Generators of large subgroups of the unit group of integral group rings

manuscripta math. 78, 303 - 315 (1993) manuscripta mathematica ~) Springet-Verlag 1993

Generators of large subgroups of the group of integral group rings*

Eric 3espers and Guilherme Leal

unit

1. I n t r o d u c t i o n

In order to prove a Milnor conjecture H. Bass proved in [3], Theorem 9.4, that if G is a finite cyclic group, then the Bass-Milnor cyclic units of the integral group ring Z G generate a subgroup of finite index in H(TZG), the unit group of 7/G. Actually Bass proved a stronger statement which implies that his result holds for all finite Abelian groups. The latter stimulated the search for new units which would generate a subgroup of finite index in the unit group of the integral group ring of a non-commutative group. In a series of papers J. PAtter and S.K. Sehgal [17,18,19,20,21] showed, for several classes of finite groups including nilpotent groups of odd order, that the Bass-Milnor cyclic units together with the bicylic units generate a subgroup of finite index in H(TZG).

3. PAtter and S.K. Sehgal [17,19] showed that under some restrictions the normal closure of the group generated by the Bass-Milnor cyclic units together with the bicycHc units is a subgroup of finite index in the unit group of the integral group ring. We will improve this theorem by giving explicit generators (and thus avoiding taking the normal closure). The reader is also refered to [12] for some related work of E. Kleinert.

In the papers quoted above, all the groups G considered are such that the following conditions are satisfied:

1. if a simple component of QG is a division ring, then its Schur index is less or equal than two. And if the Schur index is two, then the center must be a real field, i.e. this component is a totally definite quaternion algebra.

2. if a simple component is a two-by-two matrix ring over a division ring D, then D is not the rationals or a quadratic imaginary field or a totally definite quaternion algebra.

In this paper we give, in terms of some non-central idempotents, a finite set of generators of a subgroup of finite index of H(ZG) for all groups satisfying conditions (1) and (2). Furthermore, we describe the respective idempotents for

This work is supported in part by NSERC Grant OGP0036631, Canada, and CNPq, Brasil

304 Jespers and Leal

groups which have no non-abelian homomorphic image which is fixed point free, in particular symmetric groups and (almost all) metacyclic groups.

2. P r e l i m i n a r i e s

Throughout G denotes a finite group. For a subgroup H of G we denote / t = )-']heH h, and for an element g E G we set ff = < 'g">.

Recall that a Bass-Milnor cyclic unit is a unit of the form

1 - im~ (1 + a + . . . + a~-t) m + \oral (a) /a ,

where a E G, 1 < i < ord(a), (i, ord(a)) = 1 and m = 9~(IGI), ~o the Euler p-function. Let B denote the subgroup ofb/(7/G) generated by the Bass-Milnor cyclic units.

We recall some notation (used in [19]) concerning the rational group algebra QG. Let et,"., et be the primitive central idempotents of QG. Then

QG = QGel @ ...@ QGet,

and each QGei is identified with M,~(Di), the full ni X ni matr ix ring over a divsion ring Di the center of which we denote by Ki. Let

A = AI @ " ' ~ At,

where Ai is a maximal 77-order in QGei containing 77Gei. For every i, let Oi be some maximal order in Di. Then Mni(Oi) is a second maximal order in QGei. We shall write GLi or GL(ni,Oi) for its group of units, and SLI or SL(ni ,Oi) for the subgroup in GLi that consists of all elements having reduced norm one.

For an ideal Q of Oi we denote by E(Q) the subgroup of SLi generated by all Q-elementary matrices I + qetm, where q E Q, l • ra, etm a matr ix unit, and by J~(Q) its normal closure in SLi. We now state the celebrated theorems of Bass-Milnor-Scrre [4], Vaserstein [23] and Bak-Rehman [2] (for a survey we refer the reader to [15]).

T h e o r e m 2.1. Assume that either ni > 3 or ni = 2 and QGei is not a 2 x 2- matrix ring over a division algebra which is a totally definite quaternion algebra or the center of which is the rational field or an imaginary quadratic field. Then

i. (SLi : E(Q)) is finite for every non-zero ideal Q of Oi. 2. Every non-central subgroup of SLi normalized by a subgroup of finite index

contains E(Q) for some non-zero ideal Q of Oi.

T h e o r e m 2.2.

1. Ifn~ > 3, then p~(Q2) g E(Q), in particular ( S L ( . . O d : ECQ)) < oo.

Jespers and Lea2 305

2. If ni = 2 and if Oi is an order in the ring of integers of an algebraic number field which is not rational or imaginary quadratic, then ( SL(2, Oi) : E(Q) ) < co.

Note that in [17] it is shown that Di is a totally definite quaternion algebra if and only if SLI(Oi) is finite and Di ~ Ki. For a proof of Theorem 2.2 we refer the reader to [19].

The next lemma shows that the Bass-Mflnor cyclic units exhaust most of the centre of U(7/G). We will several times use the following fact (see [17], Lemma 1, or [22], Lemma II.2.9): if A1 and A~. are X-orders in a finite dimensional Q-algebra, then A1 n A2 is ~aso a X-order and (U(A~) : u ( a l n A2)) < oo. Consequently, if H is a subgroup of finite index in N(A2), then H NN(A~) is of finite index in / / (A1) N//(A2), and thus also in/ / (A1) .

L e m m a 2.3. Assume QG satisfies the following condition: if a simple component of QG is a division ring, then its Schur index is less or equal than two. And if the Schur index is two, then the center must be a real field, i.e. this component is a totally definite quaternion algebra.

Let J be a subgroup of H(WG) which contains a subgroup l-Iti=l Ji with Ji a subgroup of finite index in SLi for ni > 1; and let B be the subgroup generated by the Bass-Milnor cyclic units. I f J is spanned by elements of the form 1 + c~, c~ 2 = O, then the group < B , J > contains a subgroup of finite index in the centre of U(71G).

Pro@ Since (N(7/G): II(7/GNI- [ Mn,(O,)) < co there exists a positive integer m such that z m E lg(TYGfql"I Mn,(Oi)), for every central unit in 77G. The proof now continues as in the proof Lemma 3.2 in [18]. [3

Note that if for all i, (GLi : SLi) < (x) then the Bass-Milnor cyclic units axe not needed in the lemma.

L e m m a 2.4. Under the assumptions of Lemma 2.3:

< B , J >

is of finite index in U(7IG).

Proof. Since (U(A) : u ( w a ) ) < oo, and thus (Z(U(A)) : Z(U(Wa))) < oo, Lemma 2.3 yields that < B, J > contains a subgroup K I = 1-I K~ where each K~ is of finite index in Z(N(Ai)). It follows that < B, J > contains a subgroup K = I~ Ki where each Ki = K~ N Z(GLi) is of finite index in Z(GLi).

Clearly KiJi is of finite index in GLI and thus < B, J > contains a subgroup of finite index in the unit group of the order I-I Mm (Oi). Hence it also contains a subgroup of finite index in U(A). Since < B, J >C_ U(71G) C U(A) the result follows. [3

Note, from the proof of Lemma 2.4 it follows that, under the assumptions of Lemma 2.3, the group generated by J and the central units is of finite index in u ( w a ) .


3. T h e m a i n t h e o r e m

In this section we will construct a finite set of generators for a subgroup of finite index of H(Y/G). We do this via a set of non-central idempotents as follows.

For every idempotent f G QG, let n! be a positive integer such that n l f E Y/G. Then

i + n ~ fg (1 - f )

and

1 + nff(1 - f ) g f

are units in 77G for every g E G, because

( n ~ f g ( 1 - f))2 _ 0 = (n~(1 - f ) g f ) ~ .

Note also that

[1 + n~fg(1 - y)]~ [1 + n~fh(1 - y)] ' = [1 + n~y(~g + lh)(1 - f ) ] ,

for every k,l E Y/, 9, h E G. So, the group generated by these units contains all elements of the form

l + n ~ f ~ ( 1 - f ) , and l + n ~ ( 1 - f ) ~ f ,

~ E Y/G. In the next lemma and proposition we associate with every non-central idem-

potent of a simple component of QG, which is a matr ix ring, a set of matrix units and a maximal order Oi.

L e m m a 3.1. Let ei be a primitive central idempotent of QG such that QGei is a ni • hi-matrix ring over a division ring, ni > 1 . 1 f f is a non-central idempotent in QGei, then there exist matrix units E~,~, 1 < u,v < ni, such that

f = El,1 + " " + El3, 0 < l < ni,

and QGel can be considered as a matrix ring over the division ring Di which is the centralizer of all E,~,~.

Proof. Since f Q G f is a simple Artinian ring with identity f , write f = E1 + �9 "" + El, a sum of orthogonal idempotents, with EuQGE~ a division ring, 1 < u _< I. Similarly, write ei - f = Ez+I + . . . + E r a , a sum of orthogonal idempotents with EuQGEu a division r i ng , /+1 < u < ra. By [8], Theorem 2 page 59, m = ni and there exist matr ix units E~,~, 1 _< u,v <_ k with Eu = E~,~,. Furthermore, since f is non-central, 0 < l < ni. []

P r o p o s i t i o n 3.2. With notations and assumptions as in Theorem 2.2. Let Oi be a maximal Z-order in Di. Further let J! be the group generated by the elements

l + n ~ / g ( 1 - f ) and l + n ~ ( 1 - f ) g f ,

Jespers and Lea] 307

g E G. Then J! contains a subgroup of finite index in SLi = SL(ni, Oi).

Proof. Let n be a positive integer such that nei 6 7]G. Since

{1 + n~fc~(1 - f ) , 1 + n~(1 - f ) a f I a E 7]G} C_ J!,

it follows tha t

{1 + nn~fc~(1 - f)ei. 1 + nn~(1 -- f)e~fei ] o~ 6 ZG} C J].

Let u < l , n i > v > l + l . Then

fOiEu,~(1 - f ) = OiEu,v.

Hence, as Oi is a finitely generated Z-module , there exists a positive integer nu,v such tha t

1 J- nu,vOiEu,v C_ J!.

And similarly, 1 + n . , .O iE . , . C_ J],

for some positive integer nv,u. So we have shown the existence of a positive integer x with

1 J- xOiEu,v C_ J! and 1 + zOiE~,,, C_ Jl ,

for ail 1 < u < l, n i > v > l + l . Now let l < u , v _ < l , u # v a n d a 6 O i . Then

1 + n~z2e~Eu,v =

(1 - xn~aE~, l+l) (1- zn)Et+l, .)(1 + zn}aEu,,+a)(1 + xn~Et+l,~) 6 J!

Similarly, for l + 1 < u, v < nl, u # v, it follows tha t

1 + n}x2OiEu,v C J!.

The result now follows from Theorem 2.2. I"I We can now state our main theorem.

T h e o r e m 3.3. Let G be a finite group such that the rational group algebra ~G does not have simple components of the following type:

1. a non-commutative division algebra other than a totally definite quaternion algebra;

2. a 2 • 2-matrix: ring over the rationals; 3. a 2 • 2-matrix ring over a quadratic imaginary extension of the rationals; 4. a 2 • 2-matrix ring over a noncommutative division algebra.

For every primitive central idempotent ei for which (@G)ei is not a division ring, let fi be a non-central idempotent in (QG)ei. Then the group generated by the Bass-Milnor cyclic units and the units of the form

308 Jespers and LeaJ

1 + n~,f/h(1 - / / ) a n d 1 q- n~,(1 - f i )h f i ,

h G G, is of finite index in U(77G).

Proof. This follows at once from Lemma 2.4, Lemma 3.1 and Proposition 3.2. U

4. A p p l i c a t i o n s

In this section we give several applications of the theorem by showing that for 1 ^ with E G . N o t e t h a t i f many groups fi can be written in the form ord-7-d~gei, g

this is the case then, because ~(1 - g) = 0 and c~(1 - g) = ord(g) - ~ for some o~ E 77 < g >, it follows that the group

contains the group

U a = < 1 + ~ h ( 1 - g) I h ~ G >

< 1 + n,,'~h(ord(g) - "~)ei I h E G >,

where 0 ~ n,, E 77 is such that n,~ei E 77G. Then

< 1 + n0 , (ord(g) )2f /h(1 - y/) I h C > c_ Ug.

Similarly one shows that

< 1 + ne,(ord(g))2(1 - f i )hf / I h E G > C < 1 + (1 - g)hff] h E G > .

So it follows that in the Theorem we may replace the elements 1 + n ~ , f i h ( 1 - fi) (respectively 1 + n~,(1 - f i )h f i ) by 1 + ~h(1 - g) (respectively 1 + (1 - g)h'~). In the applications, we will make use of this without specifically refeting to it.

Recall (of. [19]) that a group F is said to be fixed point free if it has an irreducible complex representation p such that for every non-identity element f E F, p(f) has all eigenvalues different from one. It is known ([14], Theorem 18.1.iv) that these groups are precisely the Frobenius complements. The latter are well-known and completely described (see for example [14], Theorem 18.2 and Theorem 18.6).

C o r o l l a r y 4.1. Let G be a -finite group such that the rational group algebra QG does not have simple components of the following type:

1. a 2 • 2.matrix ring over the rationals; 2. a 2 • 2-matrix ring over a quadratic imaginary extension of the rationals; 3. a 2 • 2-matrix ring over a noncommutative division algebra.

I f G has no non-abelian homomorphic image which is fixed point free, then the group generated by the Bass-Milnor cyclic units and the units of the form

1 + (1 - g)h and 1 + h(1 - g),


g, h E G, is of finite index in LI(2tG).

Proof. Note that because of the assumption on G it follows that Q G has no noncommuta t ive division ring as a simple component (this is easy to prove, see for example [19], page 333). So by the theorem it is sufficient to show tha t if ei is a central primitive idempotent with (QG)e~ = M~, (D~) not a division ring, i.e.

1 ^ e ni > 1, then there exists g E G such tha t o--~-~(9) 9 i is a non-central idempotent

in (QG)ei. Let e be a primitive central idempotent of (ff3G)ei, and let p be the complex

irreducible non-linear represention p : G -~ (CG)e. Since the group Ge is a non- commutat ive epimorphic image of there exists g E G such that ge # p(ge) suitably we can w r i t e

1 0 0 1

p(ge)= 0 0 0 0 �9 :

.0 0

~j,. �9 (~, roots of unity, all different

"1

1 ^ P(o-7 (g)ge) =

G, it follows from the assumption on G that e and p(ge) has eigenvalue I. Diagonalizing

�9 .. 0 0 �9 .. 0 0

�9 .. 1 0 �9 . . o ( j .

�9 .. 0 0

f rom I, 2

0 . . . 0 �9 " ' 0

1 �9 ' ' 0

�9 " " 0

0 1

0 0 0 0

0 0

�9 . 0 �9 ' 0

" ' 0

�9 " � 9 0

�9 " �9 ~ n t

< j < ni. Consequently

0 ' " �9 0 "

0 . . . 0

0 . . . 0 0 . . . 0

0 . . . 0

Hence --L-̂ eor~(a)g is a non-zero idempotent of (QG)e. This yields or--~-(~(~)gell ^ ~ 0.

Furthermore 1 ^ because otherwise o~- '~g ^ ~ g e i 7 k ei, 1 ^e = ~ g e e i =eei = e, a contradiction. [3

The Corollary improves Thorem 1 in [19] where it was shown that the normal closure in N(TYG) of the group generated by the Bass-Milnor cyclic units and all the units of the form 1 + (1 - g)h'~ is of finite index. This la t ter group has the disadvantage tha t one has not explicit generators but instead only elements which generate it together with their conjugates in U(71G).

If one also assumes that Q G has no non-commuta t ive division ring as simple component , then the converse of Corollary 4.1 is also true. Indeed, if the considered group is of finite index, then it follows from the remark after Lemma 2.4 that the group generated by the central units and the units of the form 1 + (1 - g)h~, 1 + ~h(1 - g), g, h G G, is also of finite index in N ( Z G ) . Because of the latter, and because, by assumption, QG has no non-commuta t ive simple

310 Jespers and Lea]

component which is a totally definite quaternion algebra, the converse is now shown exactly as in the second part of the proof of Theorem 1 in [19].

R e m a r k 4.2. Apart from conditions 1-3 in Corollary 4.1 it is sufficient to assume that if (QG)el is not commutative then Gei is not fixed point free, and if, moreover, (QG)ei is a division ring then it must be a totally definite quaternion algebra. In the latter case the group Gei is embedded in the real quaternion algebra ]I:I(]R). The finite groups of this type are (see for example [16], page 344, and [5], page 69):

1. the generalized quaternion group of order 4n, n > 2:

< a,b I an = b2, bah-1 = a-1 >;

2. the binary tetrahedral group < 2,3,3 > of order 24:

< a, b l a 3 = b 3 = (ab) 2 >;

3. the binary octahedral group < 2, 3, 4 > of order 48:

< a, b la z = b 4 = (ab) 2 >;

4. the binary icosahedral group < 2, 3, 5 > of order 120:

< a, b l a 3 = b 5 = (ab) 2 > .

For descriptions of finite groups embedded in general division rings we refer the reader to [1].

The conditions of Corollary 4.1 are clearly satisfied if G is a group of odd order. The above comments therefore yield the following application.

C o r o l l a r y 4.3. Let G be a group of odd order. I f G has no non-abelian homomorphic image which is fixed point free, then the group generated by the Bass-Milnor cyclic units and the units of the form

l + ( 1 - g ) h ~ and l + f f h ( 1 - g ) ,

g, h E G, is of finite index in LI(7/G). The converse holds if QG has no simple components that are non-commutative division rings.

C o r o l l a r y 4.4. Let G be a finite group such that the rational group algebra QG does not have simple components of the following type:

1. a totally definite quaternion algebra; 2. a 2 • 2-matrix ring over the rationals; 3. a 2 • 2-matrix ring over a quadratic imaginary extension of the rationals; 4. a 2 • 2-matrix ring over a noncommutative division algebra.

I f G is a nilpotent group, then the Bass-Milnor cyclic units and the units of the form


l+(1-g)h~ and l+~h(1-g), g, h E G, generate a subgroup of finite index.

Proof. Because of Corollary 4.1 it is sufficient to show that no non-abelian homomorphic image H of G is fixed point free.

Suppose the contrary, i.e. H is fixed point free and non-abelian. Since H is the direct product of its Sylow subgroups, it follows from Theorem 18.1.iv in [14] that H is of even order and H = C x Q2k, where C is a cychc group of odd order and Q2~ is the quaternion group of order 2 k, k > 3. However it is well-known (see for example [18], proof of Corollary 2) that the rational group algebra QQ2k, and thus also QG, has a totally definite quaternion algebra lI-I as a simple component. This is in contradiction with the assumptions. []

Since odd groups don't have any simple components of the type mentioned in Corollary 4.4 we obtain

Coro l l a ry 4.5. I f G is an odd nilpotent group, then the Bass-Milnor cyclic units together with the units

l + ( 1 - g ) h ' ~ and l + f f h ( 1 - g ) ,

generate a subgroup of finite index in bl(7lG).

In [18] Ritter and Sehgal showed a bit stronger version of Corollaries 4.4 and 4.5. They show that the results still hold without the generators 1 + fib(1 - g). The generators of the form 1 + (1 - g)h'~ are called bicyclic units in [18].

Without the restrictions on the simple components, Corollary 4.4 falls in general. This has been shown for several examples (see for example [11] and [18]).

The next two corollaries could be stated as corollaries of Corollary 4.1. How- ever by having a closer look at the Wedderburn decomposition we can reduce the number of generators for a subgroup of finite index.

Coro l l a ry 4.6. Let G be a metacyclic group with presentation

< a, b l a r e = l , b ~ =1 , b-lab = a r > .

I f s is odd,(s,m) = 1 and

rJi = i (rood m) implies ri =. i(mod m),

for each 1 < i < m, 1 < j <_ s - 1, then the group generated by the Bass-Milnor cyclic units and the units of the form

l + ( 1 - b ) g b and l + b g ( 1 - b ) ,

g E G, is of finite index in 11(77G).

Proof. Since s is odd and (s, m) = 1 it follows that the group G~ < a > has order s. The assumptions and Theorem 47.11 in [6] imply that every complex


representation of G is either linear or equivalent to a monomial representation p given by

[i 0 o . . . 0 0 . . . 0

o . . . o

�9 : i

0 0 . . . ~,r' '

, p(b)= 0 -. . 0 1 .-- 0 �9 .

0 . . . 1

where ~^a primitive m-th root of unity and 1 < i < m. Cleaxly it follows that p ( ~ b ) is a non-trivial idempotent.

Now since s is odd, the rational group algebra does not contain non- commutat ive division rings or 2 • 2-matrices as simple components. From the

1 ^ above it is easy to see tha t if ei is a primitive central idempotent, then ~ b e i is a non-central idempotent . The result now follows from Theorem 3.3�9 []

In [20] a few special cases of Corollary 4.6 were considered. The case s = 2 has been dealt with in [9]. Note that the latter situation is more complicated because one can have simple components in the rational group algebra which are quaternion algebras or 2 x 2-by matrices over the rationals or a quadratic imaginary field extension of the rationals. It follows that one obtains generators, up to finite index, for all dihedral groups. The latter also hove been studied in [20] where it is shown tha t the Bass-Milnor cyclic units together with all bicyclic units generate a subgroup of finite index in the unit group.

As a last example we study symmetric groups S , of degree n. Again in [20] it was shown that the Bass-Milnor cyclic units together with the bicyclic units generate a subgroup of finite index in the unit group. If n = 3 then it was shown in [10] tha t the group generated by all (up to inverses) bicylic units is a rank 3 free normal complement of • in U(7/$3). So here all bicylics are needed. With the techniques used in [10] one can easily handle $4. In ease n >_ 5 we now show that one needs less elements than the number of bicyclic units�9

C o r o l l a r y 4.7. Let S,~ be the symmetric group on the set {1, 2, . . . , n} and let a be the transposition (1,2). I fn >_ 5 then the group generated by the following units is of finite index in bl(7lS,~) :

1--k-(1-I-a)g(1- a) and 1 + ( 1 - a)g(1-t-a),

gES..

Proof. It is well-known (cf. [7], Theorem 5.9) that QSn ~ @~ i=1M,,(Q) , and thus we may assume 7/Sn C_ @~I=IM,~,(Z).

It is also well-known (see [13]) that PSL(2, 7/) /F(2)) ~ $3, where F(2) is the congruence subgroup of level 2. Furthermore F(2) is a free group. Consequently all non-trivial elements of PSL(2, 7) have order either infinite or are of order 2 or 3. Hence GL(2, 7/) only contains elements of finite order 2, 3, 4, 6, 8, or 12. It follows tha t for n > 5, S , can not be embedded in M2(7/). Also n _> 5

Jespers and Leal 313

implies that Sn is the only non-commutative epimorphic image of Sn, it follows that QSn does not contain M2(Q) as a simple component.

Now Sn = < a, b >, where b = (1, 2 , . . . , n). Hence for every primitive central idempotent e E QG for which (r is non-commutative, it follows that ae is non-central. Hence �89 + a)e is a non-central idempotent. Theorem 3.3 and the comment following Lemma 2.3 yield the result. []

It is clear that our main theorem has many corollaries. We only have given a short list of applications. Other groups for which one can give generators are for example simple groups and Frobenius groups.

Finally it is worth mentioning that our proofs remain valid for group rings over coefficient rings R that are finitely generated commutative domains of characteristic zero. However in the statement of Theorem 3.3 one has to add the generators of the following type

1 + n),firh(1 - fi) and 1 + n~,(1 - fi)rhfi,

where r E R. In the case R = 77[~] (this coefficient ring has been considered in [20,21]), ~ a primitive root of unity, the statements actually become easier as some of the assumptions ate automatically satisfied. Indeed, if n > 4 then no simple component of the group algebra Q(~) can have a simple component which is 2 • 2-matrix ring over Q or a quadratic imaginary extension of Q.

In case R : 77[~], where ~ is a primitive IGI-th root of unity, one obtains a solution to our problem for all groups. This application was communicated to the authors by S.K. Sehgal.

We need some more notation and terminology. Let R : 77[~], where ~ is a root of unity, and let G be a finite group. Then the Bass-Milnor cyclic units of RG are all the units of the form:

(l +~a+(ea)2 +. . .+(ea) i -1)m + (~-~m---) ~'a,

d = ord(ea), where a runs through G, e through < ~ >, 1 < i < d, (i, d) = 1, and m = ~(IGI. ord(~)). The group generated by the Bass cyclic units is denoted by Bx R. Corresponding to the second type of units listed in Corollary 4.1 we denote by B~ the group of units generated by the elements:

l + ( 1 - e g ) S h ~ ' g and l + e ~ S h ( 1 - e g ) ,

g,h E G, ~,5 E< ~ > . It is shown in [20], Lemma 2.1, that the images of the Bass-Milnor cyclic

units of RG under the natural map U(RG) ~ KI(RG) generate a subgroup of finite index. Hence, it follows that Theorem 3.3 (and its proof) remains true for U(RG). Note that RG is a E-order in the semisimple group algebra Q(~)G.

C o r o l l a r y 4 . 8 . Let ~ be a primitive [Gl-th root of unity and let R = Z[~] . / f G is a finite group, then < B1 R, B~ > is of finite index in U(RG).

Proof. Note Q(~)G has no simple components that are of the four types listed in Theorem 3.3.


Suppose ei is a primitive central idempotent such that Q(~)Gel is non- commutative, that is it is a matrix ring of degree larger than 1. Clearly there exists a g E G so that gel is non-central. Thus the matrix gel has at least two different eigenvalues, say ~-1 and ~2. Then ~lgei has among its eigenvalues 1 and r162 r 1. Thus ~lgel has eigenvalues ord(g) and 0. It follows that fi =

1 ~ e ' ord(a)~lg z is a non-central idempotent. So the group generated by B~ and

Js, = < 1 + a + - / , ) I >, h C >

is of finite index in II(RG). As J/, is contained in B n the result follows. []

Acknowledgement. The authors thank Professor S.K. Sehgal for communicating Corol- lary 4.8 and for providing us a short proof.

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Eric Jespers Department of Mathematics and Statistics Memorial University of Newfoundland St. John's, Newfoundland AIC 5S7 - Canada

Guilherme Lea] Instituto de Matems Universidade Federal do Rio de Janeiro Ca]xa Postal 68530 21910 Rio de Janeiro Brasil

This article was processed by the author using the Springer-Verlag TEX mamath macro package 1990

(Received July 29, 1992; in revised form October i, 1992)

Generators of large subgroups of the unit group of integral group rings

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