- Email: [email protected]
The Multiplicative Jordan Decomposition in Group Rings
209, 533]542 Ž1998. JA987557
JOURNAL OF ALGEBRA ARTICLE NO.
The Multiplicative Jordan Decomposition in Group Rings Satya R. Arora Go¨ ernment College for Boys, Ludhiana, Punjab, India
A. W. Hales Center for Communications Research, 4320 Westerra Court, San Diego, California 92121 E-mail: [email protected]
and I. B. S. Passi Panjab Uni¨ ersity, Chandigarh 160014, India E-mail: [email protected] Communicated by Robert Steinberg Received August 8, 1997
1. INTRODUCTION Let G be a finite group and K a field of characteristic 0. Then every element a in the group algebra K w G x of G over K has a unique additive Jordan decomposition a s a s q a n with a s , a n g K w G x, a s semisimple, a n nilpotent, and a s a n s a n a s . Recall that an element a g K w G x is said to be semisimple if the minimal polynomial mŽ X . of a over K does not have repeated roots in the algebraic closure K of K, or, equivalently, if mŽ X . is coprime to its Žformal. derivative mX Ž X .. If the element a is a unit in K w G x, then a s is also a unit and a has a unique multiplicative Jordan decomposition a s a s a u , with a u g K w G x unipotent and a s a u s a u a s . If R is an integral domain with quotient field K and a g Rw G x, then a s need not necessarily lie in Rw G x. If this happens to be the case for all a g Rw G x Žresp. for all units in Rw G x., then we say that the additi¨ e Žresp. multiplicati¨ e . Jordan decomposition holds in Rw G x. It is easy to see that if 533 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
534
ARORA, HALES, AND PASSI
the AJD holds in Rw G x, then so does the MJD. If the unit group U Ž Rw G x. is viewed as a linear group, then the MJD is equivalent to the unit group being a splittable group w3, 13x. Finite groups G for which the AJD holds in the integral group ring Zw G x have been characterized in w7, 8x and we have given some illustrative examples in w2x for the behavior of the MJD in Zw G x. Our aim in this paper is to explore further the MJD property in integral group rings. We begin by examining GLnŽ R . for an arbitrary integral domain which is integrally closed and not a field. We note that GLnŽ R . is splittable for n s 2 and is not splittable for n G 4, while GL3 Ž R . is splittable if and only if R has the property that the difference of any two distinct units in R is again a unit in R Ži.e., the units together with 0 form a subfield.. This behavior naturally suggests, as in the case of the AJD, a severe restriction on the degrees of the Wedderburn components of the rational group algebra Qw G x for the MJD to hold in Zw G x. Indeed, we show that the degrees must all be less than or equal to 3. As a contribution toward characterizing the MJD for integral group rings, we examine several groups and show, in particular, that the MJD holds in the integral group ring Zw Q4 p x of a generalized quaternion group Q4 p , and that the only dihedral groups D 2 n of order 2 n for which the MJD holds in Zw D 2 n x are those with n s 2, 4, or an odd prime p.
2. PRELIMINARIES The decomposition of the elements of a finite group algebra K w G x, with K a field of characteristic 0, into semisimple and nilpotent components rests on the following basic result in linear algebra. PROPOSITION 2.1. Let V be a finite-dimensional ¨ ector space o¨ er a field K of characteristic 0 and let w g End K Ž V .. Then there exist unique elements ws and wn g End K Ž V . such that
w s w s q wn ,
w s wn s wn w s ,
where ws is semisimple and wn is nilpotent. Moreo¨ er, ws , wn can be expressed as polynomials in w o¨ er K. If w is in¨ ertible, then so is ws , and there is a unique factorization
w s w s wu , with wu g End K Ž V . unipotent and ws wu s wu ws . An analogous result holds for square matrices over K and for elements of GLnŽ K ..
535
JORDAN DECOMPOSITION
For the reader’s convenience, we recall a method of constructing the semisimple component ws for a given linear transformation w g End K Ž V .. r ŽSee, for instance, w4x, w5x, or w12x for more details.. Let mŽ X . s Ł is1 qie i Ž X . be the factorization in K w X x of the minimal polynomial mŽ X . of w over K r into irreducible factors. Let g Ž X . s Ł is1 qi Ž X .. Then Ž g Ž X ., g X Ž X .. s 1 and therefore there exist polynomials hŽ X . and k Ž X . g K w X x such that g X Ž X . h Ž X . q g Ž X . k Ž X . s 1. Choose the least m such that 2 m G e i for i s 1, 2, . . . , r. Let u : K w X x ª K w X x be the K-algebra homomorphism given by X ¬ X y g Ž X . hŽ X . . Then ws s sŽ w ., where sŽ X . s u m Ž X .. In particular, if w , c g End K Ž V . have the same minimal polynomial, then the polynomial sŽ X . is such that
ws s s Ž w .
and
cs s s Ž c . .
3. GENERAL LINEAR GROUPS Recall that a subgroup H of GLnŽ K . is said to be splittable if, for every A g H, the semisimple component A s of A also lies in H. Let R be an integral domain of characteristic 0 which is integrally closed in its quotient field K and R / K. Let K denote the algebraic closure of K. PROPOSITION 3.1. Ži. GL2 Ž R . is splittable. Žii. GL3 Ž R . is splittable if and only if, for e¨ ery a, b g U Ž R ., a / b, a y b g U Ž R .. Žiii. GLnŽ R . is not splittable for n G 4. Proof. Let A g GLnŽ R . and let f Ž X . be the characteristic polynomial of A. If f Ž X . has no repeated root, then, by definition, A is semisimple. If f Ž X . has a root a, say, of multiplicity n, then a g R and A s s aI. Thus Ži. follows. To prove Žii., suppose f Ž X . s Ž X y a. 2 Ž X y b ., a, b g K, a / b. Then again it is easy to see that a, b g U Ž R .. If R has the indicated property, then As s A y
Ž A y aI . Ž A y bI . ayb
.
536
ARORA, HALES, AND PASSI
Consideration of the semisimple part of
a 0 0
ž
1 a 0
0 1 b
/
, a, b g U Ž R ., a / b,
shows that condition Žii. is necessary, for the semisimple part of this matrix is
ž
a 0 0
0 a 0
y Ž a y b . y1 1 b
/
. To prove Žiii., let p / 0, p / 0, y4 be an element of
R which is not invertible, and write x s p q 2. Consider the matrix 0 y1 As 0 0
1 yx 0 0
1 0 0 y1
1 1 . 1 yx
0
The minimal polynomial of A is Ž X 2 q xX q 1. 2 and therefore As s A y
1 x y4 2
0
1
s y1
yx
0 0
0 0
Ž A2 q xA q I . Ž 2 A q xI . x
x2 y 2
x2 y 4 y2
x2 y 4 yx
x2 y 4 0 y1
x2 y 4 1 yx
0
,
which does not belong to GL4 Ž R . since p s x y 2 is not invertible in R. A 0 Finally, note that if A g GLnŽ R . and B s A0 10 , then Bs s s and 0 1 Ž . Ž . therefore if GLn R is not splittable, then GLr R is not splittable for all r G n. This completes the proof.
ž
/
ž
/
Remark 3.2. Let MnŽ R . denote the ring of n = n matrices over R. The preceding proof shows that the AJD holds in MnŽ R . if n s 2 and fails for n G 3; that is, if A g M2 Ž R ., then A s g M2 Ž R . and, for n G 3, there exist matrices over R whose semisimple parts do not have all their entries in R. Note that for the AJD to hold in M2 Ž R . one only requires that, whenever 2 a and a 2 are in R for a in K, then a is in R}while for n G 3, the AJD fails for every integral domain which is not a field. The preceding result immediately yields the following characterization for the AJD and the MJD to hold in Rw G x when the ring R of coefficients is a ring K w X x of polynomials over a field K for which K w G x is a direct sum of matrix rings over fields.
537
JORDAN DECOMPOSITION
If G is a finite group and K is a field of characteristic 0
THEOREM 3.3. such that
h
K wGx ,
[M is1
ni
Ž Ki . ,
where the K i ’s are field extensions of K, then Ži. K w X xw G x has the AJD if and only if n i F 2 for i s 1, 2, . . . , h; Žii. K w X xw G x has the MJD if and only if n i F 3 for i s 1, 2, . . . , h. 4. IRREDUCIBLE DEGREES THEOREM 4.1. Let G be a finite group and R an integral domain of characteristic 0 which is not a field. Then, for the multiplicati¨ e Jordan decomposition to hold in Rw G x, it is necessary that the degrees of the Wedderburn components of K w G x, where K is the field of fractions of R, must all be less than or equal to 3. h Proof. Let K w G x , [is1 Mn iŽ Di . be the Wedderburn decomposition of K w G x and assume that n1 G 4. Let p i : K w G x ª Mn iŽ Di . be the natural projections, i s 1, 2, . . . , h. Then we can choose nilpotent elements x, y g Rw G x and a, b g R such that ab / 0,
0 0 p 1Ž x . s 0 0
a 0 0 0
0 a 0 0
0 0 , a 0
0 0 p 1Ž y . s 0 0
0
0 0 0 0
0 0 0 b
0 0 , 0 0
0
p i Ž x . s 0 s p i Ž y . for all i G 2. Let r Ž/ 0. g R, with rab / y4. Consider the elements u1 s 1 q x, u 2 s 1 q ry, both of which are units and therefore so is their product u s u1 u 2 . Now p 1Ž u. s I q A, where a 0 0 0
0 0 As 0 0
0 a rab rb
0 0 . a 0
0
The semisimple component of A is 0 As s 0 0 0
0 0 0 0
0 a rab rb
a2rrb 0 . a 0
0
538
ARORA, HALES, AND PASSI
If u has a Jordan decomposition in Rw G x, then p 1Ž u s . g p 1Ž Rw G x. and so a2rrb lies in S s p 1Ž Rw G x.14 , the finitely generated R-submodule of D 1 consisting of the Ž1, 4.-entries of the elements in p 1Ž Rw G x.. Hence S contains all of K. Let x 1 , x 2 , . . . , x n be a set of generators for S over R. Choose a basis yl4 of D 1 over K containing 1, say with y 1 s 1. Then we have x i s Ý l c i, l yl , with c i, l g K. For each k g K ; S we have k s Ý i ri x i s Ý i ri Ý l c i, l yl s Ý lŽÝ i ri c i, l . yl . Only the l s 1 term can be nonzero and we have k s Ý i ri c i, 1. Thus K is finitely generated as an R-module and hence K s R, a contradiction, and the result is proved. Note that the latter part of the preceding argument allows us, in Theorem 3.1 of w7x, to only assume that R is an integral domain which is not a field. An implicit corollary of Theorem 3.1 of w7x is that, for groups G of odd order, Rw G x has the AJD if and only if G is abelian. The multiplicative analogue of this is as follows. COROLLARY 4.2. If neither 2 nor 3 di¨ ides the order of G, then Rw G x has the MJD if and only if G is abelian.
5. NONSPLITTABLE UNIT GROUPS Let p, q be odd primes and let Cn , D 2 n denote the cyclic group of order n and the dihedral group of order 2 n, respectively. We will determine which D 2 n have the MJD for their integral group rings. PROPOSITION 5.1. If G is any one of the following groups, then its integral group ring does not ha¨ e the multiplicati¨ e Jordan decomposition: Ži. G s ² x, t N x p q s t 2 s 1, txty1 s xy1 :. k Žii. G s ² x, y, t N x p s y p s t 2 s 1, xy s yx, txty1 s xy1 , tyty1 s yy1 :. k Žiii. G s ² x, y, t N x p s y q s t 2 s 1, xy s yx, yt s ty, txty1 s xy1 :. k Živ. G s ² x, t N x 2 p s t 2 s 1, txty1 s xy1 :. k Žv. G s ² x, t N x 8 s t 2 s 1, txty1 s xy1 :. kq 1 Žvi. G s ² x, t N x p s t 2 s 1, txty1 s x e : where the multiplicati¨ e order of e mod p is 4 Ž hence 4 must di¨ ide p y 1.. k
In each case k is an arbitrary positi¨ e integer. Proof. We will exhibit, for each group listed in Ži. ] Žv., a unit u in its integral group ring whose nilpotent component u n does not have integral coefficients. The verification in each case is straightforward, and is therefore omitted. Such verification is aided by applying the standard represen-
539
JORDAN DECOMPOSITION
tation of G in M2 ŽZw H x. where H is a normal abelian subgroup of G of index 2 Žsee, e.g., the proof of Proposition 3.1 of w8x.. p / q:
Ži. usx
1
un s
Ž 1 q Ž x q y xyq . Ž 1 q t 2 q ??? qt 2 y2 . Ž 1 q t . . , k
p
q
Ž 1 q x p q ??? qx pŽ qy1. . Ž x q y xyq . Ž 1 q t 2 q ??? qt 2
k
y2
. Ž1 q t . .
p s q: u s x p Ž 1 q Ž x y xy1 . Ž 1 q t 2 q ??? qt 2 un s
1 p
k
y2
. Ž1 q t . . ,
Ž 1 q x p q ??? qx pŽ py1. . Ž x y xy1 . Ž 1 q t 2 q ??? qt 2
k
y2
. Ž1 q t . .
Žii. u s y Ž 1 q 2 Ž x y xy1 . Ž 1 q t 2 q ??? qt 2 un s
2
k
y2
. Ž1 q t . . ,
Ž x y xy1 . Ž 1 q y q ??? qy py1 . Ž 1 q t 2 q ??? qt 2
p
k
y2
. Ž1 q t . .
Žiii. u s 1 q Ž x y xy1 . Ž 1 q t 2 q ??? qt 2
k
y2
. Ž1 q t .
= Ž 1 q Ž x y xy1 . Ž 1 q t 2 q ??? qt 2
k
y2
. Ž1 y t .
= Ž 1 q y q ??? qy qy 1 . , u n s Ž x y xy1 . Ž 1 q t 2 q ??? qt 2 = 1y
ž
1 q
k
y2
Ž 1 q y q ??? qy qy 1 .
. Ž1 q t .
/
.
Živ. u s 1 q Ž x 2 y xy2 . Ž 1 q t 2 q ??? qt 2
k
y2
. Ž1 q t .
= 1 y Ž x 2 y xy2 . Ž 1 y x p . Ž 1 q t 2 q ??? qt 2 un s
1 2
Ž x 2 y xy2 . Ž 1 q x p . Ž 1 q t 2 q ??? qt 2
k
y2
k
y2
. Ž1 y t . ,
. Ž1 q t . .
Žv. u s x 2 Ž 1 q Ž x q 2 x 3 y 2 x 5 y x 7 . Ž 1 q t 2 q ??? qt 2 un s
1 2
Ž x y x 3 q x 5 y x 7 . Ž 1 q t 2 q ??? qt 2
k
y2
k
y2
. Ž1 q t . . ,
. Ž1 q t . .
540
ARORA, HALES, AND PASSI
Žvi. When k s 1 the group ring Qw G x has an irreducible component of degree 4 Žsee w8, p. 24x.. The corresponding group ring for arbitrary k has the k s 1 case as homomorphic image so will also have such a component. Hence Theorem 4.1 applies. It has been shown in w2x that the MJD holds in the integral group ring of the dihedral group D 8 . The k s 1 case of the preceding proposition therefore immediately gives a characterization of the dihedral groups D 2 n for which Zw D 2 n x has the MJD. THEOREM 5.2. The multiplicati¨ e Jordan decomposition holds in Zw D 2 n x if and only if n s 2, 4, or an odd prime. Note that Proposition 5.1, for k s 1, shows that the MJD fails for each of the groups D 2 p q , Ž C p = C p . i C2 Žwhere the generator of C2 inverts each element of C p = C p ., C q = D 2 p , D4 p , D 16 , and C p i C4 Žwhere C4 acts faithfully on C p .. These include all but one of the groups whose integral group rings were shown not to have the AJD in Proposition 3.1 of w8x. The one missing is the generalized quaternion group dealt with below.
6. GENERALIZED QUATERNION GROUPS Again let p be an odd prime. THEOREM 6.1. The multiplicati¨ e Jordan decomposition holds in the integral group ring of Q4 p s ² x, t N x p s t 4 s 1, txty1 s xy1 :, the group of generalized quaternions of order 4 p. Proof. Consider the complex representations of Q4 p given by
r 1 : x ¬ 1,
t ¬ 1,
r 2 : x ¬ 1,
t ¬ y1,
r 3 : x ¬ 1,
t ¬ i,
r4 : x ¬ r5 : x ¬
ž ž
z
0
0
y1
z
z
0
0
y1
z
/ /
,
t¬
,
t¬
ž ž
0 y1 0 1
1 , 0
/
1 . 0
/
where i and z are primitive fourth and pth roots of unity, respectively. From w6x we have Q Q 4 p , Q [ Q [ Q Ž i . [ r4 Ž Q Q 4 p
. [ r5 Ž Q
Q4 p
..
541
JORDAN DECOMPOSITION
Let py1
us
Ý
py1
ai x i q
is0
Ý is0
py1
bi x i t q
Ý is0
py1
gi x i t 2 q
Ý
di x i t 3 ,
is0
where a i , bi , g i , d i g Z, be an arbitrary unit in Zw Q4 p x. With the help of the complex representations r i of Q4 p , one can easily show that if u is not semisimple then the nilpotent component of u is
un s
1 4
Ž py1 .r2
Ý Ž Ž a i q g i . y Ž a pyi q gpyi . . Ž x i y xyi .
is1
q
1 2p
py1
Ý Ž p Ž bi q d i . y Ž b q d . . x i t Ž 1 q t 2 . , is0
where b s b 0 q b 1 q ??? qb py1 and d s d 0 q d 1 q ??? qd py1. By considering the images of u mod ² t 2 : and mod ² x :, it can further be verified that u n g Zw Q4 p x. This involves an application of Lemma 5 of w9x to the matrix r4Ž u. to deduce that bi s d i for all i, and then working with r 5 Ž u n .. We can also show that the integral group ring of a certain group of order 16, with presentation similar to that of Q4 p , has the MJD. THEOREM 6.2. The multiplicati¨ e Jordan decomposition holds in the integral group ring of G s ² x, t N x 4 s t 4 s 1, txty1 s xy1 :. Proof. The group G has as homomorphic images both C4 = C2 Žby mapping x 2 to the identity. and the quaternion group Q8 Žby identifying x 2 and t 2 .. It is known w11, Theorem 2.7x that, for each of these latter groups, every unit in the integral group ring is of the form qg or yg for some group element g. Furthermore, we can multiply any unit u in Zw G x by one of the central units 1, x 2 , t 2 , x 2 t 2 without affecting its Jordan decomposability. This allows us to assume without loss of generality that Žup to sign. every unit in Zw G x is of the form u s ¨ q Ž a q bx q ct q dxt .Ž1 y x 2 .Ž1 q t 2 . with a, b, c, d integers and ¨ one of the elements 1, x, t, xt. We now apply the previously mentioned standard representation of G into M2 ŽZw H x., where H s ² x, t 2 :, to conclude that u will be semisimple if ¨ is either x, t, or xt. When ¨ is 1 we also conclude that u is semisimple unless a s 0 and b 2 s c 2 q d 2 . Finally, when these conditions hold, we find that u n s Ž bx q ct q dxt .Ž1 y x 2 .Ž1 q t 2 ., completing the proof since this has integral coefficients.
542
ARORA, HALES, AND PASSI
7. REMARKS Remark 1. By Theorem 4.1, given a finite group G, for the MJD to hold in Zw G x the degrees of the Wedderburn components of Qw G x must all be less than or equal to 3. The case when the degrees are all 1 is well understood Žsee w1, 10x.. If the degrees are all less than or equal to 2 and the Sylow 2-subgroups of G are abelian, then, proceeding as in w8x, we see that, in view of Proposition 5.1, G must be of the type C p i C2 m with p an odd prime. To complete the characterization in this case, one needs to consider in the first instance such groups and the finite 2-groups. Remark 2. Given a finite group G, let S Žresp. U . be the set of all semisimple Žresp. unipotent. units in Zw G x. Then the subgroup Y s ² S, U : generated by S and U is a normal subgroup and U ŽZw G x.rY is a finitely generated Žsince U ŽZw G x. is so. and periodic group. The periodicity is a consequence of the fact that every element u g U ŽZw G x. has a power u t with its Jordan components in U ŽZw G x.. It might be of interest to examine this quotient further.
REFERENCES 1. S. R. Arora, Nilpotent elements in group algebras, J. Indian Math. Soc. 58 Ž1992., 99]104. 2. S. R. Arora, A. W. Hales, and I. B. S. Passi, Jordan decomposition and hypercentral units in integral group rings, Comm. Algebra 21 Ž1993., 25]35. 3. S. R. Arora and I. B. S. Passi, Central height of the unit group of an integral group ring, Comm. Algebra 21 Ž1993., 3673]3683. 4. A. Borel, ‘‘Linear Algebraic Groups,’’ Springer-Verlag, BerlinrNew York, 1991. 5. C. Chevalley, ‘‘Theorie des Groupes de Lie II: Groupes Algebriques,’’ Hermann, Paris, ´ 1951. 6. C. W. Curtis and I. Reiner ‘‘Methods of Representation Theory,’’ Vol. I, Wiley-Interscience, New York, 1990. 7. A. W. Hales, I. S. Luthar, and I. B. S. Passi, Partial augmentations and Jordan decomposition in group rings, Comm. Algebra 18 Ž1990., 2327]2341. 8. A. W. Hales and I. B. S. Passi, Integral group rings with Jordan decomposition, Arch. Math. 57 Ž1991., 21]27. 9. E. Kleinert, Units in Zw Q p x, J. Number Theory 26 Ž1987., 227]236. 10. S. K. Sehgal, Nilpotent elements in group rings, Manuscripta Math. 15 Ž1975., 65]80. 11. S. K. Sehgal, ‘‘Units in Integral Group Rings,’’ Longman, Essex, 1993. 12. M. Shirvani and B. A. F. Wehrfritz, ‘‘Skew Linear Groups,’’ Cambridge Univ. Press, 1986. 13. B. A. F. Wehrfritz, ‘‘Infinite Linear Groups,’’ Springer-Verlag, BerlinrNew York, 1973.
JOURNAL OF ALGEBRA ARTICLE NO.
The Multiplicative Jordan Decomposition in Group Rings Satya R. Arora Go¨ ernment College for Boys, Ludhiana, Punjab, India
A. W. Hales Center for Communications Research, 4320 Westerra Court, San Diego, California 92121 E-mail: [email protected]
and I. B. S. Passi Panjab Uni¨ ersity, Chandigarh 160014, India E-mail: [email protected] Communicated by Robert Steinberg Received August 8, 1997
1. INTRODUCTION Let G be a finite group and K a field of characteristic 0. Then every element a in the group algebra K w G x of G over K has a unique additive Jordan decomposition a s a s q a n with a s , a n g K w G x, a s semisimple, a n nilpotent, and a s a n s a n a s . Recall that an element a g K w G x is said to be semisimple if the minimal polynomial mŽ X . of a over K does not have repeated roots in the algebraic closure K of K, or, equivalently, if mŽ X . is coprime to its Žformal. derivative mX Ž X .. If the element a is a unit in K w G x, then a s is also a unit and a has a unique multiplicative Jordan decomposition a s a s a u , with a u g K w G x unipotent and a s a u s a u a s . If R is an integral domain with quotient field K and a g Rw G x, then a s need not necessarily lie in Rw G x. If this happens to be the case for all a g Rw G x Žresp. for all units in Rw G x., then we say that the additi¨ e Žresp. multiplicati¨ e . Jordan decomposition holds in Rw G x. It is easy to see that if 533 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
534
ARORA, HALES, AND PASSI
the AJD holds in Rw G x, then so does the MJD. If the unit group U Ž Rw G x. is viewed as a linear group, then the MJD is equivalent to the unit group being a splittable group w3, 13x. Finite groups G for which the AJD holds in the integral group ring Zw G x have been characterized in w7, 8x and we have given some illustrative examples in w2x for the behavior of the MJD in Zw G x. Our aim in this paper is to explore further the MJD property in integral group rings. We begin by examining GLnŽ R . for an arbitrary integral domain which is integrally closed and not a field. We note that GLnŽ R . is splittable for n s 2 and is not splittable for n G 4, while GL3 Ž R . is splittable if and only if R has the property that the difference of any two distinct units in R is again a unit in R Ži.e., the units together with 0 form a subfield.. This behavior naturally suggests, as in the case of the AJD, a severe restriction on the degrees of the Wedderburn components of the rational group algebra Qw G x for the MJD to hold in Zw G x. Indeed, we show that the degrees must all be less than or equal to 3. As a contribution toward characterizing the MJD for integral group rings, we examine several groups and show, in particular, that the MJD holds in the integral group ring Zw Q4 p x of a generalized quaternion group Q4 p , and that the only dihedral groups D 2 n of order 2 n for which the MJD holds in Zw D 2 n x are those with n s 2, 4, or an odd prime p.
2. PRELIMINARIES The decomposition of the elements of a finite group algebra K w G x, with K a field of characteristic 0, into semisimple and nilpotent components rests on the following basic result in linear algebra. PROPOSITION 2.1. Let V be a finite-dimensional ¨ ector space o¨ er a field K of characteristic 0 and let w g End K Ž V .. Then there exist unique elements ws and wn g End K Ž V . such that
w s w s q wn ,
w s wn s wn w s ,
where ws is semisimple and wn is nilpotent. Moreo¨ er, ws , wn can be expressed as polynomials in w o¨ er K. If w is in¨ ertible, then so is ws , and there is a unique factorization
w s w s wu , with wu g End K Ž V . unipotent and ws wu s wu ws . An analogous result holds for square matrices over K and for elements of GLnŽ K ..
535
JORDAN DECOMPOSITION
For the reader’s convenience, we recall a method of constructing the semisimple component ws for a given linear transformation w g End K Ž V .. r ŽSee, for instance, w4x, w5x, or w12x for more details.. Let mŽ X . s Ł is1 qie i Ž X . be the factorization in K w X x of the minimal polynomial mŽ X . of w over K r into irreducible factors. Let g Ž X . s Ł is1 qi Ž X .. Then Ž g Ž X ., g X Ž X .. s 1 and therefore there exist polynomials hŽ X . and k Ž X . g K w X x such that g X Ž X . h Ž X . q g Ž X . k Ž X . s 1. Choose the least m such that 2 m G e i for i s 1, 2, . . . , r. Let u : K w X x ª K w X x be the K-algebra homomorphism given by X ¬ X y g Ž X . hŽ X . . Then ws s sŽ w ., where sŽ X . s u m Ž X .. In particular, if w , c g End K Ž V . have the same minimal polynomial, then the polynomial sŽ X . is such that
ws s s Ž w .
and
cs s s Ž c . .
3. GENERAL LINEAR GROUPS Recall that a subgroup H of GLnŽ K . is said to be splittable if, for every A g H, the semisimple component A s of A also lies in H. Let R be an integral domain of characteristic 0 which is integrally closed in its quotient field K and R / K. Let K denote the algebraic closure of K. PROPOSITION 3.1. Ži. GL2 Ž R . is splittable. Žii. GL3 Ž R . is splittable if and only if, for e¨ ery a, b g U Ž R ., a / b, a y b g U Ž R .. Žiii. GLnŽ R . is not splittable for n G 4. Proof. Let A g GLnŽ R . and let f Ž X . be the characteristic polynomial of A. If f Ž X . has no repeated root, then, by definition, A is semisimple. If f Ž X . has a root a, say, of multiplicity n, then a g R and A s s aI. Thus Ži. follows. To prove Žii., suppose f Ž X . s Ž X y a. 2 Ž X y b ., a, b g K, a / b. Then again it is easy to see that a, b g U Ž R .. If R has the indicated property, then As s A y
Ž A y aI . Ž A y bI . ayb
.
536
ARORA, HALES, AND PASSI
Consideration of the semisimple part of
a 0 0
ž
1 a 0
0 1 b
/
, a, b g U Ž R ., a / b,
shows that condition Žii. is necessary, for the semisimple part of this matrix is
ž
a 0 0
0 a 0
y Ž a y b . y1 1 b
/
. To prove Žiii., let p / 0, p / 0, y4 be an element of
R which is not invertible, and write x s p q 2. Consider the matrix 0 y1 As 0 0
1 yx 0 0
1 0 0 y1
1 1 . 1 yx
0
The minimal polynomial of A is Ž X 2 q xX q 1. 2 and therefore As s A y
1 x y4 2
0
1
s y1
yx
0 0
0 0
Ž A2 q xA q I . Ž 2 A q xI . x
x2 y 2
x2 y 4 y2
x2 y 4 yx
x2 y 4 0 y1
x2 y 4 1 yx
0
,
which does not belong to GL4 Ž R . since p s x y 2 is not invertible in R. A 0 Finally, note that if A g GLnŽ R . and B s A0 10 , then Bs s s and 0 1 Ž . Ž . therefore if GLn R is not splittable, then GLr R is not splittable for all r G n. This completes the proof.
ž
/
ž
/
Remark 3.2. Let MnŽ R . denote the ring of n = n matrices over R. The preceding proof shows that the AJD holds in MnŽ R . if n s 2 and fails for n G 3; that is, if A g M2 Ž R ., then A s g M2 Ž R . and, for n G 3, there exist matrices over R whose semisimple parts do not have all their entries in R. Note that for the AJD to hold in M2 Ž R . one only requires that, whenever 2 a and a 2 are in R for a in K, then a is in R}while for n G 3, the AJD fails for every integral domain which is not a field. The preceding result immediately yields the following characterization for the AJD and the MJD to hold in Rw G x when the ring R of coefficients is a ring K w X x of polynomials over a field K for which K w G x is a direct sum of matrix rings over fields.
537
JORDAN DECOMPOSITION
If G is a finite group and K is a field of characteristic 0
THEOREM 3.3. such that
h
K wGx ,
[M is1
ni
Ž Ki . ,
where the K i ’s are field extensions of K, then Ži. K w X xw G x has the AJD if and only if n i F 2 for i s 1, 2, . . . , h; Žii. K w X xw G x has the MJD if and only if n i F 3 for i s 1, 2, . . . , h. 4. IRREDUCIBLE DEGREES THEOREM 4.1. Let G be a finite group and R an integral domain of characteristic 0 which is not a field. Then, for the multiplicati¨ e Jordan decomposition to hold in Rw G x, it is necessary that the degrees of the Wedderburn components of K w G x, where K is the field of fractions of R, must all be less than or equal to 3. h Proof. Let K w G x , [is1 Mn iŽ Di . be the Wedderburn decomposition of K w G x and assume that n1 G 4. Let p i : K w G x ª Mn iŽ Di . be the natural projections, i s 1, 2, . . . , h. Then we can choose nilpotent elements x, y g Rw G x and a, b g R such that ab / 0,
0 0 p 1Ž x . s 0 0
a 0 0 0
0 a 0 0
0 0 , a 0
0 0 p 1Ž y . s 0 0
0
0 0 0 0
0 0 0 b
0 0 , 0 0
0
p i Ž x . s 0 s p i Ž y . for all i G 2. Let r Ž/ 0. g R, with rab / y4. Consider the elements u1 s 1 q x, u 2 s 1 q ry, both of which are units and therefore so is their product u s u1 u 2 . Now p 1Ž u. s I q A, where a 0 0 0
0 0 As 0 0
0 a rab rb
0 0 . a 0
0
The semisimple component of A is 0 As s 0 0 0
0 0 0 0
0 a rab rb
a2rrb 0 . a 0
0
538
ARORA, HALES, AND PASSI
If u has a Jordan decomposition in Rw G x, then p 1Ž u s . g p 1Ž Rw G x. and so a2rrb lies in S s p 1Ž Rw G x.14 , the finitely generated R-submodule of D 1 consisting of the Ž1, 4.-entries of the elements in p 1Ž Rw G x.. Hence S contains all of K. Let x 1 , x 2 , . . . , x n be a set of generators for S over R. Choose a basis yl4 of D 1 over K containing 1, say with y 1 s 1. Then we have x i s Ý l c i, l yl , with c i, l g K. For each k g K ; S we have k s Ý i ri x i s Ý i ri Ý l c i, l yl s Ý lŽÝ i ri c i, l . yl . Only the l s 1 term can be nonzero and we have k s Ý i ri c i, 1. Thus K is finitely generated as an R-module and hence K s R, a contradiction, and the result is proved. Note that the latter part of the preceding argument allows us, in Theorem 3.1 of w7x, to only assume that R is an integral domain which is not a field. An implicit corollary of Theorem 3.1 of w7x is that, for groups G of odd order, Rw G x has the AJD if and only if G is abelian. The multiplicative analogue of this is as follows. COROLLARY 4.2. If neither 2 nor 3 di¨ ides the order of G, then Rw G x has the MJD if and only if G is abelian.
5. NONSPLITTABLE UNIT GROUPS Let p, q be odd primes and let Cn , D 2 n denote the cyclic group of order n and the dihedral group of order 2 n, respectively. We will determine which D 2 n have the MJD for their integral group rings. PROPOSITION 5.1. If G is any one of the following groups, then its integral group ring does not ha¨ e the multiplicati¨ e Jordan decomposition: Ži. G s ² x, t N x p q s t 2 s 1, txty1 s xy1 :. k Žii. G s ² x, y, t N x p s y p s t 2 s 1, xy s yx, txty1 s xy1 , tyty1 s yy1 :. k Žiii. G s ² x, y, t N x p s y q s t 2 s 1, xy s yx, yt s ty, txty1 s xy1 :. k Živ. G s ² x, t N x 2 p s t 2 s 1, txty1 s xy1 :. k Žv. G s ² x, t N x 8 s t 2 s 1, txty1 s xy1 :. kq 1 Žvi. G s ² x, t N x p s t 2 s 1, txty1 s x e : where the multiplicati¨ e order of e mod p is 4 Ž hence 4 must di¨ ide p y 1.. k
In each case k is an arbitrary positi¨ e integer. Proof. We will exhibit, for each group listed in Ži. ] Žv., a unit u in its integral group ring whose nilpotent component u n does not have integral coefficients. The verification in each case is straightforward, and is therefore omitted. Such verification is aided by applying the standard represen-
539
JORDAN DECOMPOSITION
tation of G in M2 ŽZw H x. where H is a normal abelian subgroup of G of index 2 Žsee, e.g., the proof of Proposition 3.1 of w8x.. p / q:
Ži. usx
1
un s
Ž 1 q Ž x q y xyq . Ž 1 q t 2 q ??? qt 2 y2 . Ž 1 q t . . , k
p
q
Ž 1 q x p q ??? qx pŽ qy1. . Ž x q y xyq . Ž 1 q t 2 q ??? qt 2
k
y2
. Ž1 q t . .
p s q: u s x p Ž 1 q Ž x y xy1 . Ž 1 q t 2 q ??? qt 2 un s
1 p
k
y2
. Ž1 q t . . ,
Ž 1 q x p q ??? qx pŽ py1. . Ž x y xy1 . Ž 1 q t 2 q ??? qt 2
k
y2
. Ž1 q t . .
Žii. u s y Ž 1 q 2 Ž x y xy1 . Ž 1 q t 2 q ??? qt 2 un s
2
k
y2
. Ž1 q t . . ,
Ž x y xy1 . Ž 1 q y q ??? qy py1 . Ž 1 q t 2 q ??? qt 2
p
k
y2
. Ž1 q t . .
Žiii. u s 1 q Ž x y xy1 . Ž 1 q t 2 q ??? qt 2
k
y2
. Ž1 q t .
= Ž 1 q Ž x y xy1 . Ž 1 q t 2 q ??? qt 2
k
y2
. Ž1 y t .
= Ž 1 q y q ??? qy qy 1 . , u n s Ž x y xy1 . Ž 1 q t 2 q ??? qt 2 = 1y
ž
1 q
k
y2
Ž 1 q y q ??? qy qy 1 .
. Ž1 q t .
/
.
Živ. u s 1 q Ž x 2 y xy2 . Ž 1 q t 2 q ??? qt 2
k
y2
. Ž1 q t .
= 1 y Ž x 2 y xy2 . Ž 1 y x p . Ž 1 q t 2 q ??? qt 2 un s
1 2
Ž x 2 y xy2 . Ž 1 q x p . Ž 1 q t 2 q ??? qt 2
k
y2
k
y2
. Ž1 y t . ,
. Ž1 q t . .
Žv. u s x 2 Ž 1 q Ž x q 2 x 3 y 2 x 5 y x 7 . Ž 1 q t 2 q ??? qt 2 un s
1 2
Ž x y x 3 q x 5 y x 7 . Ž 1 q t 2 q ??? qt 2
k
y2
k
y2
. Ž1 q t . . ,
. Ž1 q t . .
540
ARORA, HALES, AND PASSI
Žvi. When k s 1 the group ring Qw G x has an irreducible component of degree 4 Žsee w8, p. 24x.. The corresponding group ring for arbitrary k has the k s 1 case as homomorphic image so will also have such a component. Hence Theorem 4.1 applies. It has been shown in w2x that the MJD holds in the integral group ring of the dihedral group D 8 . The k s 1 case of the preceding proposition therefore immediately gives a characterization of the dihedral groups D 2 n for which Zw D 2 n x has the MJD. THEOREM 5.2. The multiplicati¨ e Jordan decomposition holds in Zw D 2 n x if and only if n s 2, 4, or an odd prime. Note that Proposition 5.1, for k s 1, shows that the MJD fails for each of the groups D 2 p q , Ž C p = C p . i C2 Žwhere the generator of C2 inverts each element of C p = C p ., C q = D 2 p , D4 p , D 16 , and C p i C4 Žwhere C4 acts faithfully on C p .. These include all but one of the groups whose integral group rings were shown not to have the AJD in Proposition 3.1 of w8x. The one missing is the generalized quaternion group dealt with below.
6. GENERALIZED QUATERNION GROUPS Again let p be an odd prime. THEOREM 6.1. The multiplicati¨ e Jordan decomposition holds in the integral group ring of Q4 p s ² x, t N x p s t 4 s 1, txty1 s xy1 :, the group of generalized quaternions of order 4 p. Proof. Consider the complex representations of Q4 p given by
r 1 : x ¬ 1,
t ¬ 1,
r 2 : x ¬ 1,
t ¬ y1,
r 3 : x ¬ 1,
t ¬ i,
r4 : x ¬ r5 : x ¬
ž ž
z
0
0
y1
z
z
0
0
y1
z
/ /
,
t¬
,
t¬
ž ž
0 y1 0 1
1 , 0
/
1 . 0
/
where i and z are primitive fourth and pth roots of unity, respectively. From w6x we have Q Q 4 p , Q [ Q [ Q Ž i . [ r4 Ž Q Q 4 p
. [ r5 Ž Q
Q4 p
..
541
JORDAN DECOMPOSITION
Let py1
us
Ý
py1
ai x i q
is0
Ý is0
py1
bi x i t q
Ý is0
py1
gi x i t 2 q
Ý
di x i t 3 ,
is0
where a i , bi , g i , d i g Z, be an arbitrary unit in Zw Q4 p x. With the help of the complex representations r i of Q4 p , one can easily show that if u is not semisimple then the nilpotent component of u is
un s
1 4
Ž py1 .r2
Ý Ž Ž a i q g i . y Ž a pyi q gpyi . . Ž x i y xyi .
is1
q
1 2p
py1
Ý Ž p Ž bi q d i . y Ž b q d . . x i t Ž 1 q t 2 . , is0
where b s b 0 q b 1 q ??? qb py1 and d s d 0 q d 1 q ??? qd py1. By considering the images of u mod ² t 2 : and mod ² x :, it can further be verified that u n g Zw Q4 p x. This involves an application of Lemma 5 of w9x to the matrix r4Ž u. to deduce that bi s d i for all i, and then working with r 5 Ž u n .. We can also show that the integral group ring of a certain group of order 16, with presentation similar to that of Q4 p , has the MJD. THEOREM 6.2. The multiplicati¨ e Jordan decomposition holds in the integral group ring of G s ² x, t N x 4 s t 4 s 1, txty1 s xy1 :. Proof. The group G has as homomorphic images both C4 = C2 Žby mapping x 2 to the identity. and the quaternion group Q8 Žby identifying x 2 and t 2 .. It is known w11, Theorem 2.7x that, for each of these latter groups, every unit in the integral group ring is of the form qg or yg for some group element g. Furthermore, we can multiply any unit u in Zw G x by one of the central units 1, x 2 , t 2 , x 2 t 2 without affecting its Jordan decomposability. This allows us to assume without loss of generality that Žup to sign. every unit in Zw G x is of the form u s ¨ q Ž a q bx q ct q dxt .Ž1 y x 2 .Ž1 q t 2 . with a, b, c, d integers and ¨ one of the elements 1, x, t, xt. We now apply the previously mentioned standard representation of G into M2 ŽZw H x., where H s ² x, t 2 :, to conclude that u will be semisimple if ¨ is either x, t, or xt. When ¨ is 1 we also conclude that u is semisimple unless a s 0 and b 2 s c 2 q d 2 . Finally, when these conditions hold, we find that u n s Ž bx q ct q dxt .Ž1 y x 2 .Ž1 q t 2 ., completing the proof since this has integral coefficients.
542
ARORA, HALES, AND PASSI
7. REMARKS Remark 1. By Theorem 4.1, given a finite group G, for the MJD to hold in Zw G x the degrees of the Wedderburn components of Qw G x must all be less than or equal to 3. The case when the degrees are all 1 is well understood Žsee w1, 10x.. If the degrees are all less than or equal to 2 and the Sylow 2-subgroups of G are abelian, then, proceeding as in w8x, we see that, in view of Proposition 5.1, G must be of the type C p i C2 m with p an odd prime. To complete the characterization in this case, one needs to consider in the first instance such groups and the finite 2-groups. Remark 2. Given a finite group G, let S Žresp. U . be the set of all semisimple Žresp. unipotent. units in Zw G x. Then the subgroup Y s ² S, U : generated by S and U is a normal subgroup and U ŽZw G x.rY is a finitely generated Žsince U ŽZw G x. is so. and periodic group. The periodicity is a consequence of the fact that every element u g U ŽZw G x. has a power u t with its Jordan components in U ŽZw G x.. It might be of interest to examine this quotient further.
REFERENCES 1. S. R. Arora, Nilpotent elements in group algebras, J. Indian Math. Soc. 58 Ž1992., 99]104. 2. S. R. Arora, A. W. Hales, and I. B. S. Passi, Jordan decomposition and hypercentral units in integral group rings, Comm. Algebra 21 Ž1993., 25]35. 3. S. R. Arora and I. B. S. Passi, Central height of the unit group of an integral group ring, Comm. Algebra 21 Ž1993., 3673]3683. 4. A. Borel, ‘‘Linear Algebraic Groups,’’ Springer-Verlag, BerlinrNew York, 1991. 5. C. Chevalley, ‘‘Theorie des Groupes de Lie II: Groupes Algebriques,’’ Hermann, Paris, ´ 1951. 6. C. W. Curtis and I. Reiner ‘‘Methods of Representation Theory,’’ Vol. I, Wiley-Interscience, New York, 1990. 7. A. W. Hales, I. S. Luthar, and I. B. S. Passi, Partial augmentations and Jordan decomposition in group rings, Comm. Algebra 18 Ž1990., 2327]2341. 8. A. W. Hales and I. B. S. Passi, Integral group rings with Jordan decomposition, Arch. Math. 57 Ž1991., 21]27. 9. E. Kleinert, Units in Zw Q p x, J. Number Theory 26 Ž1987., 227]236. 10. S. K. Sehgal, Nilpotent elements in group rings, Manuscripta Math. 15 Ž1975., 65]80. 11. S. K. Sehgal, ‘‘Units in Integral Group Rings,’’ Longman, Essex, 1993. 12. M. Shirvani and B. A. F. Wehrfritz, ‘‘Skew Linear Groups,’’ Cambridge Univ. Press, 1986. 13. B. A. F. Wehrfritz, ‘‘Infinite Linear Groups,’’ Springer-Verlag, BerlinrNew York, 1973.