~q. c_ ~q, ~q~q c ~_q (4)

Here we have used the fact that the ideal a~q. is quasiregular, which implies that the elements of 1 - cqr_q are invertible in K. On the other hand,

[xq(~q),hs(1 + rs-)] = xq([(l + rs.) s'(q) - 1]~rq) C_ S(W) (q,s E @).

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Hence, by Lemma 1 of [10] the mutual commutant of the carpet subgroups S(a) and S(r) lies in S(w) and is moreover a normal subgroup both in S(a) and in S(r) . Using commutativity between the diagonal elements of the extended Chevalley group together with the relations

= qC[x(q)- c H c ~r

for h(x) 6. G(r) and the relations symmetric to them with respect to a and r, we deduce the inclusion [G(a), GCr)] C_ S(~).

It remains to prove the inclusion S([a,r]) C_ [S(a),S(r)] under the condition 2p(r = K. Assume that r, s, r + s 6. ~. The condition, imposed on the basic ring, guarantees invertibility of the constants c/j, rs in K. If �9 is not of type G2 then, up to the transposition of r and s, we have 2r -I- s ~ ~, and the right-hand side of the commutator Chevalley formula (1) involves at most two factors. Therefore,

�9 and so xr+s(arrs) C [S(a),S(r)]. The last inclusion is proved by analogy (although lengthier) also in the remaining case in which �9 is of type G2. Clearly, the tr~.sposition of a and r preserves the inclusion. Now, assume that q, s 6. ~. If the Cartan number 8*(q) differs from zero then it is a divisor of 2p(~), and by the choice of K we have

= = h (1 +

Hence, xq(wq) C [S(a), S(r)], q 6. ~, since the ideal wq is as generated by the products ~rrq-r (r ,q -- r 6. ~) and s*(q)~qrs*, s*(q)~s*rq.

By definition, the group S(w) is generated by the root subgroups Zr(Wr) and the diagonal elements of the form hr(1 q- z), where z = flU1 -{- "" % tkut and for every i we have ti E ~rq, ui 6. r_q, with q depending on i and equal to r or - r . As in [1, Section 2c] we have

t2u2 . . . 1 + 1% flu1 + " " + t tu t = (1% f lu1) 1% 1 + flu1 1 + flu1 + Z----I- tk-lUk-1

Therefore, the element h~(1 +z) equals the product of k elements of the shape hq(1 +tu) (q 6. {r, - r ) , t 6. aq, u 6. r_q). Relations (3) and (4) demonstrate that all factors in this product and, in consequence, the element hr(1 + z) itself belong to [S(a),S(r)]. The proof of the theorem is over.

Observe that for an elementary carpet {at I r 6. ~} with quasiregulax ideals ~r~r-r = ~r* we have E(a) = S(a). Therefore, Theorem 1 also describes the mutual commutants of elementary carpet subgroups.

If for �9 of type A,,-1 we return to the matrix terminology then the above carpet of type transforms into a coUection of ideals {aij, cr{id} [ i ~ j , i , j = 1 ,2 , . . . ,n} which is the extension of an elementary carpet of order n [17] (or a diagonal-free net in the terminology of Z. I. Borevich [5]) by means of ideals a{/d} satisfying the conditions

r C cr{i,j} -- ~r{j,i}, i ~ j.

Such a coUection is called a special carpet of order n. With it we associate a sta.udard carpet of ideals aij by setting crii = ~ j ~ i cr{i,j}. We extend the commutation [~, r] = ~ to arbitrary systems of ideals ~i~ and r ( i , j = 1,2 , . . . ,n) by the rule

n

m=l

Then the commutator of two special carpets of order n turns out to be equal to the commutator of the associated standard carpets. If cr and r are a special carpet and a standard carpet of ideals (or

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even systems) then their commutator is defined analogously. Also, we assume that 0- C_ r if a = a [3 r; moreover, the intersection a f3 r consists of the ideals

(0- [3 T)ij -- 0-ij 13 Tij, (0 [3 T){i,j} -- cr{i,j } 13 (Tii [3 Tjj), i # j .

We call a carpet a a radical carpet if all diagonal elements 0-ii or a{ij} are quasiregular ideals. (In [14], the corresponding standard net is said to be pseudoradicaL For an elementary carpet, here we may assume tha t 0-{i j} = 0-i10-1i, i ~ j.) Utilizing the ordinary matr ix notations, for example, those of [1], for the case of a special carpet we obtain S(0-) = g r ( t i i ( 0 - i j ) , d/i(1 + 0-{i~/}) I i ~ j ) and determine the carpet subgroup G(a) of the group GL(n, K) by using the associated carpet. In the case of a standard carpet, we have G(0-) = { e + a l aii e 0-ij}, while the definition of S(0-) is preserved on considering 0-{i,i} = 0-ii [3 0-ii" If

0-ii [3 E 0-jj -" E (0-ii [3 0-jj)) 1 < i < n, j = i + l /= i+1

in particular if all ideals 0-ii are pairwise distinct (this case is discussed in the main theorem of [1]) then S(a) coincides with the carpet subgroup G(0-)[3 SL (n ,K) of SL(n ,K) . Note that the proof of Theorem 1 extends with minor modifications to the case in which 0- or r is a s tandard carpet. Hence, the following theorem holds.

T h e o r e m 2. Let 0- be a special or standard radical carpet of order n of ideals of the ring K and let r be an analogous carpet. Suppose that [0-, r] C_ 0- [3 r . Then [0-, r] is a special radical carpet and

c_ [G(0-), c(r)] _c s@,r]);

for 2K = K the inclusions may be replaced by equalities.

w 2. D e r i v e d Ser ies a n d C e n t r a l Se r i e s of C a r p e t S u b g r o u p s .

A r i t h m e t i c a l C a r p e t s

T h e o r e m 3. Let a be a special radical carpet of type �9 or a radica/carpet of order n >_ 2 o[ ideals of the ring K, a 1 = pl(a) = a. Then

cri+l--[ai, a], i = 1 , 2 , 3 , . . . , and pi+l(0-)=[pi(a),pi(0-)], i - 1 , 2 , 3 , . . . ,

are nonincreasing chains of special radical carpets. For every subgroup F of G(0-), we have S(a) c F C G(a) and the series F D S(a 2) D S(0- s) D . . . is a central series. If 2K -- K and, for q~ of type G2, also 3K = K then this series is even the low central series and

r D s(p2(0-)) D s( / (0-)) D . . .

is the derived series of F. PROOF. For a special radical carpet 0- of type ~, by Lemma 1 we have 0-2 = p2(0-) < 0-, and 0 -2

is a special radical carpet by Theorem 1. Assume that we have already proved that , for a fixed i > 1, 0-1,0-2,..., a ' form a nonincreasing chain of special radical carpets. Then

i C ~ / - 1 0 - s C i i ~ - - - - 0-r+s = 0-r+s Q Or+s;

i-1 i-1 i _. i N O'q; 0-qalr. + Or*O~ C 0-q0-r* + 0-r*0-q C 0-q 0-q

0-q 0---q __ "-- 0-q. .

Again by Lemma I we have 0-i+I < 0-i [3 0-, and a i+I is a special radical carpet by Theorem I. By induction, a l , or2, a " , . . , and, similarly, pl (0-), p2(o.), p3 (0-),... are nonincreasing chains of special radical carpets. If a is a radical carpet of order rt then we derive the result by using Theorem 2 instead of Theorem 1. The claim of the theorem concerning the series is immediate from Theorems 1 and 2. The proof of the theorem is over.

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REMARK 1. Recurrent description for the terms of the derived and low central series of an arbi- trary carpet subgroup in [1] for the groups GL(n, K), 2K = K, was earlier found by Kh. Roloff [14, Theorems 3.6 and 4.2]. Likewise, the claim of Theorem 3 about the low central series for the sym- plectic case ~ = Un corresponds to the Main Theorem 4 of [20]. In [14, 20] it was also revealed that the ordinary nets or (standard or elementary) carpets are insufficient for characterizing the terms of the indicated series; the record of such a term involves a carpet subgroup together with some diagonal subgroup. As Theorems 1-3 demonstrate, for describing the mutual commutants of carpet subgroups it is reasonable to use the special carpets and subgroups S(~r) of w 1. We note that Theorems 1 and 3 were announced in [25].

A constructive method for arranging carpets is as follows. Fix a quasiregular ideal J of the basic ring K. Define a ~-function to be any function f on �9 U ~* with nonnegative integer values. To such a function, there corresponds the system ~r = or(f) of the ideals ar (r E �9 U ~*) ea~a of which is the product of f (r) copies of J ; moreover, j0 = K.

L e m m a 2. For a function f : ~ U ~* ~ N U {0}, the conditions (G1) 0 < f (q*) = f ( - -q*) <_ f(q) + f(--q) (q e ~); (G2) f(q) <_ f ( q - r) + f(r) ( q , r , q - r G ~)

are necessary and str ident for the system of ideals ~r(f) to be a special rarlical carpet for any choice of the ring K and a quasiregular ideal J of K.

PROOF. To prove necessity, we construct ~r(f) for a ring with the invertible element p(~)! (as was mentioned in w 1, the carpet condition (2) simplifies in this case) and a nilpotent ideal J of sufficiently high nilpotency class. Sufficiency follows from definitions.

The carpets of the form a ( f ) are called arithmetical. Henceforth, given arbitrary ~-functions f and g, we define the commutator [f, g] = w by setting w(q) equal to the least of the numbers

{ f ( r * ) + g ( q ) , f ( q ) + g ( r * ) ( r e ~h, r*(q) # 0) f (q -- r) + g(r) (q -- r , r e { ) }

and by setting w(q*) = min{f(q) + g(-q), f ( - q ) + g(q)} (q e ~).

It is easy to see that [a(f) ,g(g)] = cr([f,g]). Thus, the commutation of ~-functions is induced by the commutation of carpets. In view of Theorem 1, for ~-functions f and g satisfying the carpet conditions (G1) and (G2), fulfillment of the carpet conditions for the commutator If, g] is guaranteed by the condition [a(f) ,a(g)] C_ ~r(f) N a(g). Applying Lemma 1 to arithmetical carpets, we obtain the following

L e m m a 3. Let f and g be ~-fimctions. The inclusion [~r(f),a(g)] C a ( f ) fl ~(g) is valid for an arbitrary choice of K and J ff and only ff the fo//owing conditions are satisfied:

(I) f (q) <_ f (q -- r) -F g(r), g(q) < f ( q - r ) 4- g(r) (q , r ,q - r e ~); (2) f(q) <_ g(q) +/(r*) , g(q) <_ f(q) + g(r*) (q,r G ~h, r*(q) • 0); (3) f(q*) <_ f(q) + g(--q), g(q*) <_ f(q) + g(--q) (q E ~). Theorem 3 implies

T h e o r e m 4. Let f be an arbitrary ~-function satisfying the carpet conditions (G1) and (G2) and let s be a subgroup of G(tr(f)) including S(~r(I)). Put

/ (1) : f (1 ) __ f , / ( i + I ) : [ / ( i ) , f ] , f ( i + l ) : [ / ( i ) / ( i ) ] , i = I , 2 , 3, . . . . (5)

Then cr(f(i)) and o'(f (i)) (i -- 1,2,3 , . . . ) are nonincre~ing chains of special radical carpets for an ar- bitrary choice of the basic ring K and a quasiregular ideal J of K; for 2p(~)K = K the series of subgroups

r D . . . , r D D . . .

form the low central series and the derived series in s respectively.

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REMARK 2. The condition it equivalence. This condition and S(r) coincide and ar = rr assume that up to equivalence

S(a) = S(r) for special radical carpets is weaker than equality. We call amounts to the fact that the diagonal subgroups of the groups S(cr) (r E ~) (see [10, Theorem 21). For a special radical carpet u, we may

~rr. N trs. C_ trq* (r,s, q q g, r + s + q = 0 ) .

We consider g-functions f and g equivalent ( f ,,~ g) if, for an arbitrary choice of K and J , the cor- responding arithmetical carpets a ( f ) and u(g) are equivalent. Of course, such functions coincide on g. Theorem 4 reduces (for 2p(d2)K = K) the question of constructing the low central series and the derived series of the carpet subgroup determined by an arithmetical carpet u ( f ) to the ques- tion of constructing the sequences of ~-functions f(0 and f(0. Clearly, it suffices to construct these g-functions to within equivalence.

The ordinary congruence subgroup (whose commutator structure was studied by D. A. Suprunenko in [26], etc.) is determined by an arithmetical carpet with a constant ~-function f , f ( r ) ~ m (> 1). Induction on i and the definition of the commutator of ~-functions show that f(i)(r) = mi, f(i)(r) = m2 i-1. Obviously, an arbitrary g-function is equivalent to at most one g-function constant on g*. On the other hand, the definition of the commutator implies the following

L e m m a 4. If, for g-functions f and 9, the set

#*(f,g) = {r* g* I fCr) + g(-r) = + g(-s))}

generates a Z-module which includes ~* (in this event we call the pair f , g nonsingular) then the com- mutator [f,g] is equivalent to a g-function constant on ~*.

We denote the minimal value of a ~-function f on g* by f(0). Thus, we may always assume that a q~-function is also defined on q~0 = ~ U {0}.

L e m m a 5. Suppose that a g-function f satisfies the carpet conditions (G1) and (G2). Construct the ~-ftmction f(i) from f by formula (5). Then the function f ( r , k ) = f(t)(r) on (g0,N) satisfies the conditions

(F1) f(O,k) > 0, (F2) f ( r , k ) < f ( r , k + 1) (r E ~0), (F3') f (q , k + m) < f (q - r,k) + f ( r , m ) ( q , q - r , r E go, q # 0 o r r # 0).

PROOf. By Theorem 4 and Lemma 2, the g-functions f(t~) satisfy the carpet conditions (G1) and (G2), and the conditions (F1) and (F2) hold. Since the mutual comrnutant of the kth and ruth centrals of an arbitrary group lies in the (k + rn)th central [27, Corollary 10.3.6], we have

f(q,k "I" m) < [f(k),f(m)](q) <- f(q- r,k)-1- f(r, m).

REMARK 3. The construction of carpet subgroups in [i] (in the matrix terminology) and in [10] wed the function/: (#o, N) {0) satisfying (FI), (F2), and

(F3) inequality (F3 ~) holds; moreover, equa~ty in it is attained for arbitrary fixed q, k, and ra. It is clear that a ~-function fk(r) = f(r,k) (r E ~0) with a constant value f(0,k) on ~* satisfies

the carpet conditions (G1) and (G2) for all k. For a nonsingular pair of g-functions fk and fm (or, in the terminology of [I, 10], for a pair k, rn nonsingular with respect to f) the analog of Yu. I. Mer- zlyakov's theorem proves the equivalence [fk, fro] " fk+m; this equivalence holds even if (F3') holds in place of (F3) and we require attainability of equality only for a chosen pair k, m and an arbitrary q E ~0 [10, Remark 1]. In particular, if the pairs fl, fk are nonsingular then, for the g-function f(r) = f(r, 1) (r E g0), we have f(i) "~ fi for all i. Analogously, f(0 ,.~ f2~-I if all pairs of the form f:, (1 _< k < i) are nonsingular.

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w 3. C o m m u t a t o r S t ruc tu re of the Subgroups ~ ( K , J ) �9 U @ ( K )

The product of the unipotent subgroup Ucb(K) = gr(xr(u) [ r E r u e K) of the Chevalley group ~(K) and the congruence subgroup modulo a quasiregular ideal J for K = Zpm, J = (p), is a Sylow p-subgroup [10]. For some other K and J, the commutator structure of the group ~(K, J ) . U~(K) was described by Ph. Hall for r = A1 from the other viewpoints [21, Theorem 2, 3].

Let h be the Coxeter number of the root system ~ [22, Tables I-IX] and let ht r be the height function of a root, ht0 = 0. The following function was introduced in [10]: f ( r , k ) = - [ ( h t r - k)/h], (r fi ~0), where [.] is the whole part of a number. Originally, such functions were exploited in the matrix terminology in [1] and later in [6]. It turns out that the subgroup r = r J ) . U ~ ( K ) is a carpet subgroup and is determined by an arithmetical carpet with the ~-function f l ( r ) = f(r , 1) (r e ~0). Here the function f ( r ,k ) satisfies the conditions (F1), (F2), and (F3'). It is well knownthat the low central series of the group r for p(r = K is determined, as was mentioned in Remarks 2 and 3, by the chain of arithmetical carpets with ~-functions f(i)(r) = f ( r , i) (r E r having a constant value f (O,i) on ~* (see [10], and also [1,6] for the types An and C,) . Throughout this section we assume that the functions f(r , k) and f~ and the groups r are fixed.

For ~ = An, Yu. I. Merzlyakov [1] proved property (F3) for the chosen function f ( r ,k ) and constructed the derived series of the group F in the case when n is not a power of 2 (in this case all pairs f2k, f2 k are nonsingular). In the exceptional case of n = 2 s, the derived series was constructed in [11]; moreover, it turned out that now not each of its terms is determined by an arithmetical carpet with a ~-function constant on ~*.

On the other hand, property (F3) is as a rule violated when r is not of type An. Moreover, it turns out that for k = m = 2 there may even exists a root q E �9 such that equality in (F3 ~) is not attained. Such a root q for �9 = G2, Bn, Dn can be chosen among the roots of height 4. (For ~ = G2, see also [10, Example 2].) Surely, in these cases we cannot use the analog of Yu. I. Merzlyakov's theorem in [10] for constructing the derived series of F even if the pair of the functions f2~ and f2k is nonsingular. At the same time, the following theorem shows that the derived series of F, as well as the corresponding commutator functions, admits a uniform description for the Lie types An and Cn.

T h e o r e m 5. Let �9 be a root system o[ type An or Cn and let k and m be natural numbers. I l k and h are coprime or k q- m is not a multiple of h then [fk, fro] "~ fie+re. I[ k and m are multiples of h then [A,fm] fk+m+m. In the other cases, [A,fm] ak -- -- f~+,n" Moreover, [f~, fk+m+l i1r k and m are multiples o f h and i , j E {1 ,2 , . . . , h - 1}.

Here and in the sequel, we denote by dk the remainder of the division of k by h, and if k is a multiple of h and 1 < i < h then f~ the ~-function coinciding with fk on �9 U ( ~ k U ~dk-h). and with f~+l in the other cases.

We need a description for singular pairs fk, frn- Denote by ~i the collection of roots of height i, 0 < Iil < h, putting ~0 = ~• = O.

L e m m a 6. Let ~ be a root system of type An or Cn. I f Iil + IJl < h then (a) (~i .[_ ~ i ) Iq ~ -- r (b) Z(~iU ~i) = Z(r

PROOF. Additivity of the height function implies the inclusion (~i + ~1) rl ~ c_ ~i+i. Therefore, to prove item (a), it suffices to establish the inclusion ~i+i C r + ~i. Furthermore, we may assume that 0 < [i + j[ < h and that 0 < [j[ < i < h in view of the relation ~ i - i = _ ~ i - i .

Let ~ be of type An, and so the Coxeter number h equals n + 1. According to Table I of [22], for 0 < j < i the set ~i+i consists of the roots e l - i - i - - e l " - ( e l - i - - e l ) q- ( e l - i - i - - e l - i ) , i + j < I < n + 1, which belong to ~ i+~ i . If0 < j < i then every root in r can be written as (et--el+i)+(et+i-et+i-i) , 1 < l < n + 1 - i, or as (et- i - el+i-i) + (el - el- i) , J < l < n + 1 - i + j . In both cases the root lies in ~i + ~ - i . Item (a) is thus proven for the type An. We prove it for the type Cn by using the fact that, by Lemma 7 of [17], a symmetry of order 2 of the Coxeter graph of a root system r of type A2,-~ induces a homomorphism ( from the root system ~ onto a root system of type Cn which extends to a homomorphism of the root lattices. We only have to notice that ht(((r)) = ht r (r E ~).

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Item (b) results from applying item (a) and Euclid's algorithm for finding a greatest common divisor. The proof of the lemma is over.

The following lemma gives us a criterion for singularity of a pair of e-functions fi, and fm.

L e m m a 7. Let r be a root system of type A,, or Ca. A pair fk , fm is singular if and only if

k + m = 0 (modh) and G C D ( k , h ) • I . (6)

PROOF. Henceforth we assume that [k/h] = d I'. Then for every r E r we have

d :, d t: + 1,

h ( r ) = f ( r , k ) = d k + 2,

dt < ht r < h, d k - h < h t r < dk,

- h < ht r < dk - h.

In particular, if 0 < dk +din < h then h + , , ( 0 ) = d k + d ~ + 1 and the equality h ( r ) + f , , ( - r ) = fk+m(0) holds in exactly those cases in which ht r > dk and d m - h < - h t r < dm or - h t r > d,n and dt~ - h < ht r < dk. Therefore,

�9 ( h , fro) = (~ e v I ht r e [dk - h, -d,~] U [dk, h - dm]}. (7)

Analogously, for h < dk + d,= we find fk+,n(O) = d t + d m + 2 and

r = { rE ~ I htr e [1 - h,-dm] U [dk - h,h - d,nl u [dk, h - l]}.

Sufficiency:. If both numbers k and m are multiples of h or, equivalently, d/c + d m = 0 then

h + m ( 0 ) = d k + d ~ < d k + d ~ + 1 = h ( r ) + f m ( - r ) (r ~ r

whence q~(f~, fro) = o and so fk, : , , is a singular pair. Suppose that k + m is a multiple of h but dk + d,~ # 0. In view of (7), we obtain ~(fk , f,,,) =

~dk Uq~dk-h. Clearly, (~1)* = (r if the lengths of the roots in ~1 are pairwise equal. Therefore, if is of type A,, or C,, but k and rn are even numbers then we must have Z~*(fk, fra) = Z(~dkU~dk--h) * = Z((~*) ~* U (~*)~k-h). From Lemma 6 we infer that conditions (6) are in this case sufficient for singularity of the pair ft,, f,~.

Prove sufficiency in the remaining case when ~ is of type C,, and k and, consequently, dr, and d m = h - dk are odd numbers. Here h = 2n and if &, = n (> 1) then q~* does not lie in Z~*(ft, , fro) = Z((q~")*). Therefore, we may assume that dk < h - dr,. Representations for root systems which we use in the current section are taken from [22, Tables I-IX]. In particular, for �9 of type C,~ the co-root hr equals r or r/2 if r is a short root or a long root respectively. Using Lemma 6, we find

Z r = Z ( C ~ U Vd~-h U { - - e , , e t , } ) = Z ( V ~ , e , , e e ) = H,

where d = GCD(k, h), t = (dk + 1)[2, t' = (h + 1 - dk)[2. At the same time,

~,, = ~, - ( ~ - ~,+d) . . . . . (~, , -a - ~e) ,

so that H - Z(q~g, et). Since [~d[ = n - [d/2], for d > 3 the rank of the Z-module Z r is less than n(= the rank of r Suppose that d - 3. Then H lies in the Z-module generated by the elements

e i ( l < i < n , i - 2 (mod3)) , e i - e i + 3 ( l < j < n - 3 , j ~ 2 (rood3)).

Therefore, if a l~ l -}- a2e2 + - ' ' +anOn E H then the sum of the coordinates a i with j = 0 or 1 (mod 3) equals 3. Hence, el ~ H, and we again infer that q)* does not lie in Z~*(fk,fm).

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Necessity: Let us demonstrate that if the conditions (6) of the lemma are not satisfied then the pair fk, fra is nonsingular. In view of the equality II((I)*) = II(~)*, it is obvious that this pair is nonsingular when @(fk,fm) D (I)1 = II(~) , in particular when dk + dra > h or one of the numbers dk and dm vanishes. Therefore, we may assume that dk _< d m < h - dt~ and 0 < d/: < h/2. By Lemma 6, (I) 1 lies in the Z-module Z((I )~, ~d~+l) and in the case when GCD(k, h) = 1 (for (I) of type Cn this yields oddness of k) also in Z((I )~ , (I)d~-h). Validity of the lemma for @ of type An now follows from (7) and the fact that in this case the root system is selfdual.

For (I) of type Cn we have 7.(I)* = Z((I) 1, en), and by (7)

z # * ( A , I . , ) = z ( # ( A , f . , ) , ~ ) ~ ~1 ,

where s -- (h + 1 - d ~ ) / 2 if d~ is odd and s = ( h - d ~ ) / 2 if d~ is even. Moreover,

en = - - ( e . - ~ -- e . ) -- ( e . - 2 -- ~ . - ~ ) . . . . . (e , -- e , + l ) + ~, e Z ( ~ ( A . / , . ) .

Thus, f/~, fm is a nonsingular pair both in the case when k + m is not a multiple of h and in the case w h e n the numbers k and h are coprime. The proof of the lemma is over.

PROOF OF THEOREM 5. It is shown in [1] for the type An and in [6] for the type Cn (in the matr ix terminology) that the function f (r , k) satisfies condition (F3). This readily implies that the commuta- tor [fk, fm] agrees on @ with the function fk+,,- Moreover, by [1, 6,10] these @-functions are equivalent when f/~, fm is a nonsingular pair. Applying Lemma 7, we derive the first claim of the theorem. If k and rn are multiples of h then we check the equality [fk, fm] = fk+,a+l straightforward (also see [1, B] for q~ of type A, and k = m); so the second claim of the theorem holds too.

We are left with settling the case in which k + m is a multiple of h while k is not a multiple of h nor it is coprime to h. Immediate verification shows that fk(r) + f ,n ( - r ) attains the minimal value fk+m(0) for all r E r U @d~-h and takes the value fk+m+l(0) in the other cases. Thus, [:,, =

Prove the last equality of the theorem. The conditions on /c and m imply that the values of the functions f~, f ~ , and fk+m+l are equal to d k, d m, and d/c+m, respectively, at the positive roots

and to d k + 1, d m + 1, and d/~+m q- 1 at the negative roots; moreover, f~(r*) _> d k , f i ( r *) > d m and

f/~+m+l(0) -- d k+ra q- 1. This implies that [ f~ , f i ] (r) >_ f/~+m+l(r) for every r E ~. On the other hand, if r, r - q E �9 for some q E II(~) then

/~+m+~(~) = a k+~ + t = (d k + 0 + : = :~(~ - q) +/L(q),

where l = 0 for r E @+ and l = 1 if r is a negative root. However, if r is a simple root or a root of the least possible height then we can choose s E ~i U ~ i -h so that s*(r) ~ 0. In this case fk+m+l(r) =

j i , f~(r)-t- f~(s ). Thus, the commutator [f~, f~] agrees with the function fk+rn+l on @. Observing that

ft+m+1(0) = f~(r) + f ~ ( - r ) for every r E @, we obtain the equality [.f~,fJm] - flt+,,,+l. The proof of the theorem is over.

Theorem 5 enables us to describe the derived series of the group r - U@(K).~(K, J) in a uniform manner for @ of type An and Cn under the condition 2K - K. In view of Theorem 4, it suffices to indicate the commutator ~-functions f(k), to within equivalence, for the function f l , f l ( r ) -- f (r , 1) (r E �9 u '~*).

C o r o l l a r y 1. Suppose that @ - An or @ = On, f(1) _. f l . / f the Coxeter number h = h(@) is not a power of 2 then f(k) ... f2k_l, k = 2, 3, . . . . Assume that h = 2 t, t _> 1. Denote by q and s the quotient and remainder o[ the division of the natural number k by t + 1. Then, for k which is not a multiple of t + 1, we have

q

f(k) ~ ix(k), ~(k)= 2 "-1 ~(~ .h) i. i=O

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However, i f k is a multiple of t + 1, i.e., s = O, then

q--1 f(k) ~ fh/2 2 6 ( k - 1) = h E ( 2 h ) i . J26(k-1)'

i=0

We now establish analogs of Theorem 5 for the orthogonal types Bn and Dn. As was mentioned before Theorem 5, the function f ( r , k ) now falls to satisfy condition (V3).

Fix a root system �9 of type Bn and distinguish the subsets

P n = ( - t - e t , 4 - ( e i - e j ) [ l < i < j < n , l < l < n } , L ~ = ~ \ P n ,

{ L ~ k+l-h O {-~1 - el+d\} if 0 < dk < n,

Mk = L~ k+l-h U {--edk+Z-h,--el + eh--dk} if n < dk < h - 2,

p i = Pn fi q~i, and L / = Ln 13 ~i. Given a natural number k such that 0 < dk < h - 2, define the function ek by setting for r E ~0 { "

dk i f d k + l _ < h t r < h o r r E P ~ ,

C k ( r ) = d k + l i f d k < n , b u t d k + l - h _ < h t r _ < d k - l o r r E M k ,

o r d k > _ n , b u t d t + 2 - h _ < h t r _ < d k < h o r r E M k , dk + 2 in the other cases.

Observe that '/2 = f2, and that in [10] it is proved that [fx,fl] ~ f2.

T h e o r e m 6. Let �9 be a root system of type Bn, n > 5, and let k be a naturM number. Then (a) [r ~bk] "-' r 0 < dk < n -- 1 and 2d/~ r n; (b) [Ok, ek] ~ k+2 for n < dk < 2n - 2 and 2dk # 3n. PROOF. For n < d/c < 2n - 2 the set ~(r Ck) includes II(~), and for 0 < dk < n - 1 it contains

all roots of height n and n 4- 1 whose duals, as we can easily see, generate q~*. Therefore, the pairs of ~-functions Ck, Ck are nonsingular both in the case of (a) and in the case of (b). Put ~ = {r E q~0 [ Ck(r) < u}, u E Ck(~0). To prove items (a) and (b) of the theorem, it suffices to validate the equalities

( * k t + i " ~ * k t + j ) ~ O - - ~ 2 2 k t + i + j ( ' - d2t+i+j], / , j -- O, 1; (8)

tB2k+2 ~2k (i~kt+iq-*kt+j) f'lr ('-- d2~._l+i+j], i , j ' - O , 1,2, 0 < i + j _ < 3 . (9)

Inclusions of the left-hand sides of (8) and (9) in the right-hand sides are immediate from the definition of the function q,t. Prove the reverse inclusions. Observe that a root system of type G,, [22, Table III], on the one hand, is dual to �9 and, on the other hand, is a subset of Z@ which we also denote by C,,. For an even s the set of roots of the shape ei - ~j and height s, together with the roots in L',, +1 if s > 0 and in L s-1 if s < 0, coincides with the set of roots of height s in G,,. Put

H ~ = (C.) dt U {sn+l-dt} for dt <_ n,

H ~ = (Cn) dt for dk > n, Qt (C,,) d~+2-h U (MI~ \ d~+l-h = L n ) for 0 < d~ < h - 2 .

Given an arbitrary r E @, denote by {r} + the set of roots in @ of the shape r + q~ + --- + ql (qi ~ II(@) U {0}) for which the sums r + q~ + . . . + qi (1 < i < l) are roots as well. Also, put 0 + = q~+ U {0}. Using the definition of the sets @~ and the fact that r + q) < ek(r) (r, r + q ~ r q ~ II(@)), for an even k, 0 < i lk < h - 2, we have

k �9 Oo. + u ( U ,,'. = U

rEQ k r~H ~

L e m m a 8. The inclusion {r + s} + C_ {r} + + {s} + holds for every irreducible root system r and arbitrary r, s, r + s ~ ~0, r # - s .

301

PROOF. The inclusion is obvious when r = 0 or s = 0. Therefore, we assume that r, s fi #. It is clear that r + s ~ {r} + + {s} +. Suppose that our inclusion has already been proven for all roots of height less than i, ht(r + s) < i < h, in {(r + s} +, and let s' be an arbitrary root in {r + s} + of height i. The definition of the set { r+s} + implies that s' = p + q for somep ~ { r + s } + and q ~ II(~). Now, h t p < i, and by hypothesis p = pl + p2 ~ {r} + + {s} +. The inclusion s' q {r} + + {s} + then foUows from the fact that p # 0 and, in consequence, at least one of the sums p~ + q and ~ + q, for example the first of them, is a root and lies therefore in {r} +. The proof of the lemma is over.

If we define the sets { r} - as the sets of roots of the shape r - q~ . . . . . q~, where r q q~ and qi is either a simple root or 0; moreover, r - ql . . . . . qi too is a root for every i, 1 < i < 1; then it is easy to verify tha t an analogous assertion holds for them.

We now prove the reverse inclusions to (8). It is obvious that if 0 lies in the right-hand side then k . it also lies in the left-hand side. If i = j = 1 then we have -~1 - ~,,,~,, - ~2,~1 + e2,0 ~ ~d~+l,

whence, using Lemma 8 and the Remark after it, we successively obtain

(~-- C__ {--~1 -- ~n}-- "I t- {~n -- ~2}--, ~'f" C__ {~I "Jr" ~2} "f" -f" r ~ C__ ~kdk.l.1 "4" r . k

B y item (a) of Lemma 6, we have (C,,) d~ +(C, , )s ) f ' lC , , = (Cn) dk+s, s = d ~ , d , + 2 - h . Furthermore,

k p = (p - c , + 1 - ~ ) + c , + 1 - ~ e ~ + ~d~+~,

q -- e l = (s + q) + (-r - s 6 ~dkk + ~kd/,+l,

where l = 0, p = ~n+l-2d~, and q = -~1+2~ if 2d~ < rt and l = 1, p = --~2dk+2-n, and q = r k . This fact and Lemma 8 imply if 2d/, > ft. Therefore, H k C q ~ + ~ and Qk C ~ + ~d~+l

the inclusion k ~dk2l: C ~dkk -{-~dk/~, U {'r}'l" C r -t-~dk..I.l �9

r~Q ~

Let q be an arbitrary simple root. Then r - q ~ ~ for some r 6 H~; moreover, ht q - r E d~ - h; k Now, the equality ~+ = UreiI(r + and Lemma 8 imply the inclusion ~+ C i.e., q - r ~ ~d,+l"

k and To prove (9) for i = 1 and j = 2, it suffices to note that r + r162 - e l , e l - e2 ~ q~*+1

~' and apply Lemma 8. By item (a) of Lemma 6, for s = d~ or d~ + 2 - h t~2 -- ~:1, --~2 -- ~3, 0 ~ ~d~.+2

we have + C,, = Furthc ore,

k P --" (P -- ~n+l-dk) -[- en+ l -dk E r -~- ~ d k + l ,

k k q - ~i = (c1+d~ + q) + ( - c - c i + ~ ) ~ ~ k + 1 + ~ + I ,

w h e r e p = ~,t+l-d2~+2 a n d q = --Zl+d2k+2 if n < dk < 3r t /2 a n d p - -et t2~+2+2-n a n d q = eh-d2k+~ if

k and Q~ ~ ~ T h i s f ac t and L e m m a 8 3n/2 < dk < h - 2. Therefore, H ~ C ~ k + ~dk.{.1 C r -~" r �9 imply the inclusion

k k

r~Q k

Since ~-1 U ~2 C ~k and (~-1 + ~2) N �9 II(~), the same Lemma 8 yields the inclusion dk +1 -- /~ ~/~ T h e p roof of the t h e o r e m is over.

Let ( be a homomorphism from a root system �9 of type D,, onto a root system of type B,,-1 which is induced by a symmetry of second order of the Coxeter graph (see [17, Lemma 7]). Define the ~-function ~b~ on ~0 by setting ~b~(r) = ~b~(r and ~b~,(0) = ~b~(0). Observe that the definition is correct, since ~b~,(r) = ~bk(s) whenever r = ((s). Moreover, ~b2 = f2 " [fl , f l ] = f(~). The definition of the functions ~b~, and Theorem 6 imply

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C o r o l l a r y 2. Assume that Put ul = 1, and for i > 1

T h e o r e m 7. Let ~ be a root system of type D , , n > 6, and let k be an even natural number. Then

(a) [r Ck] "" r for 0 < dk < n - 2 and 2die # n - 1; (b) [r Ck] "" r for n < dk < 2n - 4 and 2dk # 3(n - 1).

We apply Theorems 6 and 7 to construction of the derived series of the group F = U~(K). ~(K, J) for ~ of type B2n-1 and D2n; here h = h(r = 4n - 2. In view of Theorem 4, as before it suffices to indicate the commutator functions f(k) to within equivalence.

= B2n-1 or D2n, n > 3; moreover, n - 1 is not a power of 2.

2ui f f dui < 2n, Ui+l = 2ui + 2 i f dul >__ 2n.

Then f(i) ,.~ r for r of type B2n-1 and f(i) .., r for ~ of type D2n.

PROOF. If one of the numbers d,~ i (i > 1) is a multiple of 2n - 1 then this number dui_~ must coincide with 2n - 1. However, as opposed to 2n - 1, all numbers d~i ( j > 1) are even. Suppose that ui is a multiple of 2n - 2 and is the least number with this property. Then d,,~ = 2n - 2. Using the definition of ul, we obtain

n - 1 n - 1 n - 1 dui_t_l = " - ~ "~" - - ~ + ' ' " ~- 2k---'-~ q- d, O < k j < t , O < l < t ,

where n - 1 = 2td and d > 1 is an odd number. Thereby we again arrive at a contradiction with evenness of the numbers duj. Thus, all numbers dul are not multiples of 2n - 1, 2n - 2. Therefore,

under the conditions of Corollary 2, we can always calculate the commutator functions f(i) by using items (a) and (b) of Theorems 6 and 7. The proof of. the corollary is over.

C o r o l l a r y 3. Let s be the solvability length of type r over the ring Zp-~ (p > 2) and let h = h(~) Then the following inequality holds for q~ of type An,

a Sylow p-subgroup of the Chevalley group of be the Coxeter number of the root system ~. Ca (n > 1):

- [ - log2 mhl - 1 < < - [ - Iog2 mh], (10)

The same estimate is also valid for �9 of type Ban- l , D2n (n > 3), i f n - 1 is not a power of 2.

PROOF. The upper bound of s is indicated in the main theorem of [10]. To prove the lower bound, we must find the smallest number k for which the minimal value f(k+l)(r) is at least m; clearly, ht(r) = h - 1 . It is easy to see that the numbers 5(k-I- 1) and uk+l of Corollaries 1 and 2 do not exceed 2 ~ -I- 2 t~-I - 1. From the inequality m < - [ ( h - 2 k - 2k-1)/h] we find tha t k A- logs 3 - 1 > log s mh, and so k > - [ - log s mh] - 1. The proof of the corollary is over.

Let us demonstrate that the estimates (10) for the solvability length s in Corollary 3 are unim- provable. The equality s = - [ - log 2 mh] for the solvalibity length s holds when ~ is of type As or C,, and n is not a power of 2 (see Corollary 1), and also for the groups go (zp4 ) - ~(Zp4, (p)), where ~ is of type B7 or Ds (p > 2). The equality s = - [ - l o g 2 mh] - 1 is attained, for example, for the group V~(7.ps) - ~(Zp,, (p)) with ~ of type B7 or D8 (p > 2).

R e f e r e n c e s

1. Yu. I. Merzlyakov, "Central and derived series of matr ix groups," Algebra i Logika, 3, No. 4, 49-53 (1964).

2. M. I. Kargapolov and Yu. I. Merz]yakov, Fundamentals of the Theory of Groups [in Russian], Nauka, Moscow (1977).

3. N. S. RomanovskiY, "On subgroups of general and special linear groups over a ring," Mat. Za- metki, 9, No. 6, 699-708 (1971).

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11.

"12.

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