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^o<j
So, denoting by (p the natural map QG ^ Q[G/H]^ we see that (p{e) = 1 — (cP*~^) is a primitive idempotent of Q[G/H]^ by Lemma 1.2, and so ( Q [ G / ^ ] M e ) ^ Q ( ^ p . )  Also
(QG)e = {QG)H(I  ^ J ] c^^'')  {Q[G/H]Me), 0<j
so e is primitive and (QG)e = Q(^pt). Clearly e = G is the only idempotent with (QG)e = Q = Q(^i) and there is only one subgroup H oi G with G/H of order 1. For e = H — ( ^ , C P * " ^ ) , notice that there are just two coefficients of elements in the support of e and that the larger is l/\H\, Writing e = ^g^G^id)^ ^^^ letting q be the largest rational in the set {e(^)  g G supp(€)}, it is obvious that H = {g ^ supp(e)  e(^) = q}. Thus H is uniquely determined by the idempotent e. On the other hand, under the onetoone correspondence between subgroups of G containing H and subgroups of G/H^ the subgroup {H^c^' ) corresponds to the unique cyclic subgroup of order p in G/H. Thus the idempotent e is also uniquely determined by H. Hence the number of primitive idempotents e with (QG)e = Q{^d) equals the number of subgroups H with G/H cyclic of order d. 2. We require a certain property about the tensor product of cyclotomic fields: if m and n are relatively prime positive integers, then Q(^m) ®Q Q(^^) =. Q(^mn) To see this, note that the two algebras in question have the same dimension (since the Euler function is multiplicative on relatively prime integers). Hence, the mapping Q(^rn) ®Q Q(^n) —^ Q(^mn) defined
180
VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS
by ^m ® 1 »^ Cmn ^ ^ ^ 1 ® ^n »^ Cmn ^^ ^ ^^^S epimorphism and, since the dimensions are equal, an isomorphism. By Lemma 0.1, there is a n a t u r a l isomorphism QG = QGi ® • • • ® QG^. For each i, 1 < i < ^, let { e ^ i , . . . ,e^A;,} be the primitive idempotents of QGi,
By part (1), for each i, 1 < i < ^, and for each j ^ , 1 < ji < ki^
{QGi)eij^ =. Qi^iji) for some ^^j., a pith power root of unity. In particular, for i :/: k^ the orders of ^{j. and ^kj^ ^^e relatively prime.
Using basic
properties of the tensor product (see Proposition 1.5.5), it follows t h a t
QG = (8)Li (e^:=i(QG)6,,,)  en,...,^ {^UxiQG)e,„) = e,„...,,, (0=, Q(eo'.)) = ®4GI "''Q(^'') for some a^ > 1, where ^d is a primitive ofth root of unity. To determine the number a^/, note t h a t , by part (1), each field
QiU) = Q(6,J ® • • • ® Q(6,,) = QGiUL ^in) is uniquely determined by a subgroup H = H\X'''XH£
of G^ where each
Hi is a subgroup of Gi such t h a t Gi/ Hi is cyclic of order di and ^d is a primitive root of unity of order d = d\d2'  d^. So Gj H is cyclic of order d. Since every subgroup H of G can be written H = {H O G i ) X • • • X [H flG^), and because G / / f is cyclic if and only if each Gi/{H OGi) is cyclic, it follows t h a t ad is precisely the number of subgroups H of G with G/H
cyclic of
order d. It is clear t h a t the set {ei • • e^  Ci a primitive idempotent of QGi} accounts for all the primitive idempotents, so the result follows.
D
1.5 R e m a r k . If C is a finite abelian group, then the number ad of subgroups H of G with G/ H cyclic of order d equals the number of cyclic subgroups of G of order d.
In particular, ad(f>(d) = ad[Q{^d)'' Q] is the
number of elements of order d in G, We sketch a brief outline of the proof. Since G is finite abelian, G is isomorphic to G* = Homz(G', Q / Z ) . Let d be a divisor of G. Denote by Af the set of all cyclic subgroups of order d in G, and by M the set of all subgroups H of G* with G'^/H cyclic of order d. Consider the map L:Af
^
M
1. COMMUTATIVE GROUP ALGEBRAS
181
defined by H ^
H^ = {f e G* \ f{h)
= 0 for all h G H},
It is standard to verify t h a t ± is a bijection, giving the result. For details, we refer the reader to the section on character groups in Chapter 10 of [ R o t 95]. 1.6 C o r o l l a r y . Ife is a primitive
idempotent
in the rational group algebra
of a finite abelian group Gj then supp(e) is a subgroup of G. PROOF.
Write G = Gi x • • • x G^ as the direct product of finite pi
subgroups Gi of G, for distinct primes p i , . . . ,p^. Let e be a primitive idempotent of QG. By Theorem 1.4, e = ci • • • 6^ where each e^ is a primitive idempotent of QG^. It follows t h a t supp(e) = s u p p ( € i ) x  • •xsupp(e^); thus, to prove the result, we may assume t h a t G is a pgroup for some prime p. In this case, either e = G, or e = H\ — H2 where H\ and H2 are subgroups of G, H\ ^ H2 and \H2lHi\
— p. In the first case, supp(e) = G is a subgroup. In
the second case, for a given element p G ^ 1 , the coefficient oi g in Hi diflfers from the coefficient of ^ in H2» Hence, supp(e) is the subgroup H2'
•
1.7 E x a m p l e . The support of an idempotent which is not primitive need not be a subgroup. Indeed, let G = (a  a'^ = 1) x (6  6^ = 1) and let e be the idempotent e =  ( 3 — a + 2ab + 2ab^). Then e is not primitive since it is the sum of the two orthogonal idempotents ei = ^(1 + a ) ( l + b + b^) and 62 = ^(1  a)(2 — b — 6^). Also, supp(e) = {l,a,a6,ai^^} is not a subgroup. (Notice, however, t h a t ei and 62 are primitive and the support of each of these elements is G.) As another application of Theorem 1.4, we obtain the following slightly refined version of [ S e h 7 8 , Proposition VI. 1.16]. 1.8 C o r o l l a r y . Let G = Gi X G2 x • • • X G^ be the direct product nite abelian pisubgroups primitive
idempotent
Gi of G, for distinct primes p i , . . . ,p^. If e is a
in QG, then there exist 2 ^ subgroups Hi^,,.
supp(€) = Hi, w < i, such that e is a linear combination ZTi,... , ^ 2 ^ '^iih coefficients Z'linear
ofG.
combination
of fi
of the
equal ^o ± 1 . Every idempotent
of idempotents
, H2^ in elements
of QG is a
of the form H, where H is a subgroup
182
VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS
P R O O F . Since every idempotent of QG is the sum of primitive idempotents, the last statement is a consequence of the preceding one. To prove the first statement, we observe that the result is obvious if e = G and, for e ^ G^ prove the result by induction on £. By Theorem 1.4, the case ^ = 1 is clear, so assume i > \, By Theorem 1.4 again, e — eie^, where each e^ is a primitive idempotent of G^. Since G — n t = i ^ n there exists an i such that e^ ^ Gi, Without loss of generality, i = i. Let f = ci" '€£^1 e Q[Gi X • • • X Gii], Then e = fe^ and, by Theorem 1.4, e^ = Di — D2 where Di and D2 are subgroups of G^, supp(e^) = D2 and Di ^ i^2 By the induction hypothesis,
for subgroups Hi of Hi = supp(/) and each a^ = ± 1 . Hence 2^
e = ^^a^HiDi 1=1
2^
 ^aiHiD2 i=\
2^
= ^aiHi t=i
2^
x Di  ^^a,Hi
x D2.
1=1
Clearly {Hi x Di, HiX D2 \ i = I,. ^ ^ ,2^} is a set of 2^"^^ (different) groups each contained in supp(e) = H^ x D2. Hence the result follows. D
2. Rational loop algebras of finite RA loops In this section, we give a complete description of the rational loop algebra QZ/ of a finite RA loop L. We start by recalling that any finite RA loop L is the direct product L = L2 X ^ of an indecomposable RA 2loop L2 and an abelian group A^ by Corollary V.1.2. Since QL = QL2 ®Q Q ^ and because we now have full knowledge of QA (see Theorem 1.4), it is sufficient to determine the structure of QL in the case that L is indecomposable. For an indecomposable RA loop L, we shall show that QL is a direct sum of cyclotomic extensions of Q and Cay leyDickson algebras at most one of which is a division algebra. The others, being split, are isomorphic to various Zorn's vector matrix algebras. Our description will allow us to identify that primitive central idempotent which defines any CayleyDickson division algebra. Moreover, for each split algebra in the decomposition, we
2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS
183
shall give an explicit isomorphism onto the appropriate Zorn's vector matrix algebra. We begin by describing the primitive central idempotents. 2.1 Proposition. / / L is a finite RA 2loop and L' = {l^'S}, then the primitive central idempotents of QL are L and H — {H^c^"^'^), where H is a normal subloop of L, m > 1 and either 1. the idempotent is in Q L ^ , s e H, L = {H,c) and \L/H\ = 2 ^ , or 2. the idempotent is in A(L, V), s ^ H, Z{L) = {H^ c) and \Z{L)IH\ = Furthermore
In particular, if e £ A(L,Z/') is a primitive central idempotent such that (Z(QL))e = Q, then e = H^^, where H is a subgroup of index 2 in Z(L) and s ^ H. P R O O F . Note that L/V and Z{L) are abelian 2groups. By Corollary VI.4.8, we have QL = Q L ^ © A(L, L') and, by CoroUary VI.1.3,
Z{QL) =
QL'^®Q[Z{L)y~^,
Thus there are two types of primitive central idempotents in QL, those belonging to Q L ^ ^ Q[L/L'] and those belonging to Q[Z{L)]^. Consider the case that e is an idempotent of the second type. Then supp(e) C Z{L) and es = —e. Hence e can be written in the form
e=
J2att^,
where T is a transversal for L' in Z{L). Clearly e / Z(L) since the coefficients of Z(L) are all positive. Thus, by Theorem 1.4, e = Hi — H2 where ^1 and H2 are subgroups of Z{L) with Hi ^ ^2? ^2 = {Hi^c^"^' ), {Hi,c) = Z{L) and \Z/H\ = 2 ^ > 1. If 5 G Lfj, then also s G H2, so sHi = Hi^ i = 1,2. It would foUow that 5e = e, a contradiction. Thus s ^ Hi and e has the required form.
184
VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS
On the other hand, suppose that e is a primitive central idempotent in Q/,i±£ ^ Q[LIL% Then es = e, so
ter where T is a transversal for V m L, Denoting the coset L'g by 'g and by (/?, the isomorphism QL^^ ^ QL, we have ip{e) = ^ati. If
1. If ? G ^ 2 \ ^ i , then 1 \H2\
_ \H2
while, if ? 6 i^i, then \Hi\
I//2I
\Hi\
\H2\
So e = Hi — H2', {H\^c) = L, L/H^ is cyclic of order 2^^ and H2 •ym — l I (/?i,c^ * ' ) , 2is required. In both situations. Theorem 1.4 says that {Z{QL))e = Q(^2^)For the final statement, we note that if {Z{QL))e = Q, then m = 1, Z{L)= {H,s) and e = H  {H^) = H  i  ^ ^ = H^,
D
Let K be a field of characteristic p > 0 and let L be an RA loop of finite order not divisible by p. For any primitive central idempotent e of the loop algebra A'L, Corollary VI.4.8 says that the simple ring {KL)e is either a field or a CayleyDickson algebra. We now describe these algebras in more detail. 2.2 Lemma. Let L = M(G, *,5fo) be a finite indecomposable RA loop with V — {1,'5}. Let K be a field of characteristic p > 0 and suppose p j^ \L\. If e G AK{L, V) is a primitive central idempotent of KL, then 1. {KG)e = {R^{xey^{yey)j a quaternion algebra, and 2. {KL)e = {R^{xey^(yey^goe), a CayleyDickson algebra
2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS
185
where G = {x^y^Z{L)) and R = K[Z{L)]e is a field. Moreover^ if (KG)e is not split (for example, if {KL)e is not split), then K does not contain a fourth root of unity. Furthermore, if K = Q, then R is a cyclotomic field and, if {QG)e is not split, then R = Q, Z{L) — {t\) X (^2) X (^3} X (it;) where t\ — s, and e — H^^, where H is a subgroup of index 2 in Z{L) not containing s. Remember that xZ{L) and yZ(L) generate G/Z{L)^ which is C2 X C2. Thus KG = S + Sx + Sy + Sxy, where S = K[Z(L)], Therefore {KG)e = R + Rxe + Rye + Rxye, where R = Se = {K[Z{L)])e. Noting that e e Z(AK{L,V)) = K[Z{L)]^, we have R = {K[Z{L)])e = {Z{KL))e, using Corollary VI.1.3; thus i? is a field. Both (xe)^ and {ye^ are in Z{L)e^ hence in i2, and (ye){xe) = yxe = sxye = —xye = —(xe)(ye) because e G KL^^. Since the elements e, xe, ye, xye are linearly independent over i2, it follows that (KG)e is the generalized quaternion algebra (/Z, (xe)^, (i/e)^). Now write KL = KG + KGu and remember that multiplication in KL is given by PROOF.
(a + bu){c + du) = (ac + god*b) + {da + bc^)u where a^b^c^d G KG. For r G {KG)e^ write r = ri + r2(xe) + ^3(2/^) + r4{xye) with the r, in R (and hence central). Then r* = ri + r2{sxe) + rs(sye) + r^^sxye) = ri — r2{xe) — rs{ye) — r^i^xye) so that the restriction of * to {KG)e is the standard involution on this quaternion algebra. It follows that {KL)e = {KG)e + {KGu)e is the Cay leyDickson algebra (i?,(xe)2,(ye)2,yoe). Suppose that {KG)e is not split (that is, {KG)e is a division algebra). Let a — (xe)^, /? = {y^)^ Proposition 1.3.4 says that the equation ax^ + /3y'^ = z^ has no nontrivial solutions. Since a and /? are roots of unity, so is /?/a and one of a,/3,/?/a is a square (because these elements all belong to a cyclic 2group). Assume R contains a fourth root of unity, say i, so i^ =  1 . If a = C^ let X = C"^ y = 0 and z = 1. If /? = C^ let x = 0, y = C~^ 2Lnd z = 1. If /3/a = C^, let x = i(^ y = 1 and z = 0. In all cases, we have a nontrivial solution to ax^ + /3y^ = 2:*^, a contradiction. Assume now that K = Q. Then R = {Q[Z{L)])e and thus, by Proposition 2.1^ R = 0(^2"*)? where ^2^ is a primitive root of unity of order 2 ^
186
VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS
for some m > 0. If (QG)e is not split, then m < 1 as shown above. Thus Re = Q and, by Proposition 2.1, e = H^^2 with H as described. By Corollary VI.1.3, we have Z{AiL,L'))
= Q[Z(L)]if^ = Q{hy^[{h)
X (i3) X {w)].
By Lemma 1.2, since s is the unique element of order 2 in (ti), we have Q{^i)^T^ — Q ( 0 ^^^^ ^ ^ primitive root of unity of order equal to o(^i). So every simple component of Z{A{L^L^))^ in particular Re = Q, contains Q(^) as a subfield. Thus mi = 1, o{ti) = 2 and ^ 1 = 5 . D Recall that there are two RA loops of order 16, Mie{Qs^ 2) = M(Z?4, *, 1) and Mie(Qs) = M^Q^, *,^i). The Wedderburn decompositions of their loop algebras over an arbitrary field (of characteristic not 2) are as foUows. 2.3 Corollary. Let K be a field of characteristic different from 2. 1. A^[Mi6(g8,2)]^8A^®3(A0 2. K[M^e{Qs)] = ^K®{K,\,\,\) Let L be either of the loops in question. In both cases L/L' = C2 X C2 X C2 and Z{L) = {1,«5}. By Proposition 2.1, with H = {1} and c = s^ e = ^ ^ is the only primitive central idempotent in /S.K{L^L'). By Lemma 2.2, and with reference to Table V.4, (KL)e = (A^, 1,1,1) if L = Mi6(Q8,2) and (KL)e = (A^  1 ,  1 ,  1 ) if L = M^eiQs)' Thus PROOF.
A^MieCQs, 2)] ^ K[C2 x C2 x C2] 0 (A, 1,1,1) and K[Mie{Q8)] = K[C2 X C2 X C2] ® (A^  1 ,  1 ,  1 ) . The result now follows because A^[C2 X C2 X C2] = 8A" and (A', 1,1,1) ^ 3(A^), by Corollary 1.4.17. D Note that Z{AK{L, V)) = K[Z{L)]^ = 2{AK(G, G")) and hence, if e is a primitive central idempotent in A A ' ( G , G^)^ then e is a primitive central idempotent in AK{L,V), Recall that H(A') = ( A ' ,  1 ,  1 ) and M2(A^) = (A', 1,1). With arguments similar to those used to prove Corollary 2.3, it is easy to show 2.4 Corollary. Let K be a field of characteristic different from 2. 1. A:g8 = 4A'® H(A^)
2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS
187
2. KD^ ^ AK ® M2{K) We now give a description of the rational loop algebra of a finite indecomposable RA loop. 2.5 Theorem. Let L be a finite indecomposable RA loop. Then
where 1 QL^^
^ Q[L/V] ^ ®2rfllLI^'^^^^'^^' ^^^^ ^^ ^^^ number of cyclic quotients of L/ L' of order d. 2. QL^y^ = A(L,L') is a sum of aL CayleyDickson algebras, with ai, the number of subgroups H in Z{L) such that Z(L)/H is cyclic and s^H. 3. Z(A(L,Z/')) ^ ®Q(^i/), where the direct sum runs over all the subgroups H described in 2 and ^H i^ ct primitive root of unity of order \Z{L)IH\. With reference to Table V.3, if L is in class C\ or £3 or £5, then all simple components 0/A(Z/,Z/') are split CayleyDickson algebras. If L is in C2 or £4 or Ce or C7, then all simple components of A(L^ V) but possibly one (in the case o(ti) = 2) are split Cay leyDickson algebras. Any component which is not split is determined by a primitive central idempotent of the form e = H^^, where H is as described below: Loop C2
r ^ 1
{1}
u {^1*2) (ii<2> X {UH) u 1 ^7 J{htj) X (^1^3) X {hw) The first statement follows from Corollary VI.4.8 and Theorem 1.4. Statements 2 and 3 follow from Corollary VI.4.8 and Proposition 2.1. To identify the simple components (QL)€ of A(L,Z/') which are not split, note that Lemma 2.2 says that e is of the form e = H^^^ where ^ is a subgroup of Z{L) of index 2 such that s ^ H and Z{L) = {h) X (^2) X (^3) X (w), with ti = s. Since \L/Z{L)\ = 8, for such H we have \L/H\ = 16. Since (QL)e = {QL)H^ ^ Q[L/H]^ is not spUt, we must PROOF.
188
VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS
have L/H = Mie{Qs)', by Corollary 2.3; in particular, the only element of order 2 in L/H is central. Consequently, L is not a loop of type £ i , £3 or £5 (see Table V.3) since, in these loops, the element u is noncentral and of order 2. Assuming o{ti) = 2 and that L is of type £2, £4, ^6 or £7, we now determine the subgroups H of 2{L) with s ^ H, Write Z{L) = (ti) x (^2) X (^3) X (tt;), where 0(^2)^0(^2)7o('^) > 1 For a G {^27^37^)7 define S{ti,a)=
{
tia
if o(a) > 1
1
ifo(a) = l.
Then the subgroups of index 2 in Z{L) not containing s are Hi = {t2) X (^3) X (w), H2 =
{S{h,t2))x{ts)x{w),
H3 =
{t2)x{6{ti,ts))x{w),
H4 =
{t2)x{t3)x{S{ti,w)),
Hs =
{S{tut2))x{6{tr,t3))x{w),
He =
{S{ti,t2))x{t3)x{S{tuw)),
Hj = {t2) X {6{tut3)) X {6{tuw)) and Hs = (^(^1,^2)) X {6ih,t3))
X {6{h,w)).
Because of the Classification Theorem (Theorem V.3.1) and because L/H has no noncentral element of order 2, it is straightforward to verify that H is ^ 8  Conversely, for any indecomposable finite RA loop L of type £2? ^^4, Ce or £7 and with 0(^1) = 2, one has L/Hg = MIQ{QS). By Corollary 2.3, ( Q L ) ^ ^ ^ is not split, so the result foUows. D Let // be a finite indecomposable RA loop and let e be a primitive central idempotent in A(Z/,//'). In Theorem 2.5, we showed that {QL)e is a CayleyDickson algebra and described its centre R explicitly. In the case that this algebra is split (and hence Zorn's vector matrix algebra 3 ( ^ ) ) , it is possible to exhibit a surjection from QL to 3 ( ^ )  We proceed to show how to do this.
2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS
By Corollary VI.4.7, Le is not associative. Since Z{Le) in t h e field (Q[Z(L)])e^
it follows t h a t Z{Le)
189
is contained
is cyclic and a 2group since
Le is a homomorphic image of L. Thus Le is indecomposable and so, by t h e Classification Theorem, Le is of type C\ or £2 It therefore suffices t o consider just those loops which are in one of these classes. If a loop L is of type C\ or £2? then any element of QX can be written in a standard form. Specifically, and with notation as in t h e Classification Theorem, we write L = D U Du with Z{D)
= (ti).
By Lemma VI. 1.1, any
a G A(X, L^) can be written uniquely as a = 7 + 7'u with 7 , 7 ' G A ( i ) , D^), Because of Theorem 2.5, e = ^ ^ is the only primitive central idempotent of A ( X , X ' ) . Let T be a transversal for { l , ^ } in Z(D).
Then j = je
and
Y — 7'e can be written uniquely in the form 76 = (70 + 71a: + 722/ + 73X2/)e, and 7'e = (7Q + 7 I X + 73^ + 73xy)e, with each 7^ and 7^ in Q ( / i ) and t h e supports of both 7^ and 7^ contained in T . With this understanding, we call <^^ = [(70 + 1\^ + I2V + Taa^y) + (TO + l'\^ + 72^ + the normalized choice oix^y
form
of a.
l'z^V)A^
Clearly this normalized form depends on t h e
and u and on the transversal T . When we refer to "normalized
form", we mean with respect t o such particular choices. By Lemma 1.2, the map Q(^i}e ^ Q(^2^i)? denoted 7 1^ 7^ and defined by {t\ey
= ^2"^i ? is an isomorphism. If rrii = 1, then Q{^i)e = Q. When
this is the case, we will often omit the superscript / , thereby identifying with
je
(je)^.
In the proof of Proposition 2.7 which follows, we shall require a lemma which is also of independent interest. For any algebra R^ with 1, over a field F , and possessing an involution a \^ a such t h a t a + a and aa are in Fl for all a, and for any scalar a, recall t h a t we have denoted by (iZ, a ) the algebra obtained by applying the CayleyDickson process to i2, as described in §1.3. 2.6 L e m m a . Let F be a field and let R = M2{F). map in R,
Then adj is the only involution
the identity
on scalar matrices
Let adj be the
adjoint
A ^^ A in R which restricts
and has the property
that A \ A ^ Fl
to and
A~A e Fl for all A £ R. PROOF.
Let A
H^
A be an involution on R with the assumed conditions.
We wish to prove t h a t A = adj(i4) for any A ^ R. If A is a scalar m a t r i x , this is so because the given involution and the adjoint each restrict t o t h e
VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS
190
identity on scalar matrices. Assume A is invertible but not central. Then A A central implies that A A = A A. It follows that f{A) — 0, where f{X) = X^  {A + 'A)X + A l . By assumption, f{X) G F[X], so f{X) is the characteristic polynomial of A and A A = {detA)I. Since A is invertible, we obtain that A = adj A, The proven cases show that R has a basis bi with bi = adj 6i, i = 1 , . . . ,4. Hence, for an arbitrary A = Y^t=i fi^i ^^ ^9 4
4
A = \ ^ 6t = 2_\ ^^J ^« — ^dj A, t=i
t=i
D 2.7 Proposition. Let L = M(Z),*,1) be an indecomposable RA loop of type C\ with (cyclic) centre generated by the element t\ of order 2^^, where D = {Z{L)^x^y) as in Table V.3. Lete = ^=^ be the primitive central idempotent 0/A(L,LO and let F = {Z{QL))e. If m^ > I, let i = ( 6  0 ^ " ' " ' . Then there exists an element w £ L such that L = DUDw and (we)^ = —e. For any a 6 QZ/, write ae in its normalized form o^e = [(70 + Jix + 72T/ + jsxy) + (TO + 1\^ + l2y + 732^y)H^Then the mapping ip: {QL)e ^ 3{F) given by ae yo^ +173^
( ^ 7 / + 7 2 ^ 7 o ^  f ^ 7 3 ^  ^ 7 i ^ + 72^]
[ (  t 7 i ^ f 7 2 ^  7 o ^ + ^73^ J^7(^ + 72^
70'
• 173'
if m\ > I, or
ae
70 4 72 (71 4 73,  7 o + 72, 7i  73)
(71  73, 7o + 72, 7i + 73) 70  72
if mi = 1, is an isomorphism. We have L = D U Du with u'^ = I and Z{L) = (ti). Since s is the only element of order 2 in Z{L)^ it follows that s = tf^^ . Let PROOF.
w = i
tf^^~ u
if mi > 1
xu
otherwise.
2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS
191
Then w'^ = 5, so (we)'^ = se = —e. Hence every element ae G (Q^)e can be written in its normalized form ^^ = [(TO + lix + 722/ + isxy) + (TO + Ti^ + 722/ + 73^y)H^By Theorem 2.5 and Lemma 2.2, {QL)e = A{L,L') = A{D,D') + /\{D,D')w is a split CayleyDickson algebra. Also {QD)e ^ M 2 ( Q ( 6  i ) ) since {Z{QD))e = {Q[Z(D)])e = (Q[Z(L)])e = F^ Q ( 6  0  For mi > 1, let en =
1 — Qxy e,
/lexy\ ^12 =
/ l + 0x2/\ 2/0a: ;:; e = ——e
y
T:
1 + Qxy ^ e,
622 =
and ^21 = I where 0 = tf^^ let
1 + Qxy\ /l0xy\ y + Qx 7^ y T: e= — e
e. (Note that 0^ =  e and thus {Qxy^ = e.) For mi = 1, 1 + 2/
1 — 2/
^11 = —^e
,
622 = —^e
=m<,
, .
^12 =
—77—
^21 =
—7^
.
^
y\
—77
,
xy + x e =
:^
e
and I y\
fl + y\ X
— —
X + xy =
  — € .
These are all elements oi{QD)e and it is easily verified that they are matrix units, in both cases; that is, e = en + 622 ^iid CijCki = ^jk^u for i^j^k^i £ {1,2}. In particular, these elements form a basis of {QD)e considered as a vector space over Z{{QD)e). Since {QD)€ = M2(Q(^2"^i)), these elements play the role of the classical matrix units. If Tni > 1, it is also readily verified that xe = 0ei2 — 0^21,
ye = 612 + 621
and
xye = Qeu — Qe 22
VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS
192
while, if m i = 1, xe = ei2 + e2i,
ye = eu  ^22
and
xye = eu
+ €21.
It follows t h a t the mapping (p: {QD)e > M2(Q(^2^i)) given by ae
70^ + hs^
iji^
+ 72^'
•^71 + 72*^
70^ 
ijs^
if m i > 1, and by ae
70 + 72
71  73
7i + 73
7o  72
if m i = 1, is an isomorphism, where ae = (7o + 7ia^ + 722/ + 73^y)€ ^
{QD)e
is written in normalized form. Since {we^
= —e, (p{wey
adj(
= —I.
Also, by Lemma 2.6, (fi{a*e)
=
Thus (p can be extended to an isomorphism
(QL)e ^ M 2 ( Q ( 6  i ) ) + M 2 ( Q ( 6  i )Mwe)
= (M2(Q(6i)),!).
From the last part of §1.3.5, we see t h a t the m a p tp: {QL)e ^ 3(Q(^2"^i))? given by ote = [(70 + 7 i ^ + 722/ + Isxy)
+ (7o + 7(2^ + 72^/ + 7 3 ^ y ) H ^ '"^
70^ + *73^
(t7i^ + 72^7o^ + t73^«7;^ + 72^)"
[ (  1 7 / + 7 2 ^  7 0 ^ + t73^»7i'^ + 7 2 ^
yo^ 
ij3^
if m i > 1, or by ae
70 + 72
(71  73, 7o + 72, 7J + 73)
(71 + 73,  7 o + 72, 7i  73)
70  72
D
if m i = 1, is an isomorphism.
A similar result will hold for a finite indecomposable RA loop L in the family £2? except t h a t we need not worry about the case m i = 1 because, in this case, the simple component A ( L , L') is not split, as shown in Theorem 2.5. 2.8 P r o p o s i t i o n . Let L = M ( / ) , * , ^ i ) be an indecomposable
RA loop of
type £2 '^ith (cyclic) centre generated by the element ti of order 2 ^ ^ , m i > 1, where D = {Z(L),x^y), central idempotent
as in Table V.3.
Let e
of A{L^ L') and let F — Z{{QL)e),
\s
be the
primitive
Then there exists an
2. RATIONAL LOOP ALGEBRAS OF FINITE RA LOOPS
193
element w G L such that L = DUDw with {we^ = —e. Writing ae G {QL)e in normalized form, ote = [(70 + Jix + 722/ + Jsxy) + (70 + Yi^ + 722/ +
Is^v)^]^^
the mapping if): (QL)e —> 3 ( ^ ) defined by ae »^
(  1 ^ 7 / + ^ 7 2 ^  7 0 ^ + * ^ 7 3 ^ i 7 i ^ + 72^)
70^  1^73^
is an isomorphism. Recall that L = D U Du with u^ = t\. Since s is the only element of order 2 in 2(L)^ we have s = tj"^^ , so, setting w = t^^xu, it follows that w'^ = s and (if;e)^ = se = —e. By Theorem 2.5, we see that {QL)e = A(L,i/') is a split CayleyDickson algebra and that {QD)e = M2(Q(6i)). Noting that mi > 1, let 0 = tf~\. Then 0^ =  e and {Qxy'^y = e. It is readily verified that PROOF.
1  exy'^ en
=
lQxy^\
e ,
€22 =
/ l + 0 x y  i '\
1 + 0xyi 2 ^' e=
y  0a:
e
and
e2i = (
^
j ^ (^
^
je =
)xy''^
^^
e
form a set of matrix units for {QD)e = A(/), D^). Furthermore, xe = 0ei2  0y^e2i,
ye = ei2 + y'^e2i and a:?/e = Qy'^eu  02/^^22,
and the mapping (p: {QD)e —» M2(Q(^2'"i)) given by
o^e = (70 + 71^ + 722/ + l3xy)e e {QD)e
194
VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS
is an isomorphism. Similarly, and as in the proof of Proposition 2.7, the composition of (^ with the isomorphism M 2 ( Q ( 6  i ) ) + M2(Q(6i ))v^(^e) ^ 3 ( Q ( 6  i ) ) (see §1.3.5) is an isomorphism i\)\ {QL)e ^ 3(Q(^2"*i)) sis required.
D
Chapter VIII
Units in Integral Loop Rings
The determination of the group of units in an integral group ring is a subject of continuing interest to many people. In any integral group ring ZG, the elements of diG, which are so obviously invertible, are called trivial units. Several of the early results about units in group rings give conditions under which certain types of units are trivial. For example, when G is abelian, it is known that all the units of finite order in ZG are trivial. In 1940, Graham Higman [Hig40] found necessary and sufficient conditions for all the units in the integral group ring of a torsion group to be trivial; later, S. D. Herman [Ber55] proved several theorems about torsion units, in particular, one for finite groups similar to Higman's. From this work, it follows that all the central units of finite order in the integral group ring of a finite group are trivial. (This was also shown independently by J. A. Cohn and D. Livingstone [CL65].) Forty years after Higman, H. Hartley and P. F. Pickel showed that if the unit group of the integral group ring of a finite nonabelian group is not trivial, then it contains a free subgroup of rank two [HP80]. In this chapter, we prove that the theorems just mentioned have natural extensions to alternative loop rings over the integers. Several of the proofs make use of two explicit constructions of units, the socalled Hass cyclic units and bicyclic units. In addition, in a loop ring ZL which is alternative but not associative, we prove that aU central units are trivial if and only if the centre of L and the quotient of L by its commutatorassociator subloop are abelian groups of exponent 2, 3, 4 or 6.
195
196
VIII. UNITS IN INTEGRAL LOOP RINGS
1. Trivial t o r s i o n u n i t s We begin with a proposition which generalizes a well known property about units of finite order in group rings due to S. D. Herman [ B e r 5 5 ] . Recall t h a t an element of finite order in a loop or a ring is called a torsion element or a torsion unit. 1.1 P r o p o s i t i o n . Let a = Y^^^i^otii be a torsion ternative
unit in the integral al
loop ring 2.L of a finite loop L. If ai ^ 0, then a = ai = ± 1 .
P R O O F . AS
in §VI.4, we denote by R the right regular representation
of the complex loop algebra CL. Thus, for x 6 iZ, R{x) is the right translation m a p CL ^ CL. This map is linear and, by Artin's Theorem (Corollary 1.1.10), / ? ( x ^ ) = (iZ(x))^ for any m > 1. Following Herman [ B e r 5 5 ] , we note t h a t li a = ^ otit G ZL satisfies a ^ = 1 for some m > 1, then the matrix for /Z(a) is similar over C to a diagonal matrix A — d i a g ( ^ i , . . . ,^n)? where n — \L\ and the ^i are m t h roots of unity. Thus the trace of i2(a) is, on the one hand, Y^=\ 6s while, on the other, it is ^ a ^ t r i 2 ( £ ) — noL\. Since 0 ^ n\oL\\ =  ^Y^\ 61 ^ S r = i 1^1 — ^ ^^^ because OL\ is an integer, it follows t h a t all the ^i are equal to  a i  and OL\ — ± 1 . Hence /Z(a) is right multiplication by 1 or —1 and a^ = 0 if £ / 1.
D
The obvious units in an alternative integral loop ring ZL, namely, the elements ± ^ , ^ € Z/, are known as trivial units.
Hy Proposition 1.1, it
follows quickly t h a t nontrivial torsion units have no central elements in their support. 1.2 C o r o l l a r y . Let L be a finite loop such that ZL is alternative. ^ aii is a torsion unit in ZL and a^ 7^ 0 for some i G Z{L), PROOF.
The element i~^a
If a =
then a =
±i.
is a torsion unit with 1 in its support. So
Proposition 1.1 yields t h a t i~^a = ± 1 and the result foUows.
D
For abelian groups, one can actually remove the finiteness condition in Proposition 1.1. For this, we need some terminology and a lemma. A group G is said to be ordered if the elements of G can be linearly ordered with respect to a relation < in a manner which is compatible with the group multiplication. Thus any two elements of G are comparable and, for any x^y^z
^ G^ x < y implies xz < yz and zx < zy.
1. TRIVIAL TORSION UNITS
1.3 L e m m a . Every torsionfree PROOF.
197
abelian group is an ordered
group.
Let A be a torsionfree abelian group. For a subset S of A, p u t
S~^ = {s~^ \ s £ S}, Let C be the set of all multiplicatively closed subsets of A which do not contain the identity element of A.
Note t h a t C ^ ^
because {a^  n > 0} G £ for any a G A, a 7^ 1. T h u s , by Zorn's Lemma, C contains a maximal element, say T . We claim t h a t A = T U T~^ U {1}. Suppose to the contrary t h a t there exists an element a G A, a 7^ 1, such t h a t neither a nor a~^ belongs to T, Clearly T is properly contained in the multiplicative subset 5 = T U {ta^
 ^ G r , n > 1} U {a^  n > 1}. T h e
maximality of T implies t h a t 1 G 5*. Since A is torsionfree, it follows t h a t 1 = td^ for some t ^ T and some n > 1. Hence a~^ G T . Replacing a by a~^ in the above argument, we obtain t h a t d^ Consequently, 1 = {a^)'^{a~'^)'^
^ T for some m > 1.
G T , a contradiction.
For a,6 G A, define a < 6 if a'^b
G T.
Since /I = T U T"^ U {1}
is a disjoint union, for any a.,b ^ A precisely one of the foDowing three possibilities occurs: a~^b G T, a = 6 or ba~^ G T. Hence either a < b^ a = b or 6 < a; thus < is a linear order on A. Finally, assume a < b and c ^ A. Then a~^b G T , so {ac)~^{cb) = a~^b G T. Therefore ac < be. 1.4 P r o p o s i t i o n ( P a s s m a n ) . Let A be an abelian group. sion units of the integral group ring ZA are PROOF.
Let /i be a torsion unit in ZA.
in the set ±A.
D Then the tor
trivial. We have to prove t h a t /i is
Replacing A by the group generated by the support of /i,
we may assume t h a t A is finitely generated. Therefore ^ = T X F is the direct product of a finite abelian group T and a free abelian group F. By Maschke's Theorem, Q T is semisimple artinian and thus
a finite direct sum of fields K^ — (Qr)e/j,, each e/^ a primitive idempotent in QT. Since ZA — (ZT)F^
the group ring of F over the ring ZT, we have
ZA C ( Q T ) F = ® ^ = i
K\F.
By Lemma 1.3, F is an ordered group, with linear order < , say. Then a unit v in the group ring KhF can be written in the form v — Yli=\
^ifi,
with each / , G F , 0 / //^ G KH and fi < " < fk Write z/"^ = X ) L i ^iSi
198
VIII. UNITS IN INTEGRAL LOOP RINGS
with 0 y^ Ui e Kh, 9i G F and gi <   < g^^ Then 1 = vv"^ = Y^J^iUJjifigj), Obviously, figi < figj for z 7^ 1 or j 7^ 1, and also figj < fkgi for i ^ k or j 7^ £. So the elements figi and fkge are uniquely represented in the set {figj \ I < i < k^l < j < £}. Since K^ has only trivial zero divisors, it follows that k = I = 1, So we have shown that any unit in KhF is trivial; that is, of the type af with a G Kh and / 6 F . In particular, any torsion unit of KhF belongs to K^* Since /i = ]C/^^/i ^^ ^ torsion unit in ZA, each fieh is a torsion unit in KhF. We obtain that /x G J^ A^/i C QT. Consequently fi eZAO QT = ZT, so /x G ±A by Proposition 1.1. D One can even extend Proposition 1.1 to arbitrary groups. 1.5 Proposition (Passman). Let a = ^g^o^gO ^^ ^ torsion unit in an integral group ring 2.G. If a\ ^ 0, then a =: ai = ± 1 . The proof is much harder than in the finite and the abelian cases. One first shows that if e is a nontrivial idempotent of the complex group ring CG then e(l) is a rational number with 0 < e(l) < 1, where e(l) is the coefficient of 1 in e. (For Qi4, A finite abelian, this follows from the results in Chapter VII.) Using this fact and working in the finite dimensional algebra C[a], with more elaborate technical arguments than in the proof of Proposition 1.1, one obtains the result. Since we will use this fact only once, we have not included the full proof. All details can be found in [Seh78] and [Pas77]. Suppose now that L — M{G^^^go) is an RA loop for some nonabelian group G with involution * and central element go. Let s denote the unique nonidentity commutatorassociator of L. In §111.4, we saw that the involution could be extended to ZL by defining (x + yuY = x* + 5t/n, for x, ?/ G ZG, and that r G ZL is central if and only if r* = r. 1.6 Proposition. Let L be an RA loop with commutatorassociator subloop V = {1,5}. Denote by a the image in the abelian group ring Z[L/L^] of an element a G ZL. If a is a central unit in ZL and a is trivial, then a is in the abelian group ring Z[Z{L)].
1. TRIVIAL TORSION UNITS
199
Since a = ±1 for some i ^ L and since the kernel of ZL ^ Z[L/L^] is A(L, V), we have a = ±^ + ^ for some element 6 G A{L, V), By Lemma VI.1.1, <5 = (1  s)li for some /? G ZL. Writing /? = /?i + /?2 where supp(/3i) C Z{L) and supp(/J2) H >E(L) = 0, we have a = ±^ + (1  s)(ii + (1 — 5)/32. Since a, /?i and 5 are central, clearly ±^ + (1 — s)fi2 is also central and hence invariant under the involution *. Note that /?2 = sfi2 because the elements in the support of (32 are not central. It follows that i is central for, otherwise, i* = si and PROOF.
±i + {ls)P2
=
[±^+{ls)f32r = ±se + (1  S)sf32 = ±se
 (1  S)f32
implying that ±£ ^ si = —2(1 — s)P2' Such an equation is not possible!, however, because the coefficients on the right are even and, on the left, odd. Thus i G Z{L)^ so (1 — 5)/?2 is also central and hence invariant under *. Now (l^)/?2 = [(l^)/?2r =  ( l  ^ ) / ? 2 implies (1  s)P2 = 0, so a = db^ + (1 — s)l3\. Since i and the support of /3i are in 2{L)^ the result follows. D 1.7 Corollary. Let L be an RA loop. 1. If a is a central torsion unit in ZL, then a is trivial. 2. If e is an idempotent in ZL, then e is trivial; that is^ e = 0 or e = I. prove (1), let a be a central torsion unit in ZL. Then a is a torsion unit in the abelian group ring Z[L/L^]. By Proposition 1.4, a is a trivial unit. Proposition 1.6 implies that a is a torsion unit in Z[Z{L)]. Again using Proposition 1.4, we obtain that a is indeed trivial. To prove (2), let e be an idempotent in ZL. Then e is an idempotent in Z[L/V]. Hence P R O O F . TO
( 2 e  T ) ^ = 4 e ^  4 e + T = T. So 2€ — 1 is a torsion unit in the commutative group ring Z[L/L^]. By part (1), 2€ — 1 is a trivial unit and with all coefficients even, except for the coefficient of 1. Thus 2e — 1 = ± 1 , so either 1 — 6 or e belongs to (1 — s)ZL.
200
VIII. UNITS IN INTEGRAL LOOP RINGS
However, (1 ~ s)ZL contains only the zero idempotent because
{^[{\s)ZLT=P\2^\\s)ZL
= {Q}.
Therefore, e = O o r e = l.
D
2. Bicyclic and Bass cyclic units Recall that for a finite subloop TV of a Moufang loop L, we write N = S n € i v ^ and note that N G ZL. (See §VI.l.) For an element g e L of order n, we write ^ rather than (g). Thus
? = l + ^ + ^' +   + ^""'. 2.1 Definition. A bicyclic unit in an alternative loop ring ZL is an element of the form Uh^g = 1 + (1  g)hg where g^h £ L and g has finite order. Bicyclic units were studied in several papers on generators of subgroups of finite index in the unit group of integral group rings of finite groups (see, for example, [RS91, Seh93, JL93a, JL95]). Since (1 — g)g = 0, the element Uh,g is of the type I + a with a^ = 0, a G ZL. Hence a bicyclic unit is indeed a unit, with (1 + a)~^ = 1 — a. Similarly, elements of the form 1 + gh{ I — g)y g^h £ L with g of finite order, are units. It is important to know when such units are trivial. 2.2 Lemma. Let L be a loop such that ZL is alternative. If gth £ L and g has finite order, then the bicyclic unit Uh^g = 1 + (1 — g)h^ is trivial if and only if h~^gh = g^ for some j . If L is an RA loop, if g^h £ L do not commute and if u^^g is trivial, then (g) is normal in L. Let n be the order of ^. Since ^ = 1 + ^ + ^ ^ +  • H^f^'^ g^^ = ^ for any j . Thus, if h~^gh = g^^ then gh = hg^^ so gh^ = h^ and u^^g = 1. Conversely, suppose u^^g is trivial. Since this element has augmentation + 1 , we have u^^g — i for some i £ L, Thus PROOF.
1 + /i(l + y + . . . + y "  i ) = i + gh{\ + ^ + .. • + ^ "  ^ ) .
2. BICYCLIC AND BASS CYCLIC UNITS
Since h is in the support of I + h(l + g +
201
h g'^^^)^ it follows t h a t either
h = i OT h = ghg^ for some i > 0. T h e latter implies h~^g~^h
= ^% as
desired. In the case h = i^ y/e obtain
l + hig + g^ + '" + g^') = gh{l + ^ + • • • + ^"^) from which it follows t h a t 1 = ghg^ for some i. So /i is a power of g and h~^gh = g^ again as desired. Finally assume t h a t L is an RA loop and t h a t g and h are noncommuting elements of L such t h a t u^^g is trivial. We have seen t h a t h~^gh = g^ for some J > 1. Since L has a unique c o m m u t a t o r , 5, we also have h~^gh = sg. Hence s = g^~^ G (^), so {g) is normal in L by Corollary IV. 1.11.
D
Recall t h a t we use the symbol to denote the Euler function. 2.3 D e f i n i t i o n . Let L be a loop such t h a t ZL is alternative. Then a Bass cyclic unit is an element of ZL of the form
n where g £ L has finite order n, 1 < i < n, and gcd(i, n) = 1. T h a t a Bass cyclic unit /i, written as above, is indeed a unit can easily be verified by showing directly t h a t M^ = ( 1 + 5 ' + • • • + P'<''))^(") +  ^ ^ ^ ^ ? , n where k is an integer such t h a t 1 < A: < n and n  (1 — ik). In order to give a more informative proof of this fact, we introduce more terminology and establish some elementary lemmas.
Recall t h a t a
unital subring of a ring A with unity is a subring with the same unity. 2.4 D e f i n i t i o n . A unital subring A of a Qalgebra A is called a Zorder^ or simply an order^ if it is a finitely generated Zsubmodule such t h a t QA = A. Clearly an order A in a Qalgebra yl is a finitely generated additive torsionfree Zmodule, so it is a free Zmodule with rank equal to dimg A.
202
VIII. UNITS IN INTEGRAL LOOP RINGS
2.5 Examples. 1. If L is a finite group or an RA loop, then ZL is an order in the rational loop algebra QL and Z{ZL) is an order in Z{QL), The latter follows from Theorem III.1.3. Furthermore, if ZL is not associative, then Z{ZL) C ZL^^
© Z{ZL)^
C Z{QL),
so ZL^^®Z{ZL)^ is also an order in Z{QL) (see CoroUary VI.1.3). 2. An algebraic number field is a finite field extension of Q, this being a field necessarily of the form Q{OL) for some a G C. An element of Q(a) is an algebraic integer if it is the root of a monic polynomial with coefficients in Z. It is known that the set of algebraic integers of an algebraic number field K forms a ring O, Furthermore, O is an order in K and Mn{0) is an order in Mn(K). If a is an algebraic integer, then the subring Z[a] of Q(o;) generated by a is an order in Q(a). (See [CR88, §17].) Let H he di subloop of a Moufang loop L, Since L is an inverse property loop, as observed in §11.2.5, there is a well defined notion of the index of H in L. Recall that H has finite index if L can be covered by a finite number of left (equivalently, right) cosets of H, When this occurs, the minimal number of covering cosets is called the index of H in L. We denote this number [L: H]. The following lemma summarizes some well known facts about orders in associative algebras, but we state them here for alternative algebras. Recall that we denote by H{R) the loop of units in an alternative ring R, 2.6 Lemma. Let Ai and A2 be orders in the alternative Qalgebra A. Then the following statements hold. 1. Ai n A2 is an order in A. 2. //A2 C Ai, the index [ZY(Ai): ZY(A2)] is finite. 3. //A3 is a unital subring of A and also a finitely generated Zmodule such that Ai C A3, then A3 is an order in A. 4. //A2 C Ai and /x G A2 is invertible in Ai, then fx~^ G A2. (1) and (3) are clear. (2) Since Ai is a finitely generated Zsubmodule of A and because QA2 = i4, there exists a positive integer i such that M i C A2. Furthermore, PROOF.
2. BICYCLIC AND BASS CYCLIC UNITS
203
considering Ai and ^Ai as additive groups, the index [Ai: iAi] is finite. We now show t h a t the (multiplicative) index [ZY(Ai): U{A2)] is bounded by t h e additive index [Ai:
iAi].
Suppose t h a t x,t/ G ZY(Ai) are such t h a t x + M i
= y + iA\,
1 denote the common unity of A and A2. Then x~^y — 1 G ^ i
Let C A2.
Thus x~^y G A2. Similarly y~^x G A2 and therefore x~^y G ZY(A2). T h u s y G xU{K2), so xU{K2) = t/ZY(A2). (4) Observe t h a t Ai = /xAi. Hence, considering all algebras as additive groups, we see t h a t [Ai: //A2] = [/^Ai: /iA2]. Since /x is invertible in A i , it follows t h a t [Ai:/iA2]<[Ai:A2]. Since /XA2 C A2, the opposite inequality obviously holds. Hence 11K2 = A2; thus fi^ G A2.
•
Let L be a finite RA loop, g an element of order n in L and ^n ^ primitive n t h root of unity. Let i > 1 be an integer with g c d ( i , n ) = 1. Obviously,
has an inverse in Z[^7i], namely P _ 1 ~ P _ 1 ~
^ ^^ ^
^ ^^
'
where ifc = 1 (mod n). Take ^' = 1 + y + • • • + ^^"^ G Z(^). From Theorem V n . 1 . 4 , we know t h a t
and
Set M == 0
1 Z[^^]. Then Z{g) C M and the projection of 1/ in every
component of M is a unit except when d == n, in which case the projection is i. Since g c d ( i , n ) = 1, z^(^) = 1 + ^n, for some i £ Z, So we modify u^ replacing it with
;, = ( l + 5 + ... + 5l)^W_£p,
204
VIII. UNITS IN INTEGRAL LOOP RINGS
where ^ = ^{l+g\
h^^~^). Then the projection of// in every component
of M is a unit, so /x is a unit in M . Since Z{g) and M are Zorders in Q(^) and because fi G Z(^) C M , it follows from Lemma 2.6 t h a t /i is a unit in Z{g).
Hence all Bass cyclic units are indeed units in ZL, but are they
trivial? 2.7 L e m m a . Let L be a torsion
loop such that ZL is alternative
g be an element
Let i be an integer with I < i < n and
of order n in L.
gcd(z,n) = 1. If i ^ ± 1 (mod n), then the Bass cyclic ^^ is
(1+^+^2^.
..^^^l)0(n)^l
and let
unit i4>(n) n
nontrivial. PROOF.
We prove the result by contradiction. Since the augmentation
of /x is 1, if // is trivial, then ji ^ L, Because the support of fi contains only powers of ^, it follows that /j, = g^ for some t. Let m = (n) and assume t h a t ( 1 + ^ + • • • + ^'"^ )'^ + ^ ^ ? = 9^ for some t. Multiplying by (1  ^ ) ^ , we get (1  g')"^ = (1  g)'^9^, so (1)
l  m ^ ^ + (  ) ^ 2 . + ... + ( _ i ) Y = g'
mg'^'
+ (^)^^+'^ + • • • + (  1 )  ^ ^ +  .
Since 1 < i < n, clearly n > 2, so m = (f>{n) is even. The assumption z ^ ± 1 (mod n) implies t h a t m > 2. Since the powers of g on the left side of (1) are distinct, we have g^ = I or g^'^'^ = 1. Assume first t h a t g^ = I. Since g^ ^ g^ we must have g^ — g^~^^
but
then ^^* = ^2m2 _^ ^ ^  2 (^sij^ce n )( m) and thus g'^^ = g^. This says t h a t n I (2m — 4) forcing n — 2m — 4 since 2 < m < n. Thus n is even and therefore m = 0(TI) < ^ , giving a contradiction. On the other hand, if ^^+^ = 1, we must have g'^ — g^^^ — g^~'^ (since g^ ^ g~^),
b u t t h e n g'^^ = g22m
^ p ^  m ^ g^ g2i _ g2
n I (2m — 4), a contradiction.
rpj^jg 2tg3^ij^ S2^ys
D
3. Trivial u n i t s It is a classical result due to G. Higman [Hig40] t h a t all the units are trivial in certain integral group rings, specifically, in the integral group ring of an
3. TRIVIAL UNITS
205
abelian group of exponent 1, 2, 3, 4 or 6 and in the integral group ring of a hamiltonian 2group. Recall t h a t a hamiltonian 2group is (isomorphic to) Qs X Ey where Qs is the quaternion group of order eight and E is a, (possibly trivial) elementary abelian 2group. We now present an elementary proof of a more general statement [ J P S 9 5 ] . 3.1 T h e o r e m . Let L be an abelian group of exponent
1, 2, 3^ 4 or 6, or a
hamiltonian
Then U{2.L) =
2group or a hamiltonian
Moufang 2loop.
±L,
P R O O F . We establish the result in stages. Step 1. U{ZL) = ±L implies l/(Z[L
x C2]) = ±{1 x C2).
To see this, assume C2 = {x) and suppose (a + (ix)(^ + 8x) = 1 for a,P,j,S
G ZL, This means aj+pS
= 1 and a6+/3j
 0, so ( a + /3)(7 + ^) =
1 and ( a — /3)(7 — (5) = 1. Thus a + /? = ±^1 and OL — (i — ±^2 for some 91^92 ^ ^ So 2 a = ±^1 ± ^2 It follows t h a t g\ — ^2 ^nd either a = 0 or /3 = 0, proving the statement. Step 2. ZY(Z[Mi6(g8)]) = ± M i 6 ( Q 8 ) and U{ZQ^)
=
±Qs.
It is sufficient to prove just the first of these two assertions since Qg C MieiQs)^
T h u s , let L = M^eiQs)
= M{Qs,^,t,)
with Qs =
(We employ the notation of §V.4.) Note t h a t Z{L)
{Z{L),x,y),
— {\^x'^] and t h a t the
nonidentity commutatorassociator of L is 5 = x^. Let z/ be a unit in ZL. Since L/{x'^)
is C2 x C2 x C2, ll{Z[L/{x'^)])
is
trivial, as already shown, so, multiplying by ±g for some g in L, we may assume t h a t 1/ = 1 + (1  x'^){ao + aix + a2y + ot^xy) + (1  x2)(/?o + (i\x + /322/ + for some a^, A* G Z, 0 < i < 3. Write a — {\ 
X^)(Q;O
(izxy)u
+ otix + 0^22/ + «3^2/)
and /? = (1  x2)(/3o + fiix + /32y + /^a^^y). Then z/ = 1 + a + /3n and z/* = 1 + a* + x^/3tt, so z/z/* = [(1 + a ) ( l + a*) + x2(x2/?)*/3] + [{x^fi){\ + a ) + ^ ( 1 + a)]t^.
206
VIII. UNITS IN INTEGRAL LOOP RINGS
Since x^ = 1 and ab + ba = (1 + x^)a6 for any a^b ^ {^^y^^y}^ it follows t h a t x'^/3 = —/3 and t h a t i/i/* = ( l + a ) ( l + a*) + /?/3* = 1 + 2(1  x^){ao + al + al + alial
+ fi^ + /3^ + (3j + / J  ) .
Since Z(x^) has only trivial units (Step 1), we conclude t h a t
But ao + ^Q > 0, so this forces all a^ and all /?^ to be 0 except possibly ao, which could be —1. So i/ = 1 or i/ = x^ and Step 2 is complete. Step 3. If L = C4 X C4 X • • • X C4, then ZY(ZL) = We proceed by induction on the rank of L.
±L,
If the rank is 1 (so t h a t
L is cyclic of order 4), then L is a subgroup of Qg and Step 2 tells us ZY(ZL) = ±L.
Assume then t h a t rank(Z/) = n > I and t h a t the result
is true for all such groups of rank less than or equal to n — 1. We have L  A X {a) X (b) where a^ = 6^ = 1 and rank(>l)  n2.
Let 1/ G U{2.L).
The induction hypothesis, together with the result of Step 1, tell us t h a t ZY(Z[L/(6^)]) is trivial, so (multiplying by a trivial unit) we can assume t h a t u— 1 + (1  62)(ao + a i a + a^a^ + aaa^ + M
+ ^^ab + /Jsa^i + ^z^i^b)
for some a^,/?^ G lA. Now ZY(Z[L/(a'^)]) is also trivial. Applying this to i/, we obtain a\ \ OL^  0, /?o + /32 = 0, /3i + /33 = 0 and QQ + a2 G { 0 ,  1 } . Similarly ZY(Z[Z//(a^6^)]) is trivial. Applying this to v^ we obtain OL\ —03 = 0, /Jo  /32 = 0, /3i  /Ja = 0 and QQ — a2 G { 0 ,  1 } . Combining the last two facts, we obtain OL\ — otz — ^^^ — ^2 — ^\ — ^'^ — ^2 — 0, and ao = 0 or — 1. This means ^' = 1 or ^/ = 6^, so ^' is certainly trivial. Step 4. If L = C3 X C3 X • • • X C3, then U(ZV) =
±L.
Let ^3 be a primitive cube root of unity and note t h a t ± ^ 3 , i = 0 , 1 , 2 , are the only units of Z[^3]. To see this, suppose a + 6^3 is a unit in Z[^3]. Since the m a p (p: Z[^3] ^ Z[^3] defined by v^(^3) = ^  is an automorphism, a + 6^3 is also a unit, hence so is (a + b^3){a + 6^1) = a^ + b'^ + ab^s + ab^l = a^ + b'^  ab.
3. TRIVIAL UNITS
207
Thus a^ + b^ab = ±1 and, in fact, +1 because o? + b'^ab = ( a  i 6 ) ^ +  6 ^ . Since then 3(a — b)^ + (a + 6)^^ = 4, it follows that a = 6 or a = 0 or 6 = 0. Thus a + b^s = ±^3 for some i, as claimed. Again, we proceed by induction on the rank of L and first assume that L = (a) is cyclic of order 3. Let 1/ G K(ZL). Since the augmentation of 1/ is ± 1 , multiplying by ± 1 , if necessary, we may assume that 1/ = 1 + {1 — a)(/3o + Pia) for some /?o,/3i ^ Z. The quotient ring ZL/{1 + a + a^), being isomorphic to Z[^s]^ has only ±^3 as units. In this quotient, 1/ becomes F = 1 + (1  ^s){0o + /3i6) = 1 + /3o + /?! + (2/Ji  /3o)6It is easily seen that all possibilities for 17 = ±^^ correspond to trivial units for I/. Now assume rank(L) = n > 1 and that the result is true for all such groups of rank less than or equal to n — 1. We have L = Ax (a) x (6) where a^ = 6^ = 1 and rank(yl) = n  2 . Let 1/ G U{ZL), The induction hypothesis tells us that ZY(Z[L/(6)]) is trivial so (multiplying, if necessary, by a trivial unit) we can assume z/  1 + (1  6)[(7o + 7i« + 120^) + (^0 + O^a + e2a?)b] for some 7i, ^t G ZA. Similarly, U(2[LI{ab)]) is trivial, so the image of v in Z[L/(a6)] must be trivial. This image is 1 + (1  a^)[7o + 7i« + 120? + (^0 + O^a + ^2^^)^^] = 1 + To  71 + ^1  ^2 + (71  72 + ^2  Oo)a + (72  70 + ^0  ^i)a^ It follows that precisely two of the above coefficients (in TA) must equal 0. This observation leads to three cases. Case 1: 71  72 + ^2  ^0 = 0 and 7 2  7 0 + ^ 0  ^ 1 = 0. In this case, the induction hypothesis tells us that the image of u in Z[L/(a^6)] is trivial. This image is 1 + (1  a)[7o + 7i« + 72^^ + (^0 + O^a + 92a})a] = 1 + 70  72 + ^2  ^1 + (71  70 + ^0  02)a + (72  7i + ^1  Oo)o?  1 + ^0  2^1 + ^2 + (^0 + ^1  2^2)a + (  2 ^ 0 + ^1 + 02)0?,
208
VIII. UNITS IN INTEGRAL LOOP RINGS
Either the second or third coefficient must equal 0 and so the first is congruent to 1 (mod 3). It follows that (9o + <9i  2(92 = 0 = 29^ + ^i + ^2, forcing ^Q = ^1 = ^2 stud 70 = 71 =72 Finally, we note that the image of p in Z[L/(a)] is also trivial and that this image is 1 + (1 — i)(37o + 3^o^) It follows that 7o = ^0 = 0; we conclude that 1/ = 1. Case 2: 1 + 70  7i + <9i  ^2 = 0 and 71  72 + <92  ^0 = 0. In this case, the image of u in Z[Ll{o?'b)] is ^0  2^1 + ^2 + (1 + ^0 + ^1  2^2)^ + (  2 ^ 0 + ^1 + ^ 2 ) a ^
As in Case 1, we conclude that ^0  2^i + ^2 = 0 = 29o + ^1 + ^2 It follows that ^0 = ^1 = ^2 stud 71 = 72 = 1 + To The image of u in Z[L/{a)] is now 1 + (1 — 6)(2 + 370 + 3^0^)5 so we conclude that 70 = — 1, ^o = 0 and u = b. Case 3: 1 + 70  7i + o  ^1 = 0. Here the image of u in Z[L/(a^6)] is 1 + flo  2^1 4 ^2 + (1 + So + ^1  2^2)a + (  1  200 + ^1 + 02)0^. Again, as in Case 1, 1 + ^o  2l9i + (92 = 0 = 1 + (9o + (9i  2^2, so 62 = Oi — I + 0Q and 72 = 7i = 1 + 7o. Finally, the image of ^ in Z[L/{a)] is 1 + (1  6)[2 + 370 + (2 + 3(9o)6], so we conclude that 70 ^^ ^o =  1 and In all cases, we have shown that u is trivial, so our proof is complete.
D
G. Higman found necessary and sufficient conditions for U{ZG) to be trivial in the case of a torsion group G [Hig40]. The classical proof ultimately comes down to determining when the ring of cyclotomic integers Z[^], ^ a primitive root of unity, has no units of infinite order. The answer to this comes from the Dirichlet Unit Theorem, which we will state in Chapter XII. We have chosen, however, to present an elementary proof due to E. Jespers, M. M. Parmenter and P. F. Smith [JPS95]. 3.2 Theorem (Higman). Let L be a torsion group or a torsion RA loop. Then U{ZL) = ±L if and only if L is an abelian group of exponent 1, 2, 3^ 4 or 6, or a hamiltonian 2group, or a hamiltonian Moufang 2loop, In Theorem 3.1, we established the result in one direction. On the other hand, because of Lemma 2.2, if L is not an abelian group and PROOF.
3. TRIVIAL UNITS
209
not hamiltonian, there must exist a^b E L such t h a t the bicyclic unit Ua^h is nontrivial. If L is an abelian group or hamiltonian, but not of the types stated, then L contains an element g of order n which is either 5 or greater t h a n 6. In these cases, (f)(n) > 2, so a nontrivial Bass cyclic unit exists by Lemma 2.7.
D
We next settle the question, for finite loops, of when merely all the torsion units are trivial.
T h e result is due t o S. D. Berman in the case
of group rings [Ber55] and to E. G. Goodaire and M. M. Parmenter for alternative loop rings [ G P 8 6 ] . T h e elementary proof given is due to M. M. Parmenter [Par95, Theorem 3.3]. Some of the ideas used were introduced by A. WiUiamson [Wil78]. 3.3 T h e o r e m . Let L be a finite group or an RA loop.
Then all
units in ZL are trivial if and only if L is an abelian group or a 2group or a hamiltonian PROOF.
Moufang
torsion
hamiltonian
2loop.
If L is an abelian group, CoroUary 1.2 says t h a t all torsion units
are trivial. If L is a hamiltonian 2group or a hamiltonian Moufang 2loop then, by Theorem 3.1, all the units (and so, indeed, all the torsion units) of ZL are trivial. For the converse, we first prove t h a t every subloop of L is normal. For this, it is sufficient to show t h a t every cyclic group contained in L is normal. Suppose the contrary and let (g) be a cyclic group which is not normal. By Lemma 2.2, L contains a nontrivial bicyclic unit fi = 1 + (1 — g)h^' figfi'
= (1 + (1  g)hg)g{l
Clearly,
 (1  g)hg) = g + {I  2g + g^)hg.
Because U{ZL) is diassociative, fJ^gfi'^ is a torsion unit, but it is not trivial; otherwise, since h ^ (g)^ it would follow t h a t h = ghg^ for some z, so /i would be trivial by Lemma 2.2. This contradiction proves t h a t every subloop of L is indeed normal in L. Next assume L is not an abelian group and not a 2group.
Then L
contains a subgroup of the form Qs X (a:), where x has odd prime order p and Qs = (a, 6 I a^ = 1, a^ = 6 ^ ba = a^b)
210
VIII. UNITS IN INTEGRAL LOOP RINGS
is the quaternion group of order eight. Consider t h e element y = ax of order 4p in L. Since (4p) = 2{p — 1), t h e element
1/ = (1 + y + • • • + 2/^
) ^^ ^ +
y 4p
is a Bass cyclic unit which is nontrivial by Lemma 2.7. We show t h a t h~^ub ^ u. Assume t h e contrary. Since b~^yb — y'^P'^^ and y is central, we obtain
(l + y + y'+ • • • +
y'^'f^^'^ = ( 1 + j,2p+l ^ ^2(2p+l) ^. . . . ^ y(2p2)(2p+l)^2(pl)_
Multiplying b o t h sides by (1  y)2(pi)(l  t/2p+i)2{pi), we obtain (1 _ ^2pl)2(pl)(i _ ^2p+l)2(pl) ^ (1 _ y)2(pl)(l _ j/l)2{pl).
Hence (2  J/2P1 _ j^2p+l)2(pl) = (2  2/  yl)2(Pl) and, because y^^ = y~^^^ also {2y'r> _ y _ y  i ) 2 ( p  i ) = (2 _ y  y  i ) 2 ( p  i ) . Note, however, t h a t t h e coefficient of y"^^"^ on the right hand side is 0 while t h e coefficient of y^^~^ on the left side is negative (and nonzero), a contradiction. Next we claim t h a t i/ = y'^ + (I — y'^P)a for some a G Z(2/}. To see this, let H — {y)/{y^^)
and let ^ denote the n a t u r a l image in H of an element
g e {y). Then
2p Since y = 2j9^, it follows t h a t
3. TRIVIAL UNITS
211
2p r 2(pl)
iWi^)'^''^'+ —^ \.2p =f E »=0
'
y
2(pl)
=f
+ —^ % ^
=f +
'
y
2p
j=i
l.[(2plf(P'^l] y +
1(2J31)2(P1)^
2p
= y
Therefore, vy~^ — 1 € A ( ( y ) , (y^^)), so t* is indeed of the form v = y^ + {l — y^^)a,
a € Z{y), as claimed.
We complete the proof by showing t h a t uhu~^ is a nontrivial unit. For this, write v = ^'ij/ + ^'2 and u~^ = uj\y+u>2^ ^i^'^i € Z(2/^) for i = 1,2. Since 2/^ is central, ^2^ = ^^"2 and, since yb = 6y^''+^ and u = j/'^ + (1 — y^'^)a, it follows t h a t v^y = 1 — y^^a\y
for some a i G Z(y^). Thus
(z/it/)6 = (1  y ^ ^ K y f t = (1  y2^)ai62/''+^ = 2/^^(1  y^^K^^/ =  ( 1  y'^^)aiby
=  6 ( 1  y^^)aiy
=
buiy.
Hence ubv
= b{uiyujiy
 U\yu2 + ^2^\y + ^2^2)
= b{uiujiy'^
 v\UJ2y + i^2^iy + ^2^2)
But (^it/ + ^2)(^iy + ^2) = 1 implies i/\Uiy'^\U2(J^2 = 1 ^.nd ^ia;2 + ^2^i = 0. Consequently i/bu~^ = 6 (  l + i/2(^2 + ^2^\y
+ ^2^12/ + ^2^2)
= 6 [  l + 2(1/20^2 + ^2^12/)]; therefore, 6 G supp(i/6i/~^). Since vbu~^ has augmentation 1, if ubu~^ is trivial unit, then ubu'^
= 6, contradicting what we showed earlier.
D
212
VIII. UNITS IN INTEGRAL LOOP RINGS
4. Trivial central u n i t s In this section, we give necessary and sufficient conditions for all the central units in an alternative (not associative) loop ring to be trivial. This question has been settled for group rings by J. Ritter and S. K. Sehgal [ R S 9 0 ] and for alternative loop rings t h a t are not associative by E. G. Goodaire and M. M. Parmenter [ G P 8 6 ] . First we introduce the notion of local finiteness. As with groups, we say t h a t a loop L is locally finite if any finitely generated subloop of L is finite. 4.1 L e m m a . Let T denote the set of torsion Then T is a locally finite normal then T is
subloop of L.
elements
of an RA loop L.
If L is finitely generated^
finite.
P R O O F . TO
prove that T is a subloop, it is sufficient to show t h a t T is
closed under inverses and products, by Corollary II.2.3. Clearly T is closed under inverses. Also, if a and b are elements of T which commute, then certainly ab € T. On the other hand, if a and 6 do not commute, then ab is again of finite order since
{
a^6^
if n = 0 or 1
(mod 4)
5a^6^
if n = 2 or 3
(mod 4)
where s is the unique nonidentity commutatorassociator of L. So T is a subloop. Because of Corollary IV. 1.11, to prove the normality of T , it is sufficient to show t h a t for ^ G T and £ G L, t'^ti
has the same order as t.
Since the loop generated by t and / is a group (Moufang's Theorem), this fact is obvious. Suppose t h a t K is a finitely generated subloop of T . If A' is commutative, then it is a group (see Corollary IV. 1.3) and, since it is generated by a finite number of torsion elements, it has finite order. If K is not commutative, then it contains V — { 1 , ^ } . Since s is both a unique nonidentity commutator and associator in L, KjV group of the abelian group LfV, so is K.
is a finitely generated torsion sub
Hence KIV
is finite. Since V is finite,
Finally, note t h a t s ^ T since 5^ = 1. If L is finitely generated,
then T / L ' is finitely generated as a subgroup of the abelian group L/L\ before, we see t h a t T is finite.
As D
5. FREE SUBGROUPS
4 . 2 T h e o r e m . Suppose L is a torsion
Moufang
native loop ring which is not associative.
213
loop and 2.L is an
Then the central units in ZL are
trivial if and only if all units in 7.[Z{L)] and Z[L/V] and only if both Z{L) PROOF.
and LjV
are trivial; that is, if
are abelian groups of exponents
If all units of Z[Z(L)]
alter
and Z[L/V]
2, 4 or 6.
are trivial, then Proposi
tion 1.6 implies at once t h a t the central units of ZL are trivial. Conversely, assume t h a t all central units of ZL are trivial. Clearly t h e units in Z[Z{L)]
are trivial. Because L is torsion, replacing L by a finitely
generated RA subloop containing the support of a central unit, we may assume t h a t L is finite, by Lemma 4.1. Therefore, as noted in §2.5, and ZL^
0 Z{ZL)^
are both Zorders in Z{QL) Z{ZL)
By Lemma 2.6, U{Z{ZL)) U{Z{ZL))
C ZL^^
0
LIL'
and
Z{ZL)^.
is of finite index in U{ZL^^®
is trivial and hence finite, so also is U{ZL^^
Therefore the unit loop of Z[L/L'] = ZL^^
Z{ZL)
Z{ZL)^).
Since
0 Z{ZL)^)
finite.
is finite as well; in particular,
is finite. Since the torsion units of the integral group ring of a finite
abelian group are trivial (CoroUary 1.2), all the units in Z[L/L'] are trivial.
D In concluding this section, we draw attention to the fact t h a t J. Ritter and S. K. Sehgal have proven that the central units of the integral group ring of a finite group G are trivial if and only if, for each g £ G and each j relatively prime to G, g^ is conjugate to g or g"^ [RS90]. This result holds for alternative loop rings in general and an elementary proof in the not associative case can be deduced from Theorem 4.2. The proof given by Ritter and Sehgal for the integral group ring of a finite group G ultimately relies on the Dirichlet Unit Theorem and on showing t h a t the character fields of the absolutely irreducible representations of G must be either rational or quadratic imaginary extensions of Q.
5. Free s u b g r o u p s In this section, we show t h a t for most finite loops L with ZL alternative, the unit loop U{ZL) contains a noncyclic free group.
VIII. UNITS IN INTEGRAL LOOP RINGS
214
We begin with the construction of free subgroups of SL(2, C), the special linear group of degree two over C. Although the result is stated with more generality t h a n Theorem 14.2.1 in [ K M 7 9 ] , the proof is almost identical. 5.1 T h e o r e m . Let ci and C2 be complex Define elements
numbers
with \ci\ = \c2\ > 2.
t\2 = ^12(^1) and ^21 = ^2i(<^2) ^^ SL(2, C) by
tuici) =
1
ci
0 1
and
^21(^2) =
1
0
C2
1
Then ti2 o,nd ^21 generate a free subgroup o / S L ( 2 , C ) ; that is, there are no nontrivial
relations
PROOF.
on these two
elements.
Let A — ^12(^1) ctnd B = ^2i(<^2) Let W be an alternating
product of nonzero powers of A and B. We have to show t h a t W ^ I^ the identity matrix. Of course, this fact is obvious if W is simply a nonzero power of A or B. So we assume henceforth t h a t W contains nonzero powers of both A and B. If W begins with a power of 5 , then conjugate W by this power of 5 , and consider instead the resulting product, which begins with a power of A. Thus we may assume t h a t
where C is A or B and each a^ ^ 0. For each i = 1 , . . . , r , let Ri denote the first row of the matrix A^^ B^^ • • • C^'. R2k = R2k\B''^^
If R2k\
— (^2A:i > ^2A;)? then
= {x2k{UX2k)
R2k^\ = /22A:^''"^"^' =
{x2k^l,X2k^2),
where X2k\\ = X2k\ + C2a2kX2k and X2A:f2 = X2k + cia2k\iX2k\i' The last two equations can be combined in the single equation Xi^2 = Xi +
mai^iXi^i,
1 < i < r — 1, where m G C and m = ci = c2 > 2. To prove the result, it is sufficient to show t h a t the number \xi\ increases as i increases from 1 to r + 1. Obviously 1 =  x i  <  a i   c i  = x2. The inductive step k t + 2  > mai4ix,^.i  x, > 2\xi^i\ yields the result.
 \xi\ = \xi^i\ + (x,+i 
\xi\). D
5. FREE SUBGROUPS
Let r ( 2 ) be the principal
215
congruence group of level two; t h a t is, the set
of matrices in SL(2, Z) of the type
with a^b^c^d
G Z.
1 + 4a
26
2c
l + 4d
Denote by / the identity m a t r i x of SL(2,Z).
PSL(2, Z) = SL(2, Z ) / { / , —/} is the projective
Then
special linear group of degree
two over the integers. It is easily verified t h a t r ( 2 ) is the kernel of the n a t u r a l epimorphism P S L ( 2 , Z ) ^ PSL(2,Z2). For more information on principal congruence groups, we refer the reader to [ N e w 7 2 ] . 5.2 C o r o l l a r y , The group T{2) is a free subgroup o/SL(2, Z) and by the matrices PROOF.
generated
^12(2) and ^21(2).
Let A = ^12(2) and B = ^21(2). T h a t A and B generate a
free group follows from Theorem 5.1. Clearly (A^B)
C r ( 2 ) . To prove the
reverse inclusion, let X = [^21 xH ] ^ r ( 2 ) . We have to show t h a t X can be decomposed as a product of powers of A and B. Consider first the case xi2 > a;ii. Noting t h a t xn ^ 0, we can write ^12 + k i l l = v{2xu) with v^r £Z
+ r
and 0 < r < 2a:ii. Note t h a t r / 0 since X12 is even and x n
is odd. It follows t h a t Xii
—2x111; + a:i2
Xii
X21
 2 x 2 i t ; + X22
2^21
r
\xu\
2x2it^ + X22
with \x[2\ < \x[i\ =  x i i  . Secondly, consider the case  x i i  > xi2. If X12 = 0, then x n = X22 = 1 and so X G ( 5 ) C ( ^ , B). On the other hand, if X12 ^ 0, write xu + k i 2  = 'w;(2xi2) + r with ii;,r G Z and 0 < r < 2xi2. Let
X" = XB"^ = Then X ' ' =
11 •*'12 L*'21 ^ 2 2 J
x i i • 2it;xi2 X21 — 2wx22
X12 2:22
with \x'li\ < [x'l^l = xi2.
r  xi2
X12
Xii — 2WX22 2:22
216
VIII. UNITS IN INTEGRAL LOOP RINGS
Applying the previous steps several times, if necessary, it follows that X is indeed a product of powers of A and B. D Let L be a finite RA loop which is not hamiltonian and let g and h be elements of L such that the cyclic group (g) is not normal in the group (^,/i); in particular, by Corollary IV. 1.11, L' = {l,'^} ^ (g). By Lemma 2.2, the bicyclic unit u^^g is not trivial. 5.3 Theorem. Let L be a finite RA loop which is not hamiltonian. Let g^h ^ L be such that {g) is not normal in the group (g^h). Then the group generated by the units Uh,g = 1 + (1  g)hg
and
u'^g = 1 + gh{l  g)
is a free group of rank two. Let s be the unique nonidentity commutatorassociator of L. Clearly s ^ (g). Let G be the group generated by g and h and recall that G/Z{G) ^ C2 X C2. Any group homomorphism (f:G^>^H determines a ring homomorphism (f: ZG ^ ZH and it is easily verified that
^i^h^g) = l + k{\
(f(g))ip{h)(f{g)
= (1 + (1  ^{g))
= {u^ik)M9))'
and, similarly,
¥ ' « , , ) = (1 + ^Mh){l
 fig))' = Kiki^^g))'.
So, to prove the result, we may replace G by any homomorphic image ^{G) in which {(f{g)) is not normal in {(p(g)^(f{h)); in particular, since s ^ (^r^), we may replace G by G/{g^) and assume that g has order 2. Since (g) is not normal in G, the idempotent ^ = ^ is not central in QG. Hence, there exists a primitive central idempotent e in QG^^ so that the idempotent  ^ e is not central in (QG)e, which is a quaternion algebra, by Lemma VIL2.2. Note that  ^ e is a zero divisor in {QG)e because ^^ = 1, so {QG)e is split and hence isomorphic to a 2 x 2 matrix ring over a field
5. FREE SUBGROUPS
217
F , by Corollary 1.4.17. By Theorem VII. 1.4, F — Q(
^21 = V ^ ^ ^ ,
^22  V ^ .
It is easily verified that these elements form a set of matrix units in (QG)e, so we can represent the elements of (QG')e as 2 x 2 matrices with respect to these matrix units. To do so, we write /? G (QG)e as /? = S K ^ 7<2 EaPEjj. Hence and
ge
he =
0 a 1 0
for some a G Q(0* I^ follows that a 0 0 a Since h^ is a central torsion unit and the only torsion units in Q ( 0 ^^e powers of ^ or the additive inverses of such elements, we see that a = ±^^ for some i > 0. Therefore Uh.ge
=
[1 0 0
1
+
"0 0 0 2
0 1
0
2 0 0 0
and, similarly. %g'
1 ±4f 0 1
By Theorem 5.1, the elements Uh,,ge and u^ e generate a free group; therefore Uh.,g and w^ generate a free group in H{Z{g,h)). D 5.4 Corollary. Let G be a finite group such that G/Z{G) S C2 X C2. If Ufi^g = 1 + (1 — g)hg is a nontrivial unit, then u^^g and u^ = 1 + 'gh{l — g) generate a free subgroup ofU{ZG). Since GIZ{G) = C2 x C2, the group G has the LC property and a unique nontrivial commutator (Proposition III.3.6). Hence G can be embedded in an RA loop (Theorem IV.3.1). The result now follows at once from Theorem 5.3. D PROOF.
VIII. UNITS IN INTEGRAL LOOP RINGS
218
5.5 Remark. Z. S. Marciniak and S. K. Sehgal have shown that if g and h are elements of an arbitrary group G with g of finite order and if the bicyclic unit Uh^g G ZG is nontrivial, then u^^g and 1 +^/i~^(l — g~^) generate a free group [MS]. To construct free groups in integral loop rings of hamiltonian Moufang loops, we need the following application of a more general result of J. Tits [Tit72] on the construction of free subgroups in linear groups. (See also the proof of Lemma 5.3 in [Seh93].) 5.6 Proposition. Let ^p denote a primitive pth root of unity. Then there exists a positive integer n such that the subgroup o/GL2(C) generated by
[ 1 [ip
n
^p' 1 _
and
" 1  ^pi 0
0 l + ^pi
is free of rank two. Recall that for a finite subloop TV of a Moufang loop L, we write N — m\ ^neN ^ ^ Q^* Also, for g e L of finite order n, we write ^ rather than (g). Thus g= 1 ( 1 + ^ + ^ 2 ^ . . . + ^   ! ) . 5.7 Theorem. Let L be a finite hamiltonian is not a power of 2. Thus there exist a^b^c quaternion group of order 8 and c is a central p. Let fii and ^2 denote the following elements
Moufang loop whose order £ L such that (a,6) is the element of odd prime order of QL:
//i = 1 + (1 — a)(l  Z)ac ;X2 = 1 + ( 1  S ) ( 1  ? ) 6 C . Let m be the order of the unit group of the finite ring Z[^4p]/2pZ[^4p], where ^4p is a primitive Apth root of unity. Then^ for some positive integral multiple n ofm, the group generated by ji^ and ^'2 is a noncydie free group contained inUiZL). PROOF.
Let G  {a,b,c) = Q^x Cp^ where Qs = {a,b\a^
^ b'^.a'^ = l,ba = aH)
5. FREE SUBGROUPS
219
and Cp = (c) is cyclic of order p. Then e = i ( l  a 2 ) [ l  l ( l + c +  . . + cPi)] = ( l  a 2 ) ( l  c ) is a central idempotent in the rational group algebra QG. Because of Corollary VII.2.4, QQs has only one noncommutative simple component, namely, H(Q). By Lemma VIL2.2, the corresponding primitive central idempotent is ^ , so ( Q ^ s ) ^ = H(Q). Because of Lemma VIL1.2,
where ^p is a primitive pth root of unity. Thus, (QG')e ^ QQs'^
®Q QCp[l  i ( l + c + • • • +
CP^)]
^ H(Q) ®Q QC^p) ^ H(Q(^p)). Consequently, (CG)e ^ C ®Q (QG)e ^ H(C ®Q QC^p)) = ©H(C), as shown in §1.5.12. By Theorem 1.3.4, H(C) is split and hence isomorphic to M2(C), by Corollary 1.4.17. Thus (CG)e ^ ®M2{C). Let / be a primitive central idempotent of (CG)€, Then {CG)f = M2(C). Let X = fa^ y = fb and z = fc. Noting that a^e = —e and hence a? f = —f^ii follows that x{iy) = {iy)x^,
x^ = /
and
{iyf
= /.
Hence, in (CG)/, the elements x and iy generate a group isomorphic to 2)4, the dihedral group of order eight. It follows that ^11 = U^+iy)
and
E22 = U^  iy)
are noncentral idempotents of (CG)/. Together with ^12 = 1(1 + iy)xl{l
 iy)
and
E21 =  ^ ( 1  iy)xl{l
+ iy),
it is easily verified that these four elements form a set of matrix units of (CG)/ = M2(C). The representations of a:, y and z as matrices with respect to these matrix units are 0 1 ' 1 0
' i ,
y =
0
0 1 i \
anc
z =
^p
0
0
ep
VIII. UNITS IN INTEGRAL LOOP RINGS
220
Hence the matrix representations of / / i / and /X2/ are 1
Ml/ =
^p
and
/i2/ =
1  ^pi
0
I(pi
0
By Proposition 5.6, there exists a positive integer k such that the group generated by (//i/)^ and (/X2/)^ is free of rank two. Since the group iif^if)^^^i/^2f)^^) is a homomorphic image of the group ( M I ' ^ , / X 2 ^ ) , it remains to show that /xf^ and /X2^ are units in ZG. By symmetry, it is sufficient to prove this for/xf'". Because of Theorem VII. 1.4, e = ( l  a 2 ) ( l  2 ) is also a primitive idempotent of Q(ac) with (Q(ac))e ^ Q(e4p) = Q(t^p), where ^4p is a 47?th root of unity. Under these isomorphisms, the element ace corresponds to i^p, thus (Z{ac))e = Z[i^p]. Clearly, 1
(i + i^p)(i»ep) = i + ep' = £^'2  1 ' Sp
and the latter has inverse ep 1 1
^'^ tp'  1  = l + e p ^ + . . . + e^^(^^)GZ[iep],
where g' is a positive integer such that 2q = I (mod p). Hence 1 + i(p is invertible in Z[i^p], so (1 + ac)e is invertible in (Z(ac))e and thus also in (ZG)e. It follows that /Xi = 1 + ecic = ( 1 — e)e(l ac) is invertible in the order (ZG)e © (ZG)(1 — e). The definition of m shows that (1 + i^p)^ G 1 + 2pZ[i(p]. Consequently, /x^^^ = (1  e) + e(l + 2j9a) for some a 6 Z{ac). Hence //^^ = 1 + 2pae 6 ZG. Finally, Lemma 2.6 (part 4) yields that /x^^ is a unit of ZG. D Since RA loops are not commutative, the foDowing corollary is the loop analogue of the HartleyPickel result for group rings [HP8O]. 5.8 Corollary. Let L be a finite RA loop. ThenU{ZL) only ifU{ZL) contains a free group of rank two.
is nontrivial if and
5. F R E E SUBGROUPS
221
Clearly, iiU(ZL) contains a free subgroup of finite index, then U(ZL) is not trivial. For the converse. Theorem 3.2 implies that L is not a hamiltonian 2loop. The result then follows from Theorems 5.3 and 5.7. D PROOF.
Chapter IX
Isomorphisms of Integral Alternative Loop Rings
In this chapter also, most group rings and loop rings will be integral; that is, the coefficient ring wiU generally be the ring Z of rational integers, and our results specifically apply to alternative loop rings which are not associative. Our objectives are threefold. First we settle the "isomorphism problem" for RA loops: we show that a torsion RA loop is determined up to isomorphism by its integral loop ring. Secondly, we show that if L is an RA loop, then every normalized automorphism of ZL is the composition of an inner automorphism of the rational loop algebra QL and an automorphism of L. Finally, we establish the validity for integral loop rings of RA loops of three conjectures about group rings made by H. J. Zassenhaus in the 1960s. For instance, we show that every torsion unit a € ZL is of the form ±7.^^ (71"^071)72 for some a ^ L and units 71,72 in QL. Most of this chapter is based upon [GM88, GM89, GM96b]. Our consideration of normalized automorphisms and the Zassenhaus conjectures requires knowing that every automorphism of a composition algebra is inner. This is a theorem of N. Jacobson, discussion of which forms the basis of Section 2. Let i2 be a commutative, associative ring with unity and of characteristic different from 2. Let L = M{G, *?5'o) be an RA loop. Thus G is a nonabelian group with an involution * and go is a central element of G. The centres of G and L are the same and //' = G' = {1,6}, where s is necessarily central of order 2. Recall that the involution * extends to RL in such a way that a G RL is central if and only if a* = a. Let e: RL ^ Rhe the augmentation map. If TV is a normal subloop of //, AR(L^N) denotes the kernel of the natural homomorphism €j\j: RL ^ 223
224
IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS
R[L/N], (We often write A(L,iV) when R is clear from the context.) The identity r = e{r) + {r — €{r)) for r G RL implies that RL = R\ A(iy), where A(L) = A(L,L) = ker€. Since A{L^N) is the ideal of RL generated by all elements of the form n  1, n G TV, it follows that A{L, N) = RL(A(N)) = A(N) + A(L)A(7V). From this it is easy to see that (1)
[^{L,N)f
C A{L)A{N)
C A(L,7V).
Since TV* = TV, we have also
(2)
[A(L, TV)]* C A(L, TV)
and
[A(TV)]* C A(TV).
Throughout this chapter, we rely heavily on the fact that a central torsion unit in an integral alternative loop ring is trivial (see Corollary VIIL1.7). We also make extensive use of the following two lemmas, the second one a result of A. Whitcomb which we state here for integral alternative loop rings. 0.1 Lemma. Let L be an RA loop or a group with V = { l , ^ } .
Then
L n ( l + Az(L)Az(L')) = { l } . Since Az{L') is the ideal of ZV generated by 5 — 1, if £ G L n (1 + Az{L)Az{V)), then i  \ ^ r{s  \) for some r G Az{L). Thus {(.  \){s + 1) = 0 and so ^ + 5^ = 5 + 1. It follows that i G { l , ^ } . Now Az{L) is the Zlinear span of {p — 1  p G Z/} because of the identity h{g  1) = {kg  \)  {h 1). In particular, Az{L') = Z{s  1). Since ^  1 G Az{L)Az{L'), we have PROOF.
(3)
£_i = ^a,(yl)(6l),
a, G Z .
geL
Letting T be a transversal for L' in L (with 1 G T) and writing each g ^ L in the form g = ti\ t £T^ i^ £ L', the map g ^^ t' extends to a Zlinear map r\ZL^ZL'. Now apply r to (3). Smc^T{g\){s\) = r{tt'stt's+l) = t'se  s ^ \ = {e  l){s  1), which is 0 or (5  1)2 = 2(5  1), we obtain an equation of the form t — \ = 2a{s — 1) for some integer a. Since the coefficients on the right are even and those on the left are odd, we obtain £ = 1, as claimed. D 0.2 Lemma (Whitcomb). Let L be an RA loop or a group and let TV be a normal subloop of L. If x G ZL and x = t (mod A(L, TV)) for some i ^ L,
1. THE ISOMORPHISM THEOREM
225
then there exists an element £o £ L such that x = io (mod A{L)A{N)). N = L', then IQ is unique. PROOF.
If
With e the augmentation map ZL ^ Z, we have X = i+ ^
an{n  1),
= £+Y^{{an
an G ZL
 e(an))(n  1) + e{an){n  1)}
neN
= £+J2<^n){nl)
(mod A(L)A(7V))
neN
since a^ — ^(<^n) G A(Z/). For ni,n2 € iV, observe that (nil) + (n2l) = (nms  1)  (ni  l)(n2  1)  (^1/12  1)
(mod A(L)A(A^)).
Hence, applying this to nj and 712 with nin2 = 1, we obtain that —(n—l) = {n~^ — 1) (mod A{L)A{N)) for all n ^ N. Since €(a) is an integer, we obtain, for some UQ £ N^ x = i+{nol)
= ino{i
l)(no  1) = irio
(mod A(L)A(A^)).
With £0 = £^0? we obtain an element of the required type. If A^ = L', uniqueness of io is an immediate consequence of Lemma 0.1. D 1. T h e i s o m o r p h i s m t h e o r e m The isomorphism problem for group rings, posed by Graham Higman in his 1940 thesis, asks if a group is determined by its group ring; that is, given a group G and a ring /2, does RG = RH for some group H imply that G = /T? It is well known that the strongest context in which to study this question is that of integral group rings because if ZG = ZH for groups G and H^ then RG = RH for every commutative and associative ring R with unity since RG = R®z ZG. G. Higman himself proved that if G is a finite abelian group and H is another group such that ZG = ZH^ then G = H. Later, A. Whitcomb proved that finite metabelian groups are determined by their integral group rings [Whi68]. (A group G is metabelian if G has an abelian normal subgroup N with an abelian quotient G/N; equivalently, if both G' and G/G^
226
IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS
are abelian.) An interesting result of R. Sandling states t h a t a finite group which can be realized as the group of units of a ring is determined by its integral group ring [ S a n 7 4 ] . If G and H are finite, the hypothesis ZG = 2.H implies t h a t G and H have the same table of characters. Since finite permutation groups are determined by their complex character tables [ J K 8 1 , Theorem 2.6.1], it follows t h a t Sn is also determined by its integral group ring. K. W. Roggenkamp and L. Scott showed t h a t finite nilpotent groups are determined by their integral group rings, a result which also follows from a recent result of A. Weiss (see Section 4). The isomorphism problem over fields was first considered by S. Perils and G. L. Walker who proved t h a t if G is a finite abelian group and H is any group with QG = QH, then G = H [ P W 5 0 ] . A similar result was obtained by W. Deskins who proved t h a t if G is a finite abelian pgroup, if F is a field of characteristic p > 0 and if ^ is a group such t h a t FG = FH ^ then G = H [Des56]. At this point, we draw the attention of the optimistic reader to a striking counterexample of E. Dade who exhibited two metacyclic groups G and H which are not isomorphic but are such t h a t FG =. FH for all fields F [ D a d T l ] . (A group G is metacyclic that both A^ and GjN
if G has a normal subgroup A^ such
are cyclic.)
Finally, we should mention t h a t it has been a longstanding conjecture in this area t h a t if G and H are pgroups and F^ is the field of p elements, then Fj,G ^ F^H implies G ^ H (see, for example, [ Z M 7 5 ] ) . R. Sandling has shown this to be the case when G is a centralelementarybyabelian pgroup [ S a n 8 9 ] . For a detailed history of the isomorphism problem, the reader is referred to the survey of R. Sandling [ S a n 8 5 ] . T h e purpose of this section is to show t h a t the isomorphism problem over Z has a positive solution for torsion RA loops.
1.1 T h e o r e m . Let L\ and L he RA loops with L\ torsion morphic
and TL\
iso
to 2.L. Then Li = L.
PROOF.
normalized
Let 0: ZL\ in
^ ZL be an isomorphism.
We assume t h a t 6 is
the sense t h a t it preserves augmentation. There is no loss
of generality in so doing since, if p: ZLi ^ ZL is any isomorphism, then
1. THE ISOMORPHISM THEOREM
227
ep{tj = ±1 for any ^ G Z/i, so the map p: ZL\ ^ ZL defined by p(i) = [€p(£)]~^p(£) extends linearly to a normalized isomorphism ZLi ^ ZL. Denote by x the image in Z[LlV] of an element x ^ ZL under the natural homomorphism e^i: ZL ^ Z[L/L^], Let £ E L^, Then 9{i) is a torsion unit whose image 6(1) in the abelian group ring Z[L/L^] is a torsion unit and hence trivial, by Corollary VIII. 1.4. Since its augmentation is + 1, 0(i) = t^ for some t £ L. Since kereit = A(i/,iy'), we have 0(£) = t (mod A(Z/,L')) and so, by Whitcomb's Lemma, e{i) = i,
(mod A(L)A(L'))
for some unique ip G L, Thus p: i \^ ip is di well defined map Li —^ L. We complete the proof by showing that p is, in fact, an isomorphism. For this, note first that if e{l>^) = hp
(mod A(L)A(LO)
and ^(£2)  ^2p
(mod A(L)A(LO),
then
e{i^i^) 3Z e{t,)e{t2) = iiphp (mod
A(L)A(L')).
By uniqueness, ^(^1^2) = hp^2p = p{(^\)p{('2)'> so /> is a homomorphism. Suppose p{t) — 1. Then (4)
e{i)
= 1
(mod
A(L)A(L0),
so e(e'^) = 9{if = 1 (mod A(L)A(LO). Since i'^ is central in Lu 0{e^) is a central torsion unit in ZL, and it is a trivial unit. Since its augmentation is l,e(P) = a e Z{L). Since a\ G A(L)A(L'), Lemma 0.1 shows that a = 1; that is, 9{iy' = 1. Consider again equation (4). Since A(L)A(L') is closed under the involution * on ZL, we have 9{iy = 1 (mod A(L)A(L')) and hence 9{i)9{iy = 1 (mod A(L)A(LO). Now 9{t)9{ty is a central torsion unit in ZL, and of augmentation 1, so it is in L. Another application of Lemma 0.1 shows that 9(i)9{iy = 1. Comparison with 9(i)'^ — 1 shows that 9(tj — 9{iy, Hence 9{JC) is central. As a central torsion unit of augmentation 1, 9{Ji) G L, hence 9(J[) — 1, by (4). Since 9 is onetoone, i — 1, so /> is onetoone.
228
IX. ISOMORPHISMS O F INTEGRAL ALTERNATIVE LOOP RINGS
Suppose t G Z{Li),
Then 6(i) is a central torsion unit in ZL, hence
trivial and, because it has augmentation 1, in Z{L). p(i)
= e{i).
Thus p{Z{Li))
C Z{L),
Also, if a G Z{L),
central, torsion and of augmentation 1, hence in Z{L\). just observed, p{6~^{a)) Z{L).
— 0{6~^{a))
By definition of p, then ^ " ^ ( a ) is
By what we have
— a. It is clear then t h a t p{Z{L\))
=
Since /9 is a onetoone m a p from one RA loop to another, and
since an RA loop is generated by its centre and any three elements which do not associate (see §IV.3) it is immediate t h a t p is onto and hence an isomorphism.
D
2. I n n e r a u t o m o r p h i s m s of a l t e r n a t i v e a l g e b r a s Let A be an alternative ring with unity and, for x G ^ , let R{x) and
L{x)
denote the usual right and left translation maps respectively. Thus aR{x) = ax
and
aL{x) = xa
for a ^ A. For consistency with other parts of this book, whenever these maps arise, all maps will be written on the right. If a , x G v4 with x invertible, we showed in §11.5 t h a t a, x and x~^ generate an associative subring of A. It follows t h a t R(x) and L{x) are invertible maps with inverses and L{x~^)
R{x~^)
respectively.
2.1 D e f i n i t i o n . An inner automorphism
of A is an automorphism which
is in the group generated by {/2(x), L{x) \ x ^ A invertible}. If A is associative, any automorphism 0 which is inner in the above sense, is of the form 6{a) = uav for some invertible elements u^v £ A, Since 6(1) = l^ u = v~^ and so 9 is inner in the associative sense. We wish to characterize the inner automorphisms of an alternative ring in a different way. To do so, we introduce in A three maps which we have previously defined on a Moufang loop. For invertible elements x , y G ^4, T{x) = R{x,y)=
R{x)L{x)^ R{x)R{y)R{xy)'
L{x,y)=L{x)L{y)L(yx)\
2. INNER AUTOMORPHISMS OF ALTERNATIVE ALGEBRAS
229
2.2 Proposition. Let Aut(A) and Inn(A) denote, respectively, the group of automorphisms and the group of inner automorphisms of an alternative ring A. Let H be the group generated by {T(x), i2(x,y), L(a:,y)} for all invertible elements x and y in A. Then Inn(A) = Aut{A) 0 H. PROOF.
We present a proof suggested by D. A. Robinson. Let G  {R(x), L{x) \x e A invertible)
so that, by definition, Inn(A) = Aut(A) fl G. Since 77 is a subgroup of G, it is clear that Aut(A) f) H C Inn(^). Thus it suffices to prove that Inn(^) C H. For this, we consider the set
s = {0eG\ee
HR{\6)]
and show that SG C S. To see why, let 0 E S^ let x be an invertible element in A and let r = 6R(x). Then 0 = pR{\0) for some y9 G H^ so T = 0R{x) = pR{\0)R{x)
= pR{l0,x)R{l0'x)
=
pR{l0,x)R{\T);
hence r £ S. Similarly, with r = 0L{x) and 0 = pR{l0)^ p £ H^ we have r = 0L{x) =
pR{l0)L{x)
= pT{10)L(l0)L{x), =
since R{10) = T{10)L{10)
pT{l0)L{l0,x)L{x'l0)
= pT{10)L{l0,x)T\x
' \0)R{x ' 10)
= pT{10)L{l0,x)T\x
. 10)/?(lr);
hence r G 5. Since G is generated by all R{x) and L(x)^ it is clear that SG C 5, as asserted. Since the identity map id^ on A is in H^ id^ = id^ R{1 id^i) belongs to 5, so G = S. Finally, if ^ G Inn(/l), then 0 e G = S, so 0 e HR{\0) = H because \0 = 1. This completes the proof. D Suppose now that A is a composition algebra over a field F of characteristic different from 2. We refer the reader to §1.4 for a general discussion of composition algebras and, in particular, to the proof of Hurwitz's Theorem, Theorem 1.4.11. We denote by q{a) and a i^ a, respectively, the multiplicative quadratic norm and involution on A] thus q{a)\ = aa for a e A. Writing A = Fl ® (Fl)^ and a = a l f a' with a e F and
230
IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS
o! e (Fl)^,
recall t h a t a = al  a\
a^ = —aa G Fl.
If a G (Fl)^,
then a ^ F l , but
Conversely, if a = al + a^ ^ Fl and a^ e Fl^ then a' ^ Q
and a? = Q^l + 2aa' + a'^ = ( a ^ l  a V ) + 2aa' shows t h a t a = 0. Thus ( F l ) ^ = {a G A I a ^ F l , a 2 G F l } . It follows t h a t (5)
e{Fl)^
for any Fautomorphism 6{a\)
C (Fl)^
9 oi A ( t h a t is, any automorphism of A satisfying
= al for a G F ) . Let a = a l + a' with a G F and a' G ( F l )  ^ and let
e be an Fautomorphism of A, Then e{a) = al + 0{a') with e(a') G ( F l )  ^ , so '0(a) = a l  d{a') = 9{a)\ thus, e{a) = e{a) for any a ^ A. Furthermore
(6)
q{a) = qiOia))
for any a e A because q{a)l = 6{q{a)l)
= 6{aa) = 6{a)9{a) —
q{9{a))\.
T h e primary aim of this section is to prove a theorem of N. Jacobson showing t h a t any Fautomorphism of a composition algebra with centre F is inner. To do this, we follow the general pattern of proof in [ J a c 5 8 ] . An Fautomorphism r of yl is a reflection if r'^ = i d ^ , the identity map on A, Let J5 be a subalgebra of A of dimension ^ dim A and on which the restriction of q is nondegenerate. Thus A = B ® B^ and B^
= Bi^ with
q{i) = —a ^ 0. Moreover, multiplication in A is defined by (7)
(x + yi){z + w(,) = (xz + awy) + {wx + y'z)i
for x,t/ G B,
Define r: A ^ Ahy
r^^ = i d ^ , T\ ^ = — i d ^ ± . Clearly r
is onetoone and onto, and r'^ = id^. Moreover, it follows directly from (7) t h a t r is a homomorphism; thus r is a reflection. We call such r the reflection
in B, In fact, every reflection r is reflection in some subspace of
dimension  dim A and on which the restriction of r is nondegenerate, as we proceed to show. Suppose t h a t r is any reflection of A, The equation a= \{a + T{a)) + \{a 
T{a))
2. INNER AUTOMORPHISMS OF ALTERNATIVE ALGEBRAS
231
shows that A = A^ © A_ is the direct sum of subspaces where A^ = {a e A \ T{a) = a}
and
A = {a e A \ T{a) =  a } .
Clearly T{A^) C A ^ and r(A_) C A_. Let / be the bilinear form associated with q. Thus / ( x , y)l = (q(x + y)
q{x)  q(y))l = xy + yx
for any x,y e A. Let x e A^ and y G A. Since r fixes F l , (8)
xy + yx = T{xy + yx) = T{x)T{y) + r(2/)r(x) = xy  yx
so that f{x,y) = 0. It follows that / is nondegenerate on A^ and also that A = (yl__)^. Again, from the proof of Theorem L4.11, we see that for some nonzero i G ^  , A^i C y l  = A. Since >l = A^ © >1_ and because dimA^i = dimi4_, we have A^ = A^l. In particular, dim A4. = ^ dim >1 and r is the reflection in A^. 2.3 Proposition. Every reflection of a quaternion or Cay leyDickson algebra over a field of characteristic different from 2 is inner. Let A he a, quaternion or CayleyDickson algebra over a field F of characteristic diflFerent from 2. Denote by q and / , respectively, the norm on A and its associated bilinear form. Suppose r is a reflection of A in a subalgebra 5 , where A = B ® B^, B = A^ and B^ = A^. If A is quaternion, then B = Fl + Fi for some i G (^l)"*" with q{i) ^ 0. (Thus i is invertible.) For any a G B^^ we have 0 — f(i^a^ — ia + ai — —ia — ai^ so aT(i) = i~^ai — —a — ar. Thus T{i) and r agree on B^, They also agree on 5 , since B = A^ is commutative. Thus r = T(i) is inner. If yl is a CayleyDickson algebra, then B = Fl + Fi + Fj + Fij is quaternion, where i and j are invertible and ij + ji = 0. In view of (7), for any a G B^^ we have aL{j^i) = {ij)~^{i jo) = {ij)~^(ji * ct.) = —a = ar^ so r and L{j,i) agree on 5^. They also agree on B because B = A^ is associative, so r = L{j^ i) is inner. D PROOF.
2.4 Theorem (Jacobson [Jac58]). Over a field F of characteristic different from 2, any Fautomorphism of a quaternion or CayleyDickson algebra is the product of reflections and hence inner.
232
IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS PROOF.
Let A be a quaternion or CayleyDickson algebra over F and
let 6 be an Fautomorphism of A, Let q{a) — aa be the quadratic form on A^ where a \^ a denotes the involution on A^ and let / be the bilinear form associated with q. Thus f{x^y)\
= xy + yx for x^y £ A. Let i G
be such t h a t q{i) ^ 0. Then 0 / V = i — 6{i). Since 6{F\)^
i^ =  n G F l .
C {F\)^
(Fl)^
Set u = i + (9(i) and
by (5), each of u and v is an element
of (Fl)'. This implies u = —u and v = —v.
Since i^ G F l , we have
i^ =z 6(P)^ thus uv + vu = 0, so f(u^v)
= 0. Since q{0{i)) = g(i), by (6),
we have q{u) — q{i) + q{0{i)) + f(i^0(i))
— 2q{i) + f(i^6{i))
q{v) = 2q{i)  f{i,e{i)).
and, similarly,
Thus either q{u) ^^ 0 or q{v) ^ 0.
Assume first t h a t q{u) ^ 0. If A is quaternion, let B — F\ \ Fu and let r be the reflection in B. Note t h a t T{U) = u and T{V) = —v because v G B^. Now the restriction of / to (Fl)^
is nondegenerate by Corollary L4.6 and dim(Fl)' = 7 by
Proposition L4.4. If A is CayleyDickson, let Bi be the subspace of which is orthogonal to u^ v and uv.
(Fl)^
Since u^ v^ uv span a subspace of
(Fl)* of dimension at most 3, it follows again from Proposition 1.4.4 t h a t d i r n d l > 4. Since dim(Fl)' = 7, the maximum dimension of a totally isotropic subspace is 3, by Corollary 1.4.3, so there exists an element j G 5 i with q{j) /
0. Since f{j^u)
= 0, the subalgebra B2 generated by j and
u is quaternion.
Let r be the reflection in B2.
f{l,v)
= f{u,v)
= f{j,v)
= 0 and f{ju,v)
Note t h a t v G B2 since
= f{vuj)
= f{vuj)
= 0.
(See (1.32).) Again we note t h a t T{U) — u and T(V) = —v. Thus, whether A is quaternion or CayleyDickson, we have i + e{i) = u = T{U) = T{i) +
T(0(i))
i  0(i) = v= T{V)
+ r((9(0),
and = T{i)
and hence T(0(i)) = i. Now assume t h a t q{v) 7^ 0. As above, there exists a reflection r^ such t h a t Ti{v) = V and TI(U) = —u. Thus T\6{i) = —i. Since q{i) 7^ 0, i is part of a basis provided by the CayleyDickson process; hence yl = S3 © B^ for some B^ with i G ^ 3  . With T2 the reflection in ^ 3 , we have T2r\d{i) = i.
3. AUTOMORPHISMS OF ALTERNATIVE LOOP ALGEBRAS
233
T h u s , whether q(u) 7^ 0 or q{v) / 0, there exists a reflection or a product of reflections r such t h a t T9(i) = i. Replacing 6 by r^, it suflRces to prove t h a t 0 is the product of reflections in the case t h a t 0{i) ~ i. So assume 6{i) = i and let j G ( ^ 1 + Fi)^ Then ij + ji = 0, so also iO(j) + 0{j)i 0{j) e (Fl + Fi)^,
Let X = j+e{j)
be any element with q{j) ^ 0.
= 0 (because 0{i) = i) and hence
and y = je(j).
Then x,y e (Fl +
Fi)^,
f(^x^y) = 0 and, as before, q{x) 7^ 0 or q(y) 7^ 0. Suppose A is quaternion. If q{x) 7^ 0, F l + Fx is nondegenerate and we let r be the reflection in Fl + Fx.
Then T(X) = x and T{y) = —y, so
T0{j) = j . Also T6{i) = T{i) = —i, so T9 is the reflection in F l + Fj.
If
q(^y^ 7^ 0, then F l + Fy is nondegenerate and we let r be the reflection in F l + Fy.
This time we have TO(J) = —j. Since again T9(i) = —i, we see
t h a t T6 is the reflection in F l + Fij.
This completes the case t h a t A is
quaternion. Suppose t h a t >1 is a CayleyDickson algebra. Then the subalgebra generated by i and x, if q(x) 7^ 0, or by i and y^ if q(y) 7^ 0, is quaternion. Let r be the reflection in this subalgebra. Thus T6{i) = i and T6{J) — ±j. TO(J)
If
= —J, choose £ with ^(^) 7^ 0 orthogonal to the quaternion algebra
generated by i and j and let ri be the reflection in the quaternion subalgebra generated by i and i. Then T\T6{i) = i and
TIT9(J)
= j . Replacing 0
by r r i ^ , we can assume 6{i) = i and 0{j) — j . Now choose ^ = ( F l + Fi + Fj + Fij)^ 3indz = ie{i).
Theni,e{i),z,te
with q{i) 7^ 0. Set w = i + 9(1)
(Fl + Fi+Fj+Fij)^
^^nd f{w, z) = 0. If
q{w) 7^ 0, let T be the reflection in the quaternion subalgebra generated by i and w and, otherwise, the reflection in the quaternion subalgebra generated by i and z. Then T6(i) = z, r0(j') = —j and
T6((.)
= ± ^ , so
T6
is a reflection
in a quaternion subalgebra containing i. This completes the proof.
D
3. A u t o m o r p h i s m s of a l t e r n a t i v e l o o p algebras Naturally related to the isomorphism problem, there is another question, which we first discuss within the context of group rings.
W h a t are the
automorphisms of an integral group ring ZG? Any automorphism of the group G can be extended linearly to an automorphism of ZG. Also, if 7 is an invertible element in the rational group algebra QG such t h a t j~^gj
G ZG
234
IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS
for all ^ G G, then the map (f^: 2.G ^ ZG given by (p^{g) = J~^gj is also an automorphism of ZG [Seh93, Lemma 34.4]. It has been conjectured that all normalized automorphisms of ZG are compositions of automorphisms of these two types. 3.1 Conjecture ( A u t ) . If 0 is a normalized automorphism of ZG, then there exists an automorphism p of G and a unit 7 in QG and such that % ) == 7 " V ( ^ ) 7 for aU ^ e G. S. K. Sehgal has established this conjecture for finite groups which are nilpotent of class two [Seh69]. G. Peterson confirmed the conjecture for finite symmetric groups Sn [Pet76] and extended his results to some classes of metacyclic groups [Pet77]. The conjecture has also been verified by A. Giambruno, S. K. Sehgal and A. Valenti for the wreath product ^4 i ^n of a finite abelian group A with Sn [GSV91]. It has also been verified for P I Sn, where P is a finite pgroup for some prime p. The result for odd p is due to A. Giambruno and S. K. Sehgal [GS92] and, for p = 2, to M. M. Parmenter and S. K. Sehgal. In general, however, the problem remains open. It is unresolved for alternating groups, for instance, though it holds for ^5 [LP89] and, in fact, for An li n < 8 [Gia]. In this section, we prove that, if L is a finite RA loop, then any normalized automorphism of ZL is the composition of an inner automorphism of QL and an automorphism of L, 3.2 Theorem. Let L be a finite RA loop and let 9 be a normalized automorphism ofZL. Then there exists an inner automorphism ip of the rational loop algebra QL and an automorphism p of L such that 6 = ip 0 p. Our first observation is that 9(s) = s where, as always, s denotes the unique nonidentity commutatorassociator of L, To see this, let k and £ be any elements of L which do not commute. Then PROOF.
(9)
ki  ik = {I  s)ke
and it follows that the (ring) commutator of any two elements of ZL is contained in (1 — s)ZL. Also, applying 6 to each side of (9), we obtain (l9{s))0{ke) = 9{k)9{i)9{t)9{k) G {\s)ZL, Multiplying by I + 5 gives (1 + 5)(1  9{s))9{ki) = 0. Since both 1 + 5 and 1  9{s) are central and
3. AUTOMORPHISMS OF ALTERNATIVE LOOP ALGEBRAS
235
0{k£) is invertible, we obtain (1 + 5)(1 — 0(s)) = 0. Hence (10)
l + s = e{s) + s0{s).
As a central torsion unit, 6{s) G L (Corollary VIII.1.7), so each side of (10) is a sum of loop elements. Since 0(s) ^ 1, necessarily 9{s) = s. As in the proof of Theorem 1.1, there exists an automorphism p of L defined by p(£) = ^p, where i^ is the unique element of L satisfying e{i)
= ip
(mod
A(L)A(L0).
We denote also by p the linear extension of this map to ZL and note that this extension is a ring automorphism of ZL. We now claim that 6{i) is central if and only if p{i) is central and, in this case, 9{i) = p{i). For this, we first note that, if 6{i) is central, then 6{i) G L (as a central torsion unit of augmentation 1), so p{t) — t^ — 6(i) (by uniqueness) is central. On the other hand, suppose that p{£) =: £p is a central element of L. Then 0{i) = i^ (mod A{L)A{L')) implies e{e)^ = ij (mod A{L)A(L')). But 6{i)^ = 9{(?) is a central torsion unit (because i'^ is a central torsion unit) and of augmentation 1. By uniqueness,
(11)
e{if^tl.
Now d{iy = £; (mod A{L)A{L')) since A(L)A(L') is closed under the involution * on ZL and, since ip is central, i* = i^. Thus d(ty = ip (mod A(L)A(L')) and 9{i)9{ey = i] (mod A{L)A{L')). Since rr'^ is central for any r G ZL, we see that 9{i)9{iy is a central torsion unit of ZL, and of augmentation 1. By a now familiar argument, 9{i)9{iy = £^. Comparison with (11) shows that 9{i) = 9{iy^ so 9{(.) is central. As a central torsion unit of augmentation 1, 9{i) G L and, by uniqueness, 9(i) = £p. This estabUshes our claim. As one consequence, observe that p(s) — s. Since 9{i) = p{£) (mod A(L)A(L')) and A(L') = ZL'(1  s), we have (12)
^i + s)9{i) =
{\^s)p{t)
for any C £ L. Recall that the centre of a loop ring RL is spanned by Z{L) and {£ + si \ i £ L} (Corollary III.1.5). The claim above shows that for £ G 2 ( L ) , 9{£) rr p{£), while (12) shows that 9{{1 + s)£) = p{{l + s)£), since p and 9 fix s. The point is that 9p'~^ fixes elementwise the centre of ZL and.
236
IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS
more importantly, the centre of the rational loop algebra QL,
This loop
algebra is the direct sum of simple alternative algebras A^, which are either fields or CayleyDickson algebras (see Theorem VI.4.8). Since the centre of QL is fixed by Op~^^ the restriction of this m a p to any commutative component is the identity and to any component which is a CayleyDickson algebra an inner automorphism, by Theorem 2.4. Define ipi'. QL > QL by
M^j) where ai G Ai.
Op^{ai)
i = j
aj
i 7^ j
It is easy to see t h a t tpi is an inner automorphism of QL
for each i and hence t h a t tp = Yii^i is an inner automorphism such t h a t ip == Op~^ ^ which gives 6 = tp o p dis desired.
D
4. S o m e c o n j e c t u r e s of H. J. Z a s s e n h a u s We denote the loop of units in the integral alternative loop ring ZL by U{ZL) and the loop of normalized
units by U\{7.L)\ t h a t is,
U^{ZL) = {aeUiZL)
I 6(a) = 1}
where e: ZL ^ Z \s the augmentation m a p . By TU{ZL)
and TU\{ZL)
we
denote, respectively, the loops of torsion units and normalized torsion units
of ZL. If G is an abelian group, then we know t h a t the units of finite order in ZG are trivial. (See Chapter VIII.) When G is not abelian, an obvious way to exhibit new torsion units in ZG is to compute conjugates of the form ± 7 ~ ^ ^ 7 , with g ^ G and 7 G U{ZG),
In fact, one could take 7 from the
rational group algebra QG, provided that 7~^6'7 C ZG. In the mid 1960s, H . J . Zassenhaus conjectured t h a t all torsion units in ZG can be constructed in this way. More precisely, he formulated 4,1 C o n j e c t u r e ( Z C l ) . Let a G ZG be a normalized unit of finite order. Then there exists an invertible 7 G QG and an element a ^ G such t h a t 7 ~ ^ a 7 = a. If a G ZY(ZG), 7 G U{QG) and 7 ~ ^ a 7 = a G C , we say t h a t a is rationally conjugate conjugate
to a. More generally, a subgroup H of ZY(ZG) is
to a subgroup K of G if j~^Hj
— K for some 7 G U{QG).
rationally There
4. SOME CONJECTURES O F H. J. ZASSENHAUS
237
are two stronger versions of Z C l dealing with subgroups of normalized units. 4.2 Conjecture (ZC2). Let ^ be a subgroup of normalized units in ZG such that 1^1 = \G\. Then H is rationally conjugate to G. 4.3 Conjecture (ZC3). Let H be any finite subgroup of normalized units in ZG. Then H is rationally conjugate to a subgroup of G. The reason for restricting attention to subgroups of normalized units should be apparent since, for /^ G ^ , 7 G U{QG) and j'^hj
G G, then
1 = €(71/^7) = e{h). Clearly Z C 3 implies t h e first two conjectures and a positive answer to Z C 2 implies a solution to the isomorphism problem. It is also easy to verify t h a t Z C 2 implies the conjecture A u t described in §3.1 and t h a t a positive answer to both A u t and the isomorphism problem implies Z C 2 . The Zassenhaus conjectures have been established for various kinds of groups. All of them have long been known to be true for finite nilpotent groups of class two [Seh69].
T h e most far reaching result to date is a
theorem due to A. Weiss [Wei91] (see also [Seh93]) which says that Z C 3 holds for the group ring of any finite nilpotent group. Z C l was proved for metacyclic groups of the form G = (a) XI ( x ) , where gcd(o(a),o(x)) = 1, by C. Polcino Milies, J. Ritter and S. K. Sehgal [ M R S 8 6 ] . A. Valenti has shown t h a t Z C 3 holds for a group of the form G = (a) Xl X^ where gcd(o(a),  X  ) = 1 and X is abelian (see also S. 0 . Juriaans [Jur94] ). T h e book by S. K. Sehgal [Seh93] contains an exposition of most of the known results on this subject. In another direction, it should be mentioned t h a t Z C l was proved for 5*4 by N. A. Fernandes [Fer87], for A4 by I. S. Luthar and I. B. S. Passi [LP89] and, for ^ 5 , by I. S. Luthar and P. T r a m a [LT]. On a negative note, K. W . Roggenkamp and L. Scott found a metabelian group of order 2^ • 3 • 5 • 7 which is a counterexample to Z C 2 [Rog91] and later L. Klinger found another counterexample with a group of the same order, but using different methods [ K l i 9 l ] . The goal in this section is to show t h a t , with minor modifications, all three conjectures hold for alternative loop rings which are not associative.
238
IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS
4 . 4 P r o p o s i t i o n . Let L he an RA loop with centre Z{L). torsion
unit in ZL.
Then there exists a unique a £ L such that a = a
(mod A ( L ) A ( L ' ) ) . Moreover, PROOF.
Let a be a
a^ = a^ G ^ ( L ) .
Since the image a of a in the abelian group ring Z[L/L^] is
a torsion unit, a is trivial.
Thus a = ±t (mod A(Z/,L')) for some t G
L and, by Whitcomb's Lemma, there is a unique a £ L such t h a t a = ±a (mod A{L)A{V)).
Thus a^ = a^ (mod A{L)A{V)),
(mod A ( L ) A ( L O ) . Write a ' ^ a ^ = i + S(ls),6
so a^a^
e ZL,supp{6)
= 1
C T , where
T is a transversal for { l , ^ } in L and 1 G T . It follows t h a t 1 or 5 is in the support of a~^a^, hence a~^a^ G // by Corollary VIII.1.2. Now Lemma 0.1 shows t h a t a^ = a^.
D
If // is a finite RA loop, the alternative loop algebra CL of L over the complex numbers is the direct sum of simple alternative algebras which are either fields or CayleyDickson algebras. (See Chapter VI.) In a CayleyDickson algebra, every element is either invertible or a zero divisor because a is invertible if and only if aa /
0, where a \^ a denotes the canonical
involution. Thus every element of CL is either invertible or a zero divisor. An argument of C. P. Milies and S. K. Sehgal can therefore be adapted in the alternative case and gives us a key lemma. 4.5 L e m m a . [MS84] Let k C K be infinite whose loop algebra over k is semisimple fi zn 7 ~ ^ a 7 for some invertible invertible
fields.
Let L be a finite loop
and alternative.
element 7 G KL,
If ct^P G kL and
then (3 = 7 ~ ^ a 7 for
some
element 7 G kL.
PROOF.
Consider the equation ax — x(i which, relative to the basis
L (of either vector space kL or A'L), can be expressed in matrix form as x M = 0, where M is the matrix of the linear transformation L{a) — R{l3) and X is the coordinate vector of x. Note that M is an n x n matrix, n = Z/, with entries in /c, and singular since xM — 0 has a solution x corresponding to an invertible element x G KL.
Let x i , . . . ,Xi be a basis for the nuUspace
of M in k'^ and let Xi G kL correspond to x^. We prove t h a t some A;linear combination of these vectors corresponds to an invertible element of kL. If this is not so, then, for any A i , . . . ,At G A;, the element x = Yl^i^i zero divisor in kL; equivalently, the matrix of R{x) is singular.
^^ ^
Consider
4. SOME CONJECTURES OF H. J. ZASSENHAUS
239
the determinant det Z KR{xi)
= / ( A i , . . . , At) G A:[Ai,... , A,].
Since k is infinite and we are assuming t h a t / ( A i , . . . ,At) = 0 for all Xi G k, the polynomial / must be identically zero. But this implies t h a t / ( c i , . . . , Ct) = 0 for all Ci G K, which is not true. Thus / is not identically zero, so some solution of x M = 0 in Z;;^ corresponds to an invertible element in kL.
n
Let a and /3 be elements of ZL. In what follows, it will be convenient t o say t h a t a and /3 have a common
conjugate
in QL (or CL) provided there
exist 7i and 72 in QL (or CL) such t h a t 7 f ^ a 7 i = 12^(^12' T h e following corollary is immediate. 4.6 C o r o l l a r y . If a and fi are elements of the integral alternative ZL which have a common conjugate
conjugate
loop ring
in CL, then a and (3 have a
common
in QL.
We now present an analogue of Z C l for alternative loop rings which are not associative. Clearly our result reduces immediately to Z C l in the associative case. 4 . 7 T h e o r e m . Let L be a finite
RA loop and suppose a is a
torsion unit in ILL. Then there exists an element QL such that J2^{j^^aji)j2 PROOF.
normalized
a E L and units 71,72 in
= CL
By Proposition 4.4, there exists a unique a ^ L such t h a t a = a
(mod A ( i / ) A ( L ' ) ) and a^ — a}. As in the proof of Theorem 3.2, o. is central if and only if a is central. If this is the case, then a = a by Lemma 0.1 and there is nothing to prove. Therefore, we assume henceforth t h a t neither a nor a is central. By Corollary VIII. 1.2, a has no central elements in its support, so a* = ^o;. Since also a* = ^a, it follows t h a t aa^ota = (aoL\adY is central (Corollary III.4.3) and, since OL — a <^ A(Z/)A(L') and A ( L ' ) = Z(l  5), t h a t a ( l + 5) = a{\ + s). We will show t h a t a and a have a common conjugate in CL. Because of Corollary 4.6, it will follow t h a t a and a have a common conjugate in QL, which is the assertion of the theorem. Write CL — A\®"
•© An, where the
Ai are simple alternative algebras. Denoting by oti and a^, respectively, the
240
IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS
components of a and a in A^, it is sufficient to show t h a t a^ and a^ have a common conjugate in Ai. Suppose Ai is commutative and hence a field. By Proposition VI.4.6, the projection Si of s in Ai is the unity of Ai. Since (1 + s)a = (1 + s)a^ we have 2ai = 2ai^ so a^ = a^; thus a^ and ai have a common conjugate. Suppose t h a t Ai is a noncommutative component; thus (by Proposition VI.4.6) Si ^ I; hence Si = ~1. In this case, since a is not central, ai is not central, by Corollary VI.4.7. Since a^a^ + ct^a^ is central in Ai^ it follows t h a t ai is not central in Ai (because ai is a unit). For convenience, we now drop the subscripts on a^ and a^. Let B be the (associative) subalgebra of Ai generated by a and a. Since the centre of a simple finite dimensional algebra over C is C, and since a^ — a} and aa\aa
are central, we can write
a^ = o? — } ^ z^ ^ and aa\aa — // with A, /i G C. Every element in 5 is a linear combination of 1, a, a and aa. Set j \ —  ( l    a / A ) and J2 — \{\ — ajX). Then j \ and /2 are orthogonal idempotents different from 0 and 1 such t h a t / i + /2  1 and a = A(/i  h \ Thus B = {h + f2)B{h Defining fij — fiafj
and
/2i=/2a+—/2.
and / 2 ^ / 2 each have dimension 1 (with bases f\ and
/2 respectively) and, for i / j , t h a t d\m{fiBfj) in which case fiBfj
has basis fij.
< 1, with equality if fij ^ 0,
Moreover, 2
/?2 = / l i = 0 ,
UBf,.
we have, by straightforward computation, t h a t
f\2 = f\a—fi It follows t h a t f\Bf\
+ /2)  E
/,2/2i = (A^  £ ^ ) / ,
2
and
/2i/,2 = ( A 2  i ^ ) / 2 .
If A^ — ij?IAX^ / 0, replacing /12 and /21 with /12 = c~Vi2 and /21 = c"V2i 2
respectively, where c G C is a root of X'^ — {X^ — j^)
= 0, it is apparent
t h a t { / i , / 2 , / i 2 5 / 2 1 } form a set of matrix units and hence t h a t B = M2(C). Since the minimal polynomial of both a and a over C is X'^ — A^, each element is similar to [ Q J \ ] , so a and a have a common conjugate in B and hence in Ai. If A^  /i2/4A2 = 0, then fi/2X^ = ± 1 . If fi/2X'^ = 1, defining p = ( l / 2 A ) ( a h a ) , it can be checked directly t h a t p^ = I and p~^ap
= a.
Thus a is a common conjugate of a and a in ^ i . If ///2A^ = — 1 , define g = ( l / 2 A ) ( a — a ) and check t h a t q'^ = I and q~^aq
= —a. Since a is
4. SOME CONJECTURES OF H. J. ZASSENHAUS
241
not central, for some h in the projection of CL onto Ai^ we must have h~^ah =z sa = —a, so it follows here too that a and a have a common conjugate in Ai, namely —a. D We now turn our attention to the other two conjectures of H. J. Zassenhaus, ZC2 and ZC3. The following proposition generalizes in a completely straightforward manner a known result for group rings. 4.8 Proposition. Let L be a group or an RA loop. Then any set of normalized torsion units in ZL which forms a loop, Li, is linearly independent overZ. In particular, if L is finite, then \Li\ < \L\, P R O O F . If X)r=i ^«7« — ^' ^^^^ ^« ^ ^ ^^^ 7i? • • • ^ Tn distinct normalized torsion units contained in a loop Li, then we can write n
(13)
ail =  5 ] a , ( 7 r S i ) . t=2
Each 7f ^7t is a torsion unit and different from 1, hence, when expressed as a linear combination of loop elements, the coefficient of 1 is 0, by Proposition VIII.1.1. Comparing coefficients of 1 on each side of (13), we obtain ai = 0. Similarly, all a^ = 0. D Recall that an RA loop L has the LC property; that is, if x,y G L and xy = yx, then one of x, i/, xy must be central. 4.9 Lemma. If LQ is a subloop of an RA loop and LQ is not commutative, then Z{Lo) C Z{L), Let a G Z{LQ) and suppose that a ^ Z{L). Let x^y e LQ be such that xy ^ yx; thus, none of x, y, xy is in the centre of L. Since ax = xa but neither a nor x is in 2 ( L ) , it must be that ax is central in L and, similarly, ay and a{xy) are central. But then PROOF.
a{xy) = a{{a~^ • ax){a~^ • ay)} = a~^{ax)(ay) impfies that a~^ is central in L, contradicting a ^ Z{L),
D
4.10 Remark. Suppose that L is a torsion RA loop with V = {1,^5}. If a is any normafized torsion unit in ZL, then, as noted Proposition 4.4, there is a unique element p(a) G L such that a = p(a) (mod A(L)A{V)) and
242
IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS
cP' — p{oif'. Noting the uniqueness of pia) as an element of L, it is clear that /9(a) = a if a G //. As in the proof of Theorem 3.2, a G TZYi(ZL) is central if and only if p{o) is central and, in this case, P(OL) — a. In particular, p(^s) — s. Suppose a i and a2 are normalized torsion units in ZL such that aia2 is also a torsion unit. Then ai = />(<^i) (mod A(L)A(L')) and a2 = pi(^2) (mod A(L)A(L')), so aia2 = p{ai)p{a2) (mod A(L)A(L')). On the other hand, aia2 = p(tti0^2)? so p(aia2) = p{ai)p{a2) by uniqueness. It follows that the restriction of p to any subloop of normalized torsion units is a homomorphism; moreover, this restriction is onetoone because \{ p{a) = 1, then a = 1 because 1 is central. We remark, in passing, that these comments essentially provide another proof of the isomorphism theorem for alternative integral loop rings of torsion RA loops. 4.11 Lemma. Let L be a finite RA loop with L' = {1,^} and let H be a finite subloop ofU\{TL). Let a^/3^j be elements of H. If the loop commutator (a,/?) is not I, then (a,/3) = s and, if the loop associator (a,/3^j) is not I, then (a,/?,7) = s. If H is not commutative, then Z{H) C Z[L). Let p: H —^ L he the homomorphism discussed above. Since p{a^f5) = (/>(a),/9(/3)) and p is onetoone, (a,/3) / 1 implies p{a^(i) ^ 1; hence p{a^f3) == 5 is central. As noted, it follows that (a,/?) = p(a^/3) = s, A similar argument establishes the second statement of the lemma. If H is not commutative and a G Z(H)^ then p(a) is central in the subloop of L which is the image of p. By Lemma 4.9, p{a) is central in L, so a = p{a) G 2 ( 1 ) . D PROOF.
4.12 Lemma. Let H be a finite subloop of U\{7L) containing s and let p: H ^ L be the homomorphism described in ^4^0, Then H* = H and p commutes with *. If a G ^ is central, then it is trivial, so a* = a G fl^. If a £ H \s not central, then it has no central elements in its support, by Corollary VIII.1.2, so a* = 5a G H. In any event, ^ * C H, so 7?* = H. Let a E H, If a is central in ZL, then a is trivial, so p{a) = a and p(a*) = p{a) = a = a* = />(<^)* If (^ is not central, then a* = sa and also p(a) is not central. So again p{a*) = p{sa) = p{s)p(a) = sp{a) = />(«)*• Thus p commutes with *. D PROOF.
4. SOME CONJECTURES OF H. J. ZASSENHAUS
243
For alternative loop rings which are not associative, the second and third conjectures of Zassenhaus take the following form. 4.13 Theorem. Let L be a finite RA loop. If H is a finite subloop of normalized units in ZL, then H is isomorphic to a subloop of L, Moreover, 1. if H is an abelian group, then there exists an inner map tj) ofQL such that tp{H) C L and the restriction of ip to QH is a monomorphism; 2. ifH is not associative, then there exists an inner automorphism ip of the rational loop algebra QL such that i^{H) C L; and 3. if H is a group, then there exists an inner automorphism ip of the complex loop algebra QL such that ip{H) C L. Let /T be a finite subloop of Ui{ZL), Since HZ{L) is also a torsion subloop of Ui{ZL)^ we may assume Z[L) C H^ so, if H is not commutative, we may also assume that 2{H) — Z[L)^ by Lemma 4.11. Let p: H ^ L he the homomorphism described in §4.10 and set Hp = p(H). Since p fixes Z{L)^ our assumption about H implies that Z[L) C Hp and, if H is not commutative, that Z[Hp) = Z{L). If H is an abelian group, then Hp is an abelian group contained in L. If H is central, H = Hp C L and there is nothing to prove. Thus we assume that H is not central and select h £ H and i E L such that (h^i) ^ 1. Since h is not central, hp = p(h) is an element of Hp which is not central in L. If x is any other element in Hp^ since xhp = hpX^ either hp or x or hpX is central since L has property LC. (See §IV.2.) In the latter case, x = x'^{hpx)~^hp is a central multiple of hp. It follows that Hp = Z{L) U Z{L)hp and that H — Z{L) U Z{L)h. By Theorem 4.7, there exists an inner map ^0 on QL such that ip(h) = hp — p{h). Clearly 0 fixes central elements of L, thus ip{H) = p{H) C L. This proves statement (1). Next we shall show that there exists an inner automorphism V^, of QL if H is not associative and of CL if ^ is a nonabelian group, such that PROOF.
(14)
i^{a) = p{a)
for dX\ a^ H. Assume H is not associative. Expressing QL as the sum of fields and CayleyDickson algebras, it is sufficient to show that in each simple component A = (QL)e of QL, e a primitive central idempotent, there exists an
244
IX. ISOMORPHISMS OF INTEGRAL ALTERNATIVE LOOP RINGS
inner m a p 'tpA such t h a t (14) is valid in t h a t component (for all a G ^ ) . Let TT: Q L ^ Ahe
the natural projection. Since, for any a £ H^ a = p(a)
(mod A ( L ) A ( L O ) and A ( L ' ) = Z ( l  5), we have ( 1 + 5)a = {l + s)p{a).
If
A is commutative, then 7r{s) = e, the unity of A, by Proposition VI.4.6, so 7r(a) = 7r{p{a)) and we can take ipA ^o be the identity m a p on A. It remains only t o consider components which are CayleyDickson algebras. Assume henceforth t h a t A is such a component; it follows t h a t 7r(s) = — e, again from Proposition VI.4.6. We know t h a t Z{L)
C H and t h a t H = H^ =
p{H)
is a subloop of L which is not associative. Since L is generated by
Z{L)
and any three elements which do not associate (Theorem IV.3.1), Hp = L. Thus H is isomorphic to L. Since the Qsubalgebra of QL spanned by H is isomorphic to Q ^ , by Proposition 4.8, clearly p extends to an isomorphism Q ^ ^ QL, t h a t is, to an automorphism of QL. Since p fixes 2{L)
and
hence, in particular, p{l — s) = 1 — s^ we see t h a t p defines an automorphism oi A which fixes its centre. (See Corollary III.1.5.) By Theorem 2.4 (Jacobson's Theorem), this automorphism of A is necessarily inner. This establishes statement (2), the case t h a t H is not associative. Assume t h a t H is a, nonabelian group. Expressing CL as the sum of fields (each isomorphic to C) and CayleyDickson algebras with centre C (each isomorphic to Zorn's vector matrix algebra over C), as before, it is sufficient to show t h a t , in each simple component A = {CL)e of CL, e a primitive central idempotent, there exists an inner map ipA such t h a t (14) is valid in t h a t component (for aD a G ^ ) . If A is commutative, we can again take tl)A to be the identity map on A, so assume A is not commutative. Again, 2{H)
= Z{L)C
H ^ Hp = p{H)C
L,so
L = Hp[jHpU
= M{Hp,^,u^)
for
some element u ^ L. By Corollary VI. 1.3, e is in CH and it is a primitive central idempotent of this algebra. know t h a t (CH)e
Also, because of Lemma VII.2.2, we
is a split quaternion algebra and hence isomorphic to
M2(C). (See §1.3.3 and Theorem 1.4.16.) Since p fixes central elements. Corollary VI. 1.3 shows t h a t p defines a Cautomorphism from {CH)e {CHp)e = M2(C) C (CL)e = A.
Let ^ 1 = {CH)e.
to
By Theorem VI.4.5,
the m a p n: CL —> CL defined by n(6) = 66* restricts to a nondegenerate quadratic multiplicative form on A and on Ai. As in the proof of Hurwitz's Theorem (Theorem 1.4.11), there exists ir e Ai, t h a t A = Ai QA\i\
el =  n ( ^ i ) ^ 0, such
with the usual CayleyDickson multiplication (see §1.3).
4. SOME CONJECTURES OF H. J. ZASSENHAUS
245
Replacing ii by ( \ / n ( ^ ) ~ ^ ^ i , we may assume i\ = —\, In a similar way, A = A2® ^2^2 with A2 = {CHp)e and 1^ = I, Finally note that A* C Ai for i = 1,2 and p commutes with *, by Lemma 4.12. It is straightforward now to verify that the map A = Ai Q Ai£i ^^ ^2 © >l2^2 = ^ defined by a + bii^ p{a) + p{b)i2 is an automorphism V^ of ^ which extends p and is inner, by Jacobson's Theorem. Finally, the proof of (3) in the case that H is an abehan group is similar to the foregoing, by essentially applying the methods of Hurwitz's Theorem twice. We leave this to the reader. D
Chapter X
Isomorphisms of Commutative Group Algebras
In Chapter IX it was shown that the isomorphism problem has a positive answer for integral loop rings of torsion RA loops. At this point, our primary interest is the isomorphism problem for loop algebras of finite RA loops. Thus, for finite RA loops Li and L2, and for a field F , we investigate whether the existence of an Falgebra isomorphism between FLi and FL2 implies that Li is isomorphic to Z/2. Since abelian groups play such an important role in the structure of RA loops, we first need to study this question for group algebras of finite abelian groups. The present chapter is devoted to that goal; in Chapter XI, we apply our results to loop algebras. Since familiarity with tensor products of fields is crucial to our study, we start this chapter by establishing some results in this area. In the second section, the isomorphism problem for semisimple abelian group algebras FG is investigated. An affirmative answer is given for rational group algebras and, in general, a reduction is given to the case where G is a pgroup for some prime p. These results are consequences of the decomposition of the algebra FG as a direct sum of cyclotomic fields. In Section 3, we direct our thoughts to the case where the characteristic of F divides the order of the group G. Such group algebras are called modular. It is shown that the isomorphism problem in this situation can be reduced to the semisimple case and to the case that G is a pgroup. In the fourth and last section, we determine when two fields F and K are equivalent on a class S of finite abelian groups; that is, we find conditions which imply that FG and FH are Fisomorphic if and only if KG and KH are A'isomorphic for all G, J? € S,
247
248
X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS
1. Some results on tensor products of fields In this section, we discuss the structure of a tensor product of the form E ®F K^ where E and K are fields both of which contain F as a subfield. We are interested mainly in the case where the fields involved are cyclotomic extensions of F , that is, extensions by a primitive root of unity. We begin by summarizing the essential ingredients of Galois theory, referring the reader to [Jac74, Chapter IV] and [Jac64] for details. A field extension E/F is said to be finite if the dimension [E: F] of JS, considered as a vector space over F , is finite. The extension E/F is algebraic if every element of F is a root of a polynomial with coefficients in F . A polynomial in F[X] is separable if it has no multiple roots. An algebraic extension E/F is separable if the minimal polynomial over F of each element of E is separable. A finite extension of F is normal if it is the splitting field of a polynomial in F[X], that is, the smallest subfield of the algebraic closure of F containing all the roots of some polynomial. A finite extension E/F is Galois if it is both normal and separable. An Fautomorphism of E is an automorphism of E which fixes all the elements of F . The Galois group of the extension EjF^ denoted G a l ( F / F ) , is the set of all Fautomorphisms of F . A finite extension E/F is Galois if and only if [ F : F] = I G a l ( F / F )  and the set of elements of E which are fixed by aU the automorphisms of G a l ( F / F ) is precisely F . It is easy to see that if EjF is Galois and K is any field containing F such that both E and F are contained in a common field fi, then the composite field K{E) (defined in fi) is Galois over K. If not explicitly specified, the field fi will always be clear from the context. As in Chapter VII, we shall need to work with roots of unity. We recall the main definitions in the general case. So let F be an arbitrary field. An element ^ G F which is a root of the polynomial X"^ — 1 is called an nth root of unity \ if ^ has order n in the multiplicative group of F , then ^ is said to be a primitive nth root of unity. For n > 1, a primitive nth root of unity over F means a primitive nth root of unity in a splitting field of X'^ — 1 over F . Unless specified otherwise, such an element is consistently denoted ^n If char F does not divide n, a primitive nth root of unity over F always exists and F{^n)lF is a Galois extension called the nth cyclotomic extension
1. SOME RESULTS ON TENSOR PRODUCTS OF FIELDS
of F,
249
The elements i\^ I < i < n^ g c d ( i , n ) = 1, constitute precisely the
primitive n t h roots of unity in
F{^n)'
Let
l
Since any a 6 G a l ( i ^ ( ^ n ) / ^ ) maps ^^ to a primitive n t h root of unity, it follows t h a t * n (  ^ ) is invariant under each a and therefore $ n (  ^ ) has coefficients in F, We call $ n ( ^ ) the cyclotomic
polynomial
of order n and
note t h a t $ n (  ^ ) is a polynomial of degree <^(n) where, as always (f) denotes the Euler function. Recall t h a t the prime field of a field F is Q if char F = 0 and the field Fp of p elements if char F = p ^ 0. 1.1 L e m m a . Let F be a field containing
a subfield P. Let ^ be a
root of unity over P whose order is not a multiple o/char P, Then is isomorphic PROOF.
to a direct sum of[P{^):
P]/[F{^):
F] copies of
primitive F®pP(^)
F{^).
Let n be the order of ^ and let / G P[X] be the minimal (monic)
polynomial of ^ over P. Then / is irreducible and P{^) ^ P[X]/(f),
Thus
(See §1.5.11.) Let / = /1/2 • * • /t be the decomposition of / as a product of irreducible (monic) polynomials in F [ X ] . The assumption on the characteristic of P implies that the roots of X ^ — 1 are distinct. Thus / is separable and hence f zji f if i ^ j . Using the Chinese Remainder Theorem, we can write
\h) Now F[X]l{fi)
\Jt)
= F(0i)^ where Oi is a root of fi and thus also a root
of / , for 1 < i < t.
Since the roots of / in P{()
are all primitive n t h
roots of unity over P , each 0i is a primitive n t h root of unity over P. Thus F{ei) ^ F ( 0 . It follows t h a t
(1)
F®pP{0 =
FiO®'[email protected]{0"
>/ t summands
'
250
X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS
Since [F ®p P{^): of (1) is t[F{^):
F] = [P{0 • ^ ] ^ ^ ^ the dimension of the right hand side
F], we have t = [P(^):
P]/[F{0'
F] as required.
D
1.2 L e m m a . Letp be a prime and let F be a field of characteristic from p. If the polynomial in
X^ — a has no root in F, then X^ — a is
different irreducible
F[X]. PROOF.
Write X^  a = f{X)g{X),
where f,g
e F[X] and / is irre
ducible. Working over the splitting field of / , we have f{X)
= (X
a i ) ( X a2)'"{X
am)
for some integer m < p. Since each of the roots of / is also a root of X ^ — a, we have a^^ = a and, with C,i = a^^ai^ it is clear t h a t (f = 1 for 1 < i < m. Let b = aia2 '' am Then b £ F and 6^ = ^^1^2 '' 'O^ = CL^ If m < p^ we can find integers r and s such t h a t rp + sm = 1, so a = a^'^a'^^. Since ^5771 _ f^sp^ ^^ hdi^we a = {b^d^y X ^ — a has no root in F.
with b^a^ G F , contradicting the fact t h a t
Consequently m = p and X^ — a = f{X)
irreducible in F [ X ] .
is D
1.3 C o r o l l a r y . Let p be prime and let F be a field of characteristic ent from p which contains a primitive
pth root of unity.
differ
Let ^ be a pHh root
of unity over F. Then [ F ( ^ ) : F] is a power of p. PROOF.
Consider the following sequence of field extensions: F = F{e'~')
C F{e'~')
C . . . C F{e)
C F(0.
Since F contains a primitive pth root of unity, F contains all pth roots of unity.
By Lemma 1.2, it is easy to see t h a t for each j , j = l , . . . z , the
dimension
[F{^P^
) : F{^^^)] is either 1 or p. The result follows.
1.4 L e m m a . Let ^ be a primitive acteristic
D
rth root of unity over a field F of char
relatively prime to r. Then [F{^): F] is a divisor of (t>{r).
PROOF.
Let a G G a l ( F ( ^ ) / F ) .
Since (7(^) is also a root of unity of
order r , ( J ( ^ ) = ^* for some rational integer t relatively prime to r. It is easy to see t h a t the mapping a »^ ^ is a monomorphism from G a l ( F ( ^ ) / F ) into the multiplicative group Z* of invertible elements of the ring of integers modulo r. Since Z* has order (r), we see t h a t  G a l ( F ( ^ ) / F )  is a divisor of (r). Since F ( 0 is Galois over F , [ F ( 0 : F] =  G a l ( F ( 0 / F )  and the result follows.
D
1. SOME RESULTS ON TENSOR PRODUCTS OF FIELDS
251
1.5 Lemma. Let E and N be subfields of a field il and suppose E is a finite Galois extension of a field F. Then [E{N): N] = [E: ED N]. Let B = { e i , . . . ,6^} be a basis of E over E O N, Since B generates E{N) over iV, we have only to show that it is also linearly independent. Assume, by way of contradiction, that B is linearly dependent over N and let { e i , . . . ,6^^}, m < n, be the smallest subset which is still linearly dependent over N, Then there exist elements a i , . . . , a ^ in TV, not all 0, such that ^ ^ 1 aiei = 0. Without loss of generality, we may assume that ai = 1. Let a G Gal(E(N)/E), Since a is an Fautomorphism and since N/F is normal, we have cr(ai) £ N for 1 < i < m. Moreover, a{ai) = ai (since PROOF.
aj = 1), so m
^{ai
 cr(a,))e, = 0.
t=2
The choice of m implies that a^ — (T(ai) = 0 for z = 2 , . . . ,m. This shows that each coefficient a^ is fixed under any element a G G3l(E(N)/E). So a, G ^ n TV for 1 < i < m, a contradiction, since iB is a basis of E over EON, D 1.6 Corollary. Let E and N be subfields of a field SI and suppose E is a finite Galois extension of a field F. Then E(N) = E ®EnN ^ • It is easy to see that e ® n y^ en defines a homomorphism from E ®EnN N onto E{N), Also [E ®EnN N: N] = [E: En iV], so, by Lemma 1.5, E{N) and E^EHNN have the same dimension as vector spaces over A^. The result follows. D PROOF.
1.7 Theorem. Let p be a prime and let F be a field of characteristic different from p which contains a primitive pth root of unity. Let ^ps and ^r be roots of unity over F of orders p^ and r, respectively^ with gcd(p, (r)) = l = gcd(p,r). Let E = F{^ps), Then E ®F H^r) = E{^r) and [E{^r)' E] = [F{^r):F], Let h = [F{^r) n F{^ps): F]. Then /i is a divisor of [F(^^): F] and, since gcd(p, r) = 1, also a divisor of (r), by Lemma 1.4. On the other PROOF.
252
X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS
hand, h is also a divisor of [F(^ps): F] which, by Corollary 1.3, is a power of p. Since gcd(;>,^(r)) = 1, we see that h = I, Thus F{^r) H F{^ps) = F. From Corollary 1.6, one obtains
and, by Lemma 1.5, [^(^r): E] = [F{Cr): F].
D
2. Semisimple abelian group algebras In this section, we survey some of the more important results concerning the isomorphism problem for semisimple group algebras FG over finite abelian groups G; so F is a field whose characteristic does not divide \G\. Throughout, an isomorphism between group algebras FG and FH means an Fisomorphism. For group algebras over fields, the isomorphism problem was first formulated by T. M. Thrall at the Michigan Algebra Conference in the summer of 1947. The first results on this subject appeared in a well known paper by S. Perils and G. L. Walker [PW50]. We begin our study with the first theorem of that work, a description of the Wedderburn decomposition of a commutative group algebra, though we shall give a different proof. Recall that in Chapter VII, we solved this problem for rational group algebras. 2.1 Theorem (Perils—Walker [PW50]). Let G be a finite abelian group of order n and let F be a field whose characteristic does not divide n. Then
where a^ = ndl[F{^d)'' F]^ n^ denotes the number of elements of order d in G and ^d denotes a primitive dth root of unity over F. We proceed by induction on n. Assume first that G = (a) is cyclic and consider the map tp: F[X] ^ FG given by if^{f) = /(tt) It is easy to see that ^ is a ring epimorphism. Hence, PROOF.
ker ip Since F[X] is a principal ideal domain, kerV^ = (/o) is the ideal generated by the monic polynomial /o of least degree such that fo{a) = 0.
2. SEMISIMPLE ABELIAN GROUP ALGEBRAS
253
Clearly a^ = 1 impUes X^  1 € kerV^. Also, if f{X) = E L o ^ t ^ ' is a polynomial of degree r < n^ then / ( a ) = Si=o ^i^^ T^ ^ because the elements { l , a , a ^ , . . . ,a^} of FG are linearly independent over F, Thus kerV^ = (X^ — 1) and, therefore, FG =
F[X] (X"l)'
We know that X " — 1 = JJ \ $d(X), the product of all cyclotomic polynomials $d(X) in F[X]. For each d dividing n, let $d(X) = n":^i fd,{X) be the decomposition of $d(X) as the product of irreducible monic polynomials in F[X]. Since char F )f n, the polynomial X " — 1 is separable, hence the polynomials fj are relatively prime. By the Chinese Remainder Theorem,
where (dt denotes a root of /^^, 1 < i < adFor a fixed rf, all the elements Q^ are primitive roots of unity of order d; therefore, all the fields F(Ccf,), 1 ^ ^ ^ ^d? ^.re isomorphic, so
where ^d is a primitive root of unity of order d and adF(^d) denotes the direct sum of ad copies of F(^^). Since degfd^ = [F{^d) F]^ all the polynomials fd^ have the same degree. Since ^d{X) is a polynomial of degree (d),
ai[FiU):F].
Since G is a cyclic group of order n, for each divisor d of n, the number rid of elements of order d in G is precisely (d); hence.
This completes the case that G is cyclic. Now assume that the result holds for all abelian groups of order less than n. If G is cyclic, we have already shown that the proposition is valid. Otherwise, G = Gi X H^ where H is cyclic and Gi = rii < n. By the induction hypothesis, FGi = 0 i ad^F{^di), where a^^ = ndJ[F{^di)' F] aj
Til
and Ud^ denotes the number of elements of order di in Gi. Hence FG = F[Gi xH] = {FGr)H ^ 0 . ,
ad,F{U,)H.
254
X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS
Applying the first part of the proof to the group algebras F{^cii)H of the cyclic group H ^ we obtain
where ad^ = «d2/[^(^''i'^rf2) ^i^di)] and nd2 denotes the number of elements of order d2 in H. Let d = lcm(di,d2). Then F{^di,U2) = F{U), so
flrf = Y^C'diO.d^, the sum taken over all pairs di, ^2 such that lcm(di,d2) = d. Since [FiU): F] = [FiUr,^d,): F{U,)][F(Ur)• F], it foUows that
ad[Fi^d): F] = ^
a^. a,jF(^d,, ^^J : Fi^.Wi^d,):
d\^d2
F] = J^ ^d.Ud,. d\,d2
Since each element g ^ G — G\ x H can be written uniquely in the form g = g^h^ with gi G Gi and h e H^ and because o(^) = lcm(o(yi),o(/i)), it is clear that ^ ^ ^ '^di'^d2 — ^cf, the number of elements of order d in G, Hence _ and the result follows.
n^ D
Our proof of the result of Perils and Walker concerning the isomorphism problem for rational group algebras will require the following fact. 2.2 Lemma. Let S,di (^f^d ^d2 ^^ primitive roots of unity of order di, ^2, respectively J with di > ^2 Then Q(^di) = Q{^d2) if cirid only if di = ^2 or di = 2^2 with ^2 odd. Since [Q(^d): Q] = (l>{d) for any primitive dth root of unity over Q and since (j){di) = (f>{d2) implies di = ^2 or di = 2^2, the result is clear. D PROOF.
2.3 Theorem (PerlisWalker [PW50]). LetG be a finite abelian group. If H is any other group such that QG = QH^ then G = H.
2. SEMISIMPLE ABELIAN GROUP ALGEBRAS
255
We follow the argument of [Seh78, Theorem III.2.12]. Set n = G. If QG = QH^ then H is also a finite abelian group of order n so, by Theorem 2.1, PROOF.
where ^d denotes a primitive dth. root of unity, ^d = 77^777—7^5
^d =
and n^, rrid denote, respectively, the numbers of elements of order cJ in G and in H, So ad (respectively bd) equals the number of cyclic subgroups of order d in G (respectively H), Since Q(^d,) — Q if and only if d^ = 1 or 2, it follows that ai + a2 = &1+62. Since ai = 61 is always the case, we also obtain a2 = 62 For a 2^th root of unity, r > 2, we have by Lemma 2.2 that Q(^2^) = Q{^d) if ^^^ only if d = 2^. Thus the isomorphism of QG and Q ^ impUes that 02^ = 62'"• On the other hand, if p is an odd prime. Lemma 2.2 implies that if Q(^pr) = Q(^ci), then either d = p^ or d = 2p^. Therefore apr + a2pr = bpr + b2pr. Now a cyclic group of order 2p^ can be written uniquely as the direct product of a cyclic subgroup of order 2 and a cyclic subgroup of order p^. Hence a2pr = a2apr^ so our equality now reads apr (I + a2) = 6pr(l + ^2) Thus apr := bpr.
We have shown that, for any power p'^ of a prime dividing n, both G and H have the same number of elements of order p'^. This implies that the pprimary components of G and H are isomorphic for all p^ so G = H. D We now turn our attention to the isomorphism problem for finite abelian groups over other fields. If G is a finite abelian group. Theorem 2.1 shows that CG is a direct sum of \G\ copies of C. Hence CG = C ^ whenever G and H are two abelian groups of the same order. This shows quite clearly that the answer to the isomorphism problem for group algebras over fields depends critically on the field of coefficients under consideration. In order to give further results, we shall reduce the study of the isomorphisms of group algebras of finite abelian groups to the study of isomorphisms among the group algebras of their corresponding pprimary components. This was also done in [PW50], but there is an oversight in the arguments given there. Independent corrections to this gap appeared in
256
X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS
papers by R. Bautista [Bau67] and D. E. Cohen [Coh68]. Our proof will follow the latter, but first, we need a lemma. 2.4 Lemma. Letp be a prime and let K be a field of characteristic different from p. For n > 1, denote by ^pn a primitive p'^th root of unity over K, If [A^(^pn+i): /ir(^pn)] = p and n>lifp = 2, then [/ir(^pn+r+i): K{^pn\r] = p for every positive integer r. It will suffice to prove that [A'(^pn+2): A^(^pn+i)] = p since the full statement will then follow by induction. Clearly we may assume K{^pn) = K and thus [A"(^pn+i): A"] = p. Let a i = (^pn+2)P and a2 = (ai)P. Then a i is a primitive p^'^^ih root of unity and a2 is a primitive p^th root of unity. So a2 € A^, a i ^ K and a i G A^(^pn+i). Since 02 is not the pt\\ power of an element in A', it follows from Lemma 1.2 that f{X) = X^  OL2 is irreducible in K{X\. By Corollary 1.3, {K{ipn^i)\ A^C^^n+i] is 1 or j9, so it is sufficient to show that ^pn+2 ^ A'(
2. SEMISIMPLE ABELIAN GROUP ALGEBRAS
257
K[X] if and only if X^ — a2 is irreducible in A^[X]; however, —a2 = A;^, so A; is a root of X^ + a2? giving again a contradiction. D 2.5 Theorem. Let G and H he abelian groups of the same order n and, for each rational prime p dividing n, denote by Gp and Hp the sets of pprimary components of G and of H respectively. Let F be a field whose characteristic does not divide n. Then FG = FH (as Falgebras) if and only if FGp = FHp for all primes p dividing n. PROOF.
If G = Gp^ x Gp^ x • • • x Gp^, then, by Lemma VII.0.1, we have FG ^ FGp, ® FGp, ® • • • ® FGp,,
so the conditions of the theorem are clearly sufficient. To prove the converse, we will show that FG determines FGp for each prime divisor p of n; that is, that FG = FH for some other group H implies FGp = FHp. Write G = Go X Gp X Gi, where Go is the product of all the g^primary subgroups of G with q < p and Gi is the product of all the g'primary subgroups with q > p. Then FG = FGo ® FGp ® FG\. We prove the result by induction on the number of prime factors of Gi (so, for the first step in the induction process, Gi will be trivial). It is sufficient to establish two facts. Fact 1. Let ^ be a primitive root of unity over F of order p and let K = F ( 0 . Then KGp determines FGp, If A^ = F , that is, if F contains a primitive pth root of unity, this is obvious. So assume A' 7^ F; in particular, p is odd. Using Theorem 2.1, write ^Gp = ®,>o ««^(6)
and
KGp ^ e . > , 6,F(6),
where ^i denotes a primitive root of unity of order p\ By Lemma VII.0.2, KGp ^ K ®F FGp ^ 0 . > o a,F{0 ®F H^t) and, for i > 1, F{^) ® F{(i) ^ [A^: F]F{^i), by Lemma 1.1. Then, by Lemma 2.4, we see that 61 = ao + ai[A^: F], where ao = 1, and bi = ai[K: F] for z > 1. Clearly the coefficients bi determine the coefficients a^ for alH > 1 and KGp determines FGp,
258
X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS
Fact 2. Let \Go\ = T^ let 9^ he Si primitive rth root of unity and let E = K{9r). Then EG determines KGp. (Note in the sequel that we may replace r by any multiple which is a product of primes q with q < p.) With notation as in Fact 1, write KGp = 0^>i biF{^i). As shown in Corollary 1.3, [A^(^^): A'] = p'^ for some m. We denote by K^ the cyclotomic extension of K whose dimension over K is precisely p'^. Then we can also write KGp in the form KGp = 0^>o CiKi for some integers Q . Again, we have EGp = 0^>o CiE®K Ki. Because of the choice of Go, we have gcd((/>(r),p) = I = gcd(r,p) so, using Theorem 1.7, each Ei = E®K Ki is a field and EGp = 0^>o ^iEi. Theorem 2.1 shows that EGQ = TE^ SO, by Lemma VILO.l, EG ^ EGo ®E EGp ®E EGi ^ vE ®E ( © , > O c^^^) ®E
EGL
Since we are assuming that FG\ is determined by FG, we have that EGi ~ E ®F FG\ is determined by EG. Assume that a tensor product of the form Ej ® EG\ contains Sji summands isomorphic to Ei. Clearly Sji = Q\i j > i and Sii > 1, because EG\ contains at least one summand isomorphic to E. Thus Ei®E EG\ will again contain at least one summand isomorphic to Ei. Let fii be the number of summands in EG which are isomorphic to Ei. Then I5i = r{ciSii + Y^j^iCjSji)^ so we can determine the integers Ci inductively from this formula. Fact 2, and hence the result, follow. D
3. Modular group algebras of abelian groups In this section, we consider isomorphisms of a group algebra FG, where F is a field of prime characteristic p and G is a finite abelian group whose order is divisible by p. This subject was first studied by W. E. Deskins in [Des56]. We begin by showing that a finite abelian pgroup is determined by its group algebra over any field of characteristic p. We introduce some notation. For a given group G, define
G"" = {x^\xe
G}.
If G is an abelian group, then G^ is a subgroup and it follows from the decomposition theorems for finite abelian groups that two abelian groups G and H are isomorphic if and only if G^ ^ H^ and G/G^ ^ H/m.
3. MODULAR GROUP ALGEBRAS OF ABELIAN GROUPS
259
For a commutative algebra A over a field F of characteristic p > 0, we denote by A^ the set of all elements a'^, a £ A, Clearly this is an algebra over the field F^. As noted at the beginning of the proof of Theorem IX. 1.1, if FG = FH as Falgebras, there exists a normalized isomorphism ip: FG ^ FH; that is, an isomorphism FG ^ FH such that €('tp(g)) = 1 for any g ^ G. 3.1 Theorem. Let F be a field of characteristic p > 0 and let G and H be finite abelian pgroups. Then FG = FH (as Falgebras) if and only if G^ H, Let FG = FH. Then 1^1 = \H\ = p"" for some n and we prove the theorem by induction on n. If n = 1, both G and H are cyclic groups of order p and hence isomorphic. So assume that the result holds for pgroups whose order is less than p'^. Certainly FG ^ FH implies that (FGy ^ (FHy and, since (FG^ = F^GP, we obtain F^G^ ^ F^H^. Since F^ is a field and charF^ = p, our induction hypothesis implies that G^ = H^. On the other hand, if xp: FG ^ FH denotes any normalized isomorphism, then ip{g^ — 1) = i^igy  1 e A{H,HP), Thus V ^ ( A ( G , G P ) ) C A{H,HP), Working with ip~^ we obtain the reverse inclusion, so 'tp(A{G^G^)) = A{H^ H^), Hence '^ induces an isomorphism of the respective quotient rings and we have PROOF.
F[G/G^] ^ FG/A{G,G^)
^
FH/A{H,HP)
^
F[HIH^].
Since  G / G P  < G, the induction hypothesis yields that GJG'P = H/H^ and thus, as noted at the beginning of this section, G = H, The converse is trivial. D To study the general abelian case, notice that if G is an abelian group and p is a prime dividing G, then G = P X A^ where P is a pgroup and yl is a group such that p jf \A\. We shall show that if Gi = Pi X Ai and G2 = P2 X A2 are two abelian groups whose orders are divisible by p and which are decomposed as above, then an isomorphism FGi = FG2 splits into two others; namely, FPi = FP2 and FAi = FA2, We follow the proof of [PM78]. 3.2 Lemma. Let F be a field of prime characteristic p and letG = PxA be a finite abelian group, where P is a pgroup andp](\A\. Thenl + A(G,P)
260
X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS
is a pgroup of exponent p'^ for some integer m > 0 and U{FG) PROOF.
= (1 + A ( G , P)) X
U{FA),
Clearly the set 1 + A ( G , P) is closed under multiplication so, to
prove t h a t it is a subgroup, it suffices to show t h a t its elements are invertible with inverses contained in 1 + A ( G , P ) . For this, let u G A ( G , P ) . Since A ( G , P ) is nilpotent, by Proposition VI.2.6, there exists a positive integer t such t h a t A(G', P)^ = 0. Let n be an integer such t h a t p^ > t.
Then
(1 + u)^" = 1 + u'P'' = 1, so 1 + t^ is an invertible element of 1 + A ( G , P ) , of order a divisor of p ^ . Hence 1 + A ( G , P ) is a pgroup of exponent p ^ for some m > 0. An element in U{FA)
has in its support no elements of the form xa
with 1 / a: G P and a G >1, while every nontrivial element in 1 + A ( G , P ) necessarily contains at least one such element in its support.
Therefore
ZY(F/I)n(l + A ( G , P ) ) = { 1 } . To prove t h a t U{FG)
is actually the product of 1 + A ( G , P ) and
consider an arbitrary element a G U{FG). momorphism FG ^ F[G/P] u = a^a
Denote by ep the natural ho
=. FA and let a i = €p{a).
as an element of P C , we have QI G U{FG)
U{FA),
Thinking of a\
and ep{a'[^a)
— 1.
Hence
 1 G A ( G , P ) and a = (1 + u)a^ G (1 + A ( G , P ) ) X U{FA),
desired.
as D
3.3 T h e o r e m . Let F be a field of prime characteristic
p and, for i — 1,2,
let Gi = Pi X Ai be finite abelian groups, where Pi is a pgroup and Ai is a group such that p )( \Ai\. thus Pi ^ P2) and FAi ^ PROOF.
Then FGi = FG2 if and only if FPi = FP2
(and
FA2.
Assume t h a t FGi
= FG2 and let ip: FGi
> FG2 be a nor
malized isomorphism. By Lemma 3.2, 1 + A ( G 2 , P 2 ) has exponent p ^ for some m > 0 and thus a^
e U{FA2)
for every element a G ZY(PG2)
Since p )( \A\\^ the group homomorphism A\ ^ A\ defined by a H> a^"^ is onetoone and onto. Thus, for every a G ^ 1 , there exists b ^ Ai such t h a t a = b^ . Since iP{a  I) = ^{b^^  1) = ij(by^
 1
4. THE EQUIVALENCE PROBLEM
and since '0(6)^" G U{FA2),
it foUows t h a t
V^(6)P"
261
 1 G A(G2, A2). This
shows t h a t ' 0 ( A ( G i , A i ) ) — A(G27^2) ^ind therefore ^^ induces an isomorphism of the respective quotient rings. Thus
^ FG2IHG2IA2)
^ F[G2lA2\
^
On the other hand, if a G 1 + A ( G i , P i ) , then V^(a) G U{FG2)\ Lemma 3.2, '0(a) = xt/ with x G 1 + A(G2,P2) and y G U{FA2).
FP2. thus, by Also,
by Lemma 3.2, there exists n > 0 such that a^" = 1. Hence xl^^aY"^ — xv""yv""  i^ ^Q j.p''  yv"" 
\^ Writing y  ^aeA2
^«^' ^^ ^ ^ ' ^ ^ ^^^^
yP" _ ^ ^ ^ ^ Ta a^" = 1. Since ;? J^ yl2, it is easy to see t h a t y^" = 1 if and only if y = 1. Thus jp{a) = x G 1 + A(G2, F2). Hence V^(A(Gi, P i ) ) = A(G2,P2) and, similarly, P ^ i ^ P ^ 2 . The converse is an immediate consequence of Lemma VILO.l.
D
4. T h e e q u i v a l e n c e p r o b l e m In the previous sections, we saw t h a t the rational group algebra of a finite abelian group G determines G in the sense t h a t , if H is another finite abelian group such t h a t QG = Q/T, then G = H.
Similarly, we showed t h a t the
modular group algebra of an abelian pgroup determines the group. In this section, we carry this investigation further, developing tools which allow us to decide which other fields have similar properties. We begin by introducing a new concept. 4.1 D e f i n i t i o n . Let S denote a family of finite groups. We say t h a t two fields F , A^ are equivalent
on S if, for any G^ H £ S^ it is the case t h a t
FG ^ FH if and only if KG ^
KH.
This definition is due to E. Spiegel [Spi75], who first studied the equivalence problem in the class of finite abelian groups and who later, with A. Trojan [ S T 7 6 ] , extended this study to the class of finite 2groups. In the next chapter, we shall apply these results to the study of the isomorphism problem for loop algebras of finite RA loops. We first extend Theorem 2.5.
262
X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS
4.2 Theorem. Let G and H be abelian groups of the same order n and, for each rational prime p dividing n, denote by Gp and Hp the pprimary components ofG and of H respectively. Let F be any field. Then FG = FH if and only if FGp = FHp for all p dividing n. In the case c h a r F jf \G\^ this is precisely Theorem 2.5, so assume c h a r F = q  G. Write G = Gq X A^ where Gq is the gprimary component of G and q )f \A\^ and write H = Hq x B^ where Hq is the ^primary component of H and q )f \B\. By Theorem 3.3, FG = FH if and only if FGq ^ FHq and FA ^ FB. The result then follows from Theorem 2.5. D PROOF.
4.3 Definition. Let K be a field and p a rational prime. Let ^pn denote a primitive p'^th root of unity over K and set 7/c,p(n)[/i'(^p.+i):A'(V)]The psequence of K is the sequence {7A",p(^)}n=i,2,...These sequences can be neatly characterized. 4.4 Theorem. Let p be a prime and let K be a field of characteristic different from p. If p is odd, then the psequence of K has one of the forms 1,1,1,... 1,1,1,... , l,p,p,... p,p,p,... while 2sequences have similar forms except that the first terms can be arbitrarily I or 2. PROOF.
This follows at once from Corollary 1.3 and Lemma 2.4.
D
We now exhibit two concrete situations which show that, in the case p = 2, the arbitrary choices of 1 or 2 in the first term of a 2sequence can, indeed, occur. 4.5 Examples.
(i) Let K = R, the field of real numbers. Then [/v(62):A'(6)] = [R(0:R] = [C:R] = 2,
while
[/ai2n+0:/i'(6n)] = [C:C] = l
4. THE EQUIVALENCE PROBLEM
263
for all n > 1. Thus the 2sequence of K is 2 , 1 , 1 , — (ii) Let K = Q{V2).
Then
[Ki^2^): and, since ^2^ = ^{l
^  ( 6 ) ] = [Q{V2,i):
Q(x/2)] = 2
+ i)^ we have
It is now easy to see t h a t the 2sequence of A^ is 2 , 1 , 2 , 2 , Given a finite abelian group G of order p'^ and a field K of characteristic different from p , we have, by Theorem 2 . 1 ,
where ^p, denotes a primitive root of unity of order p^ and a^ = '^i/vi^ rii the number of elements of order p^ in G and Vi = [K(^px): K]. Notice t h a t ao = 1 and a^ = 0 if G contains no element of order p\ Associated with the field Ii\ a sequence {Q;n}n=o,i,2,..o can be defined as follows: set QQ = 1 and, for n > 0, let a^f 1 be the least integer r such t h a t Vr > '^a„ when this integer exists; otherwise, set a^+i = (^n\2 =    = oc. With this definition, we have A'(^pan) = A'(^pan+i) = ••• = A'(^ a„^.ii) if anfi 7^ 00, while if a^^ 7^ 00 and a^i+i = 00, then A'(^pan) = ^^(^p^) for /j >
a^.
We define another sequence of numbers associated with the group algeb r a KG.
Set bm = <
dam + ^ « m  h l H
^ « « m + l  l
0
if « m /
OO
if a ^ = 00.
Then
I
264
X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS
then KG = KH if and only if bi = 6^ for all i = 1 , . . . ,n; equivalently, if and only if the numbers of elements of order at most p^' in G and in H are equal. 4.6 Theorem. Letp he a prime and let K and F be fields of characteristic different from p. Then K and F are equivalent on the class S of all finite abelian pgroups if and only if they have the same psequence. PROOF.
Suppose K and F have the same psequence and, for ^ > 1, set VK{s) = [K{^^sy.K]
and
VF{S) = [F{^j,s): F].
Then 51
VK{S) = VK{\)
J J 7A>(0
51
^nd
VF[S) = VF{\)
1=1
J ^ 7F,P(01=1
Hence, for 5 > 1, VK{S) — \VF{S)^ where A = 't'A"(l)/'^F(l)Let G and H be abelian groups of order p^ •> n > 1, and suppose that FG ^ FH. We have to show that KG ^ KH. Write FG ^ ®:=ob^Fi^,.)
and
FH ^
etoKnCp^)
Then KG ^K®
e;^^, A6,A'(^pO
and
KH ^ A' 0 0 : ^ , , Xb[K{^^.).
Since FG = FH implies bi = 6' for all i = 1 , . . . , n, we have Xbi = Xb[ for all z; thus KG ^ KH. To complete the proof, we show that if the psequences of F and K differ, then there exist finite abelian groups G and H such that KG =. KH but FG^ FH. Assume that 7 F , P ( ^ ) 7^ lK,p{'^) foi* some positive integer n. First, consider the case where p is odd, or p = 2 with n > 1. Because of Theorem 4.4, we can assume, without loss of generality, that 7 F , P ( ^ ) = V and 7 A > ( ^ ) — 1Let G — Cpn+i and H  Cj,n x Cj). Then KG = KH since G and H have the same number of elements of order at most p'^ for r < n^ but FG ^ FH since G and H do not have the same number of elements of order at most
4. THE EQUIVALENCE PROBLEM
265
Now consider the case where p = 2, n = 1 and assume t h a t we have two fields F and K whose respective psequences are 2,1,1,...
and
1,1,1,... .
Let i denote a 4th root of unity and ^ an 8th root of unity. T h e hypotheses on the psequences imply t h a t [F{i): F ( —1)] = 2 and [K(i): so i e K but i i
F,
Since [ F ( 0 : F{i)\
F ( 0 = F{i) and A^(0 = K{i).
A^ —1)] = 1,
= [ A ^ O  ^^XO] = 1, we have
Let G  Cg and 77 = C4 x C2. Applying
Theorem 2.1, it is easy to see t h a t FG^2F®
3F{i)
and
KG ^ 8A^
FH ^AF®
2F(i)
and
KH ^
while
Hence, KG = KH
8K.
but FG ^ FH^ so F and K are not equivalent over
s.
a In the next chapter, we shall be interested especially in finite abelian
2groups. In order to work with such groups, we introduce here some isomorphism invariants of a field. 4 . 7 D e f i n i t i o n . Let K be a field. We define 1 ind2(A) = {n
if
00
0{K) =
t{K)
if7A',2(l) = 2 7A',2(^)
= 2 and
7A',2(^
if 7 A ' , 2 ( ^ ) = 1 for n >
 1) = 1 for n > 2
1,
1
if x^ + 1 = 0 is solvable in K
0
if x^^ + 1 = 0 is not solvable in A',
1
if x^ + y^ = — 1 is solvable in K
0
if x'^ + y^ = — 1 is not solvable in K.
With these definitions, Theorem 4.6 readily gives the following. 4.8 T h e o r e m . Let K and F be fields of characteristic
different
from 2.
Then K and F are equivalent on the class S of all finite abelian 2groups if and only ifO(K)
= 0(F)
and md2(K)
= ind2(F).
266
X. ISOMORPHISMS OF COMMUTATIVE GROUP ALGEBRAS
As mentioned earlier, E. Spiegel and A. Trojan extended this study to the class of all finite 2groups [ST76]. We quote their result here since it will be interesting to compare with its extension to 2loops, which we shall give in the next chapter. We omit the proof because we shall not need the result directly in our work. 4.9 Theorem. Let K and F be fields. Then K and F are equivalent in the class S of all finite 2groups if and only ifO{K) = 0(F), ind2(A^) = md2{F) andt{K) = t(F),
Chapter XI
Isomorphisms of Loop Algebras of Finite R A Loops
In this chapter, we turn our attention to the isomorphism problem for alternative loop algebras over fields. Given a finite RA loop L and a field F such that c h a r F )( L, we saw in Chapter VI that the loop algebra FL is a direct sum of ideals which are simple rings. We shall refer to this situation as the semisimple case. On the other hand, if char F  Zy, then such a decomposition does not exist; we call this the modular case. Since the techniques involved in the study of each of these cases are quite different, we deal with them separately. As in the previous chapter, all isomorphisms between loop algebras over the same field F are Falgebra isomorphisms. 1. S e m i s i m p l e l o o p algebras We begin with an easy lemma. 1.1 L e m m a . [dBM95a, Lemma 2.5] Let L\ and L2 be RA loops and let F be any field such that FLi ^ FL2. Then FlL^/L'^] = ^[1211'^] and A{LuL[)^A(L2.V^). Let ei^f : FLi ^ F[Li/L\] be the natural epimorphism. By Lemma VI.1.1, ker 6^/ = A(//i, L[) = F//i(l —5), where L' = {1, ^s}. Denote by [FLi^ FLi] the left ideal of FLi generated by all elements of the form a/3 Pa with a,/3 G FL^. We claim that [FLuFLi] = A{Li,L[). In fact, given two elements ^,m 6 Li which do not commute, we have immi = im{ls) G FLi{ls); thus [FLuFLi] C FLi(is). On the other hand, if i and m are any two elements of Li which do not commute, PROOF.
267
268
XI. ISOMORPHISMS OF LOOP ALGEBRAS
then 1 — 5 = 1 — (i^m)
= m~^i~^(im
— mi) G [FLi^ ^ ^ i ] ; thus t h e opposite
inclusion also follows. Now given an isomorphism ip: FL\ i^{[FLi,FLi])
^ FL2 we have ' 0 ( A ( L i , L^)) =
= [FL2, FL2] = A(L2, L'2) and thus A{Lu
L[) ^ A ( L 2 , V^)
This shows t h a t 1/^ also induces an isomorphism ip of the corresponding factor rings, so F [ L i / L ' i ] ^ F L i / A ( L i , L ; ) S FL^I^iL^,
L'^) ^ FiL^/L'^].
D
If L is an RA loop with unique nonidentity commutatorassociator s and if F is any field of characteristic different from 2, then, by Lemma VI. 1.2, FL = FL^^
® FL^
^ F[LIL']
© A ( L , V),
Thus Lemma 1.1 implies the following. 1.2 T h e o r e m . Let L\ and L2 be RA loops and let F be any field of characteristic
different from 2.
F[L2lL'^]
and A{Lu
Then FL\
= FL2 if and only if F[L\IL\]
=.
L\) ^ A(L2, L'^).
In the semisimple case, this result is not surprising since it is essentially a consequence of Corollary VI.4.8. Indeed, for a finite RA loop L and a field F such that char F j( Z/, we know that F[L/L']
is the sum of all
the simple components of FL which are both commutative and associative while A ( L , L ' ) is the sum of those simple components which are neither commutative nor associative (these are actually CayleyDickson algebras). Accordingly, it is apparent t h a t the structure of A(L,Z/') will be of fundamental importance for the study of the isomorphism problem.
We
first study the centre of this ideal. 1.3 P r o p o s i t i o n . Let L be a finite RA 2loop and let F be a field such that c h a r F 7^ 2.
Write Z{L)
i — 1 , 2 , . . . , r. Assume ZiAf{L,
L'))
= {ti) x ••• x (tr),
that V C {t\). ^
where {t,) ^ €2^^
Then
^ ^ ^ J ^ L ^  ^ F C ^ n . )[C2n2 X • • • X C a n . ] .
P R O O F . Because of Corollary VI.1.3, 2 ( A H L , L ' ) )  F[Z{L)Y^
= [F{t,y^][{t,)
X . . . X (f.)]
for
1. SEMISIMPLE LOOP ALGEBRAS
269
From Theorem X.2.1, we know t h a t ni
^<">"®in?^^«where Vi is the number of elements of order 2^ in C2«i. Thus
On the other hand, because of Lemma VI. 1.2, F{h)
= F{hy^
® ^ ( ^ i ) ^ = mi)/L']
® AFiih),
L')
and
Comparing equations, we have
and the result follows.
D
Recall t h a t any finite RA loop L can be written L — M X A d.s the direct product of an RA 2loop M and an abehan group A of odd order (Corollary V.1.2). We show that isomorphisms of semisimple alternative loop algebras "split" into two separate isomorphisms. 1.4 T h e o r e m . [ d B M 9 5 b , Theorem 2.5] Let L\ and L2 be finite RA loops and let F be a field whose characteristic
does not divide the order of either
of these loops. For i — 1,2^ write Li — Mi x Ai, where Mi is an RA and Ai is an abelian group of odd order. FMi
^ FM2 and FAi ^ PROOF.
F[Li/L\]
Then FL\
=. FL2 if and only if
FA2.
Suppose first t h a t FLi
= FL2.
By Theorem 1.2, we have
^ F[L2/L'^]] t h a t is, F [ ( M i / M O X A,] ^ F[{M2lM'2)
x A2].
As these are commutative group algebras, Theorem X.4.2 yields t h a t F[MilM[]
2loop
^ F[M2lM!^]
and
FA^ ^
FA2.
270
XI. ISOMORPHISMS OF LOOP ALGEBRAS
In view of Theorem 1.2, in order to prove that FMi = FM2 as well, it will suffice to show that A(Mi,M{) ~ A(M2,M^). By Theorem 1.2, A(Z/i,i/2) ~ A(Z/2,i/2) Moreover, for i = 1,2, denoting by Si the unique nonidentity commutatorassociator of LJ, we have A(L,, L[) = FLi^
^ FMi^
® FAi ^ A(M,, M/) ® FAi
since Si G M^ Thus A(Mi, MO ® F ^ i ^ A(Li, L[) ^ A ( L 2 , 4 ) = ^ ( ^ 2 , M^) ® FA2. By Theorem X.2.1, FAi ^ nF ® ®d'^dF{U) = FA2, where ^d is a primitive root of unity of odd order d and d runs over the set of divisors of I All such that F{U) ^ F, For i = 1,2, A(L„ L\) ^ nA{M,, M[) ® 0 , m , ( A ( M „ M[) ® F{^d))^ By Corollary VI.1.3, Z ( A ( M ,  , M ; ) ) = F [ 2 ( M , ) ] i ^ . Thus, again using Theorem X.2.1, 2(A(M,,M/)) ^ 0^/^(^a,), where the ^f^^ are primitive roots of unity of order 2^K Consequently,
z(A(M., Ml) ® F{u)) = ®, n^a,) ® n^ci)Since (d,2"') = 1, Theorem X.1.7 yields Fi^a,)® F{U) = FiUMa,)
= Fi^a.d),
where ^ajd is a primitive root of unity of order 2^^d. We claim that this field is never isomorphic to a field of the form F(^a^)' In fact, assume that FUa.d) = F{^a,) Then FiU) C F{^a.) and so n ^ a . ) ®F F{U) ^ F{^,,d) = F{^a,){U) = F{U). However, as F{^d) / F^ the tensor product F{^a,) ®F f'i^d) has dimension at least two over the field F{^a^)'> ^ contradiction. Hence the centre of the algebra A(Mi, M) is a direct sum of fields which are all different from those appearing in the decomposition of the centre of A(Mi, M!) ® F{(d) Since nA{MuM{)
® ® ^ md(A(Mi, M{) ® FiU)) ^ nA(M2, M^) ® ® ^ md(A(M2, M^) ® F{U))
2. RATIONAL LOOP ALGEBRAS
271
and because A(M^,M/) is a sum of CayleyDickson algebras over fields of the form F{^aj) while Z(A{MM) ® F{^d)) contains no such direct summands, it follows that nA{Mi,M[) ^ nA{M2,M!^). Hence A(Mi,M{) ^ A(M2,M2), as desired. The converse is straightforward. D Theorem 1.4 reduces the isomorphism problem for semisimple alternative loop algebras to the study of the problem for group algebras of finite abelian groups, which we discussed in Chapter X, and the study of isomorphic loop algebras of RA 2loops over fields of characteristic different from 2. This last question will be discussed in two stages. First, we shall study loop algebras over the field of rational numbers; then we shall see how to use the results obtained in this case to give general theorems over arbitrary fields.
2. Rational loop algebras In this section, we consider the isomorphism problem for finite RA loops L over the field of rational numbers. We shall see that the results depend on whether or not the unique nonidentity commutatorassociator of L is a square in the centre of L. For this reason, we introduce some terminology due to L. G. X. de Barros [dB93a]. 2.1 Definition. An RA loop L is of type I if the unique nonidentity commutatorassociator 5 of L is a square in Z{L); that is, if there exists a G 2(L) such that a^ = s. If there exists no such element, the loop is said to be of type 11, If L is a finite RA loop with L' = { l , ^ } , it follows from Theorem IV.3.1 and Theorem V.1.8 that the centre of L can be written in the form Z(L) = {h) X (^2) X (^3) X A, where o(^,) = 2"^*, mi > 1, m2,m3 > 0, 5 6 {^1) and A is an abelian group. It is clear that L is of type I if and only if mi > 1 and hence of type II if and only if Z{L) = V x H for some (abelian) group H. The isomorphism problem for loops of type I was studied by G. Leal and C. Polcino Milies in [LM93] and their work was completed with the study of loops of type II by L. G. X. de Barros [dB93a].
272
XI. ISOMORPHISMS OF LOOP ALGEBRAS
2.2 T h e o r e m . Let L be a finite RA loop of type I and let M be
another
loop. Then QL ^ QM if and only if L/L' PROOF.
^ M/M'
and Z{QL)
^
Z(QM).
Assume first t h a t QL ^ Q M . Then, clearly, Z{QL)
^
Z{QM),
Also, Lemma 1.1 shows t h a t A ( L , L') ^ A ( M , M ' ) and Q[L/V] so L/L'
^ M/M'
^ Q[M/M'],
by Theorem X.2.3.
By Corollary VI.4.8, QL ^ Q[L/L'] 0 A ( L , V) and Q M ^ Q [ M / M ' ] © A ( M , M ' ) where A(L,LO = A i © .  . © A n and A(M,
M')^Bi®'"®Bm
are the direct sums of CayleyDickson algebras Ai and 5 j , respectively. Comparing respective centres, we obtain n = m and, after possible reordering, Z(Ai)
= Z{Bi)
foTi=
1,.. .,n.
Let L' = { l , ' ^ } . Since L is of type I, there exists a G Z(L) a2 = s. Then a ^
such t h a t
G Z ( A ( L , L ' ) ) = 2 ( A ( M , M ' ) ) and ( c ^ ^ ) ^ =  1 ^ ,
so the projection of this element onto each simple component is a central element whose square is —1. It follows t h a t the centres of each Ai and each Bi are not Q and so, in view of Lemma V n . 2 . 2 , these simple components must be split CayleyDickson algebras and hence unique over their centres (Theorem L4.16). So A^ ^ B^{oT i = I,,.,
, n a n d Q L ^ Q M as desired.
D
This result shows that finite RA loops of type I need not be determined by their rational loop algebras. To give a specific example, it is useful to get some information about the quotient L/L\
We adopt the notation of
Theorems V. 1.7 and V.1.8. 2.3 P r o p o s i t i o n . Let L be a finite indecomposable
RA loop.
Write
Z{L) = (^i) X (^2) X (^3) X (w) ^ C2 X C26 X C2C X C2d, with a> I, b,c,d>
0, and V C (^i). If L G £1 U £ 3 U £5 U £ 7 , then L/L
= 0 2 0  1 X 0 2 6 + 1 X C2C+1 X C2d\i]
otherwise L/L
= 0 2 0 X 026+1 X 02C+1 X 0 2^*
2. RATIONAL LOOP ALGEBRAS
273
Our proof relies on the classification of indecomposable RA loops given in Theorem V.3.1. For an element x G L, we denote by x its image in the quotient L/ V, \i L eCi U £ 3 U £ 5 , then PROOF.
LjL' = (?7) X (x) X (y) X (u) = C2ai x C26+1 x C2C+1 x C2. If L G £2, we have L/L' = (y) X (uT) X (Ff) ^ C2a x C2 x C2, where i^i = uy^j ~^ and Xi = xy^j If L G >C4, we have
~^.
L/L' = {x) X (y) X (tZT) = €2^ x C26+1 x C2, where i^i = ux/j L/L'
~ \ while, if L G /^e? then = ( i l ) X ( x ) X (y)
=
C 2 a X C26+1 X C2C+1.
If // G £7, we have L/L' = (?7) X (x) X (y) X (tU) = C2ai X C26+1 x C2C+1 x C2d+i. The result follows.
D
When a = I (that is, ti = s) and L G £2 U £4 U Ce^ then d = 0, so a = d + 1 and we have the following. 2.4 Corollary. Let L be a finite indecomposable RA loop of type II and write 2{L) = L' x C26 X C2C x ^2^ with b,c,d> 0. Then L/V = C26+1 X C2C+1 X C 2 d + i •
2.5 Example. We now give an example which illustrates that finite RA loops of type I need not be determined by their rational loop algebras. Let L G £3 be such that Z{L) = C4 X C2 and let L2 G £2 be such that Z(L2) = C4. Set M = L2 X C2. Clearly LfM. On the other hand, we use Theorem 2.2 to show that QL ^ QM. By Proposition 2.3, LIV ^ C2 X C4 X C2 X C2 = M/M', so we have only to establish that Z{QL) and Z{QM) are isomorphic. We have Z(QL)SQ[L/L']®Z(A(L,I'))
274
XI. ISOMORPHISMS OF LOOP ALGEBRAS
with Z{A{L,V)) Now
= Q[Z{L)]^
and Q[L/L'] ^ Q[M/M'] ^ 12Q © 6Q(0.
Q[Z(L)] ^ Q[C4 X C2]  4Q ® 2Q(0. Since
Q[Z{L)]^Q[Z{Lr±^eQ[Z{Lr^ and I 1 + 5 rv.
we see that Q[Z{L)]^ ^ 2Q{i). So Z(QL) ^ 12Q + 8Q(0. Similar computations show that Z(QM) ^ 12Q + 8Q(i) = Z{QL), as desired. We now turn our attention to RA loops of type II. By contrast, these indeed do turn out to be determined by their rational loop algebras. We begin with a refinement of Corollary VI.4.8. 2.6 Lemma. Let L be a finite RA 2loop of type II and let N = Then
L/Z(Ly.
where A(A^, A^') is a direct sum of Cay leyDickson algebras each with centre Q and A is a direct sum of split Cay leyDickson algebras each with centre different from Q. Let L' = U,s}. Since L is of type II, Z{L) = L' x H with H ^ C2"2 X • • • X C2nt. We know that PROOF.
Z{A(L, L')) ^ Q[Z{L)Y^
^ [{QVy=^]H ^ QH,
Also, Q ^ ^ QC2n2 ® ••• ® QC2nt and, by Theorem VII.L4, QC2n. ^ 01^=0 Q ( 6 0 = Q ® Q ® ^7=2^(^2^^ where ^2* denotes a root of unity of order 2\ Hence Z{A{L,L'))^
Q®®Q
05,
2^~^ summands
where <S is a sum of fields of the form Q(^2»)? ^ ^ 2. Thus, by Corollary VI.4.8 and Lemma VIL2.2,
A{L,L')^^t\'
MQ) ® A,
2. RATIONAL LOOP ALGEBRAS
275
where each Ai(Q) is a CayleyDickson algebra over Q and A is a> sum of split CayleyDickson algebras over Q(^20? ^ > 2. Now N = LlZ{Ly is an RA 2loop of order 2*+^ and Z{N) has exponent 2. Furthermore (as above)
2*~1 summands
Thus A{N,N') QL ^ QLZ{Ly
^ 0 ^ 1 " / Ai{Q). Furthermore, using Lemma VI.1.2, ® Qi.(l  . i f ^ ^ ) ^ QL/A{L, Z{Lf)
^ Q7V © A ( l , Z{Lf) thus A{L, Z(Ly)
® A(Z, Z ( Z f )
^ Q[N/N'] ® A(JV, N') © A ( I , ^(L)^),
= !F Q A, where T is the direct sum of fields. Hence QL ^ Q[N/N'] © JT © A{N, N') © A,
so Q[N/N']®T = Q[L/L'] is the sum of the simple commutative, associative components of QL and the result follows. D 2.7 Theorem. Let L and M be finite RA 2loops of type 11. Let N and P denote the quotient loops L/Z{L)^ and M/Z{My, respectively. Then QL ^ QM if and only if L/L' ^ M/M', Z{L) ^ Z{M) and A{N, N') ^ A(P,P')Suppose L and M are RA 2loops of type II such that QL = QM. By Theorem 1.2, Q[L/L'] ^ Q[M/M'] and A(L,L') ^ A ( M , M ' ) , so Z{A{L,L')) ^ Z{A{M,M')). By Theorem X.2.3, L/L' ^ M/M'. Writing Z{L) — L' X H and Z{M) = M' x K\ where H and K are abelian 2groups, we have PROOF.
Z(A(L, L')) ^ Q[Z{L)]{1  V) ^ [{QL'){\  L')]H ^ QH and Z ( A ( M , M ' ) ) = Q[Z(M)](1  M') ^ [(QM')(1  M')]K ^ QK. Theorem X.2.3 shows that H = A', which implies Z{L) ~ Z{M). Since Lemma 2.6 shows that A(7V, N') and A(P, P') are the sums of the simple components of QL and QM which are CayleyDickson algebras with centre Q, it is also clear that A(7V, N') ^ A(F, P').
276
XI. ISOMORPHISMS OF LOOP ALGEBRAS
Conversely, assume t h a t L/V
^ M/M\
Z{L)
^ Z(M)
and A(7V, TV') ^
A ( F , P O  By Lemma 2.6, QL^Q[L/L']®A{N,N')®A and Q M ^ Q [ M / M ' ] ® A ( P , P ' ) ® i? where ^ and 13 are the sums of the simple components of QL and Q M , respectively, which are split CayleyDickson algebras with centres not Q. The first two summands in each of the above decompositions are pairwise isomorphic. Furthermore,
and hence
= Q[Z(L)]^,
Z{A(N,N')QA)
where L' = { l , ^ } , so Z{A)
is
the sum of the components of Q [ 2 ( Z / ) ] ^ ^ which are not Q, and similarly for Z[B).
Since V is a direct factor of Z{L)
and M' is a direct factor of
>Z(M), we may assume t h a t the isomorphism Z[L) M', It follows t h a t Z{A)
= Z{B),
^ Z{M)
maps L' to
hence the result.
D
In what follows, we shall require a description of the rational loop algebras of finite indecomposable RA loops with centres of exponent 2. According to the Classification Theorem (Theorem V.3.1), there are (up to isomorphism) seven such loops, which we describe below. If Z{L)
= C2, then L =
Mi6(Q8?2) or I2 =
MIQ{QS)'
16 and L is isomorphic to either /]
=
T h e corresponding rational loop algebras
were described in Corollary VII.2.3: Q/i = 8 Q ® 3 ( Q ) , Q/2 = 8 Q © ( Q ,  1 ,  1 ,  1 ) . H e r e 3 ( Q ) and (Q,  1 ,  1 ,  1 ) denote, respectively, the split CayleyDickson algebra over Q and the division algebra of Cayley numbers over the rationals. If Z{L)
= C2 X C2, then Z/ = 32 and there are two such loops (see
Table V.4): /3 = M ( 1 6 r 2 C i , * , l ) G > C 3
2. RATIONAL LOOP ALGEBRAS
277
and h = M(16r2C2,*,/i) G A , where 16r2Ci = (a:, ^,^1,^2 I ^^ = ^1 = ^2 — 1^2/^ — ^29^2 2ind ^i = (a:,i/) central) and 16r2C2 = (a:,y,^1,^2 Ml == ^2 — l?^'^ = ^1^2/^ = ^25^2 ^nd ^1 = (x^y)
central).
Since Is and I4 are of type II, using Corollary 2.4, we see t h a t Is/I^
—
C2 X C2 X C4 ^ 14/14 Since QC2 ^ Q ® Q and QC4 ^ Q ® Q 0 Q ( 0 , we have Q[/3//^] = Q[/4//4] = 8Q ® 4 Q ( 0 . According to Theorem VII.2.5, all simple components of A ( / 3 , / 3 ) are split CayleyDickson algebras, while all but one of the simple components of A ( / 4 , / 4 ) are also split. Computing dimensions, we see t h a t both A ( / 3 , / 3 ) and A ( / 4 , / 4 ) have precisely two components, so
Q/3^8Q®4Q(0®23(Q) and Q/4 ^ 8Q ® 4Q(0 ® 3(Q) ® (Q,  1 ,  1 ,  1 ) . If Z{L)
= C2 X C2 X C2, then L = 64 and, according to Table V.4,
there are two indecomposable RA loops of this order, both constructed from the same indecomposable group 32r2A: = {x,y,h,t2,t3
\ x^ = t2,y^ = h.tj
= tj = ^ = l , ( x , y ) = ^1),
with ^1,^2,^3 central. The corresponding loops are /5 = M ( 3 2 r 2 A : , * , l ) G £ 5 and
Ie =
M{32T2k,^,h)eCe.
278
XI. ISOMORPHISMS OF LOOP ALGEBRAS
Once again, using Corollary 2.4, we see that h/I's — C2 X C4 X C4 = h/I^ so, as before, we see that Q[hir^] = 8Q 0 12Q(i) ^ Qi/e/^] Applying Theorem VII.2.5, we get Q/5^8Q®12Q(0e43(Q) and Q/e ^ 8Q e 12Q(i) ® 33(Q) 0 (Q,  1 ,  1 ,  1 ) . FinaUy, if Z{L) = C2 X C2 x C2 x C2, then L ^ I7 = M{32T2k x C'2, *,'";) G Cj^ where w ^ \ generates the factor C2. Here, hll'j = C4 X C4 X C4 and Q[/7//7] = 8Q 0 28Q(z), so Theorem VII.2.5 gives Q/7 ^ 8Q © 28Q(0 0 73(Q) 0 (Q,  1 ,  1 ,  1 ) . 2.8 Lemma. Let L be a finite RA 2'loop of type 11. Write L = Li x H, where L\ is an indecomposable RA 2'loop and H is an abelian group. Set N = LlZ{Lf, Ni = L i / 2 ( L i ) 2 and p = rank{H/H^) = rank{H). Then A{N,N')^2f'A{NuN[). Observe that A^i is an indecomposable RA loop with centre of exponent 2 and that rank(2:(A^i)) = rank(2:(Li)). Since N = N^ x{H/H'^), PROOF.
A{N,N')
= QN 1 25 = Q A ^ i ^ ®Q Q[H/H^] ^ AiNuN{)
where N' = N[ = {\,s}.
So the result follows.
®Q Q[H/H% D
2.9 Lemma. Let L and M be finite RA 2loops of type IL Write L — L\X E and M — M\ x K, where L\ and M\ are indecomposable RA loops and H and K are abelian groups. If QL = QM, then rank(/r) = rank(A^) andL^IZ{Lif^ L2lZ{L2f. Let N = L/Z{Lf, P = MlZ{Mf, N^ = L^lZ^L^f and Pi = Mi/Z(Mi)2. Set p = rank(//) = T^iik{H/H^) and r = rank(/l) = rank(A7A'2). If QL ^ QM, then, by Theorem 2.7, L/V ^ M/M', Z{L) ^ Z{M) and A(7V,7V') ^ A ( P , P ' ) . so, by Lemma 2.8, 2^A{NuN[) ^ 2^A{Pi,P{). Since Ni and Pi are indecomposable RA loops with centres of exponent 2, both loops are either in 7^ = {/2,/4,/e,/?} or in I = {/i,/3,/5}, where PROOF.
2. RATIONAL LOOP ALGEBRAS
279
/^, 1 = 1 , . . . , 7 denote the seven indecomposable RA loops described above, according to whether or not (Q,—1,—1,—1) appears as a direct summand in the decompositions of QL and QM. If Ni and Pi are in H, then p = r since (Q,—1,—1,—1) appears only once as a direct summand in the decompositions of Q/2, Q/4, Q/e and Q/7. On the other hand, if Ni and Pi are both in J , then we shall show that the assumption p ^ r leads to a contradiction. For this, it is sufficient to consider three cases. Case 1: TVj ^ h and Pi ^ I3. In this case, rank(Z(i/i)) = rank(>Z(iVi)) = 1 and rank(Z(Afi)) = rank(Z(Pi)) = 2. Hence Z{Mi) ^ C2 X C^b with 6 > 0. Since Z{Li) ^ C2, Z{L) ^ Z{Li) X H ^ Z(Mi) X K ^ Z{M), we have C2 x 7^ ^ C2 x C26 X K, which implies H = C26 X K. Therefore L ^ Li x C26 X K and L/L' ^ Li/L[ X C26 X K. By CoroUary 2.4, Li/L[ ^ C2 X C2 x C2 and Mi/M{ ^ C2 X C2 X C26+1. Since L/L' ^ M / M ' ^ (Mi/M{) x A^ it follows that C2 X C2 X C2 X C26 X K = C2 X C2 X C26+1 x A% which implies that 6 = 0, a contradiction. Case 2: A^i ^ A and Pi ^ /s. Here rank(2(Li)) = 1 and rank(2(Mi)) = 3. Hence 2 ( M i ) ^ C2 X C26 X C2C with 6 , 0 0. Since Z{Li) ^ C2, 2 ( L ) ^ Z(Li) x 7/, Z ( M ) ^ 2 ( M i ) X A^ and Z{L) ^ Z{M), we have C2 X ^ ^ C2 X C26 X C2C X A^ which implies H = C26 X C2C X K. Thus L = Li x C26 x C2C X A^ and L/L' ^ Li/L[ X C26 X C2C X K. By Corollary 2.4, L I / L ; ^ C2 X C2 X C2 and Mi/M{ ^ C2 x C26+1 x C2C+1. Since L/L' ^ M / M ' , it foUows that C2 X C2 X C2 X C26 X C2C X K = C2 X C26+1 X C2C+1 X A^ which implies 6 = c = 0, a contradiction. Case 3: A^i ^ /a and Pi ^ /g. In this final case, rank(2(Li)) — 2 and rank(Z(Mi)) = 3. Hence Z{Li) ^ C2 X C2r and 2 ( M i ) ^ C2 x C26 x C2C, with r,b,c > 0. As before, we have C2 X €2^ X H = C2 X C26 X €2^ X K, If r = 6, then H ^C2cX A^ so L ^ Li X C2C X K. Thus L/L' ^ C2 x C2 x C2r+i x C2C x K and M / M ' ^ C2 X C2r+i x C2C+1 X A'. Since 1/1' ^ M/M\ we conclude that c = 0, a contradiction. The case r = c is similar, so assume
280
XL ISOMORPHISMS OF LOOP ALGEBRAS
r ^ b and r ^ c. Since €2^ X H = C26 X C2C X K^ there exist subgroups Hi and A'l of H and K, respectively, such that H = C26 X C2C X ^ 1 , K ^ C2r X Ki and i7i ^ R\. Then L ^ Li x C26 x C2C x Hi and M ^ Ml X C2r X i7i, hence L/V ^ C2 x C2 x C2r+i x C26 x C2C x ZTi and M/M' ^ C2 X C26+1 X C2C+1 x C2r x i^i. Since L/V ^ M/M', we obtain r = 6 = c = 0 , a contradiction. Finally, p = r implies that A{Ni,N{) = A(Pi,P{) and, since Ni and Fi are indecomposable RA loops with centres of exponent 2, it foUows from the discussion prior to Lemma 2.8 that Ni = Pi. D 2.10 L e m m a . Let L and M be finite RA 2loops of type II such that QL = QM. Write L = Li x H and M = Mi x K, with Li and Mi indecomposable RA loops and H and K abelian groups. If H = K, then Li = Mi. Clearly, Z{L) ^ Z{Li) x H, Z{M) ^ Z{Mi) x K and, by Theorem 2.7, Z{L) ^ Z{M). Hence, if 77 ^ A^ then Z{Li) ^ Z{Mi). By Lemma 2.9, Q[Li/Z{Li)'^] ^ Q[Mi/Z{Mi)^], so these algebras either both admit (Q,—1,—1,—1) as a direct summand or both do not. Because of Lemma 2.6, the same is true for QLi and QMi. Since Z{Li) and Z{Mi) have the same rank, and because of Theorem VII.2.5, it follows that Li and Ml are in the same family Cj described in Table V.3. Furthermore, since Z{Li) ^ 2 ( M i ) , we have Li^ Mi. D PROOF.
We now proceed with the main result of this section. 2.11 T h e o r e m . Let L and M be finite RA loops of type II. Then QL = QM if and only if L = M. One implication is obvious. For the converse, assume QL = QM. By Theorem 1.4 and Theorem X.2.3, we may assume that L and M are finite RA 2loops of type II. Write L = Li x H and M = Mi x K, where Li and Mi are indecomposable RA loops and H and K are abelian groups. Let N = L/Z{L)\ P ^ M/Z{M)\ Ni ^ LilZ{Lif and A = MilZ{Mif. By Lemma 2.9 and Theorem 2.7, LjV ^ M/M\ Z{L) ^ 2 ( M ) , A{N,N') ^ A ( P , P ' ) and Ni ^ Pi. Because of Lemma 2.10, we now need only prove that H = K. We do this by analyzing all possibilities for A^i and Pi. Recall that A^i = Pi is one of the indecomposable loops li described in the discussion prior to Lemma 2.8. PROOF.
2. RATIONAL LOOP ALGEBRAS
281
If Ni ^ Pi is isomorphic to either h or h, then 2 ( L i ) ^ C2 = So Z{L) ^ Z{M)
readily implies H ^
2:(Mi),
K,
Suppose Ni ~ Pi is isomorphic to either I3 or I4. Then r a n k ( 2 ( L i ) ) = r a n k ( 2 : ( M i ) ) = 2.
Write Z{Li)
^ C2 X Cs^ and Z{Mi)
^
C2 X C^b,
r > 0,6 > 0. By CoroUary 2.4, L I / L ; ^ C2 X C2 X C2r+i and M i / M {
^
C2 X C2 X C26+1. Since 2 ( L ) ^ 2 : ( M ) , we obtain C2r x H ^ C^^ x K,
If
r ^ b^we can find subgroups Hi and A^i of H and /iT, respectively, such t h a t H ^C2bX
Hi, K ^ C2r X Ki and Hi ^ Ki.
M ^ MixC2rxHi.
Then L ^ Li x C26 x i7i and
Since L/L' ^ {Li/L[)xH
smd M/M' ^
{Mi/M[)xR\
we have L/V
^ C2 X C2 X C2r+i X C26 X Hi
and M/M' and hence LjV
^ C2 X C2 X C26+1 X C2r X Hi,
^ MjM',
a contradiction.
Hence r = b and therefore
7/ ^ A\ Suppose A^i = Pi is isomorphic to either Is or IQ, Then r a n k ( Z ( L i ) ) = r a n k ( 2 ( M i ) ) = 3. Write Z{Li)
^ C2 X ^2^ X C2^ and Z{Mi)
^ C2 X C26 x
C2C, r, 5,c > 0. Hence, again by Corollary 2.4, Z/l/Z/i = C2 X (72r+l X C25+I and Mi/M[
^ C2 X C26+1 X C2C+1.
From 2:(L) = 2 ( M ) , we conclude t h a t C2r X C2' X H ^ A'. If {b,c}
/
C26 x C2C x
{^,'5}, simple arguments (as in the previous case) lead to
contradictions in aD possible cases, so H = K. If Ni = Pi is isomorphic to /y, we write Z{Li) and Z{Mi)
= C2 X C2r X €2^ X C29
= C2 X C26 X C2C X C2d, with r,s,q,b,c,d> Z/l/Li = C2r1 X C2.'+l X C2qr+1
and Mi/M[
= C26+1 X C2C+1 X C2d+i.
0. Then
282
XL ISOMORPHISMS OF LOOP ALGEBRAS
Since Z(L) = Z{M), we obtain C2r X C23 xC2
c{K) = I
1
\i x'^ { y^ { z^ \ w'^ = \ is solvable in A'
0
if x^ + t/"^ + 2:^ + tt;^ = — 1 is not solvable in K
As shown in Theorem 1.3.4, c{K) = 0 if and only if the Cay leyDickson algebra (A', —1, —1, —1) is a division algebra. Hence, by Corollary VII.2.3, K[Mie{Q)] ^ A^[Mi6(g,2)] if and only if c{K)  1. 3.3 Proposition. Let p he a prime and let Fp denote the field with p elements. Then t{Fp) = 1 and thus also c{Fp) = 1. P R O O F . We have to show that —1 is a sum of two squares in Fp. This is obvious for p = 2^ so assume that p is odd. Let ^i and 5*2 be the subsets
3. THE EQUIVALENCE PROBLEM
283
of Fp defined by 5l = { o ^ l ^ 2 ^ . . . , ( 2 ^ ) ' } and 52 = {  0 ^  1 ,  1 2  1 ,  2 2  l , . . . ,  ( H f i ) 2  l } . Since x'^ = t/^ (mod p) implies p\ {x — y)oTp\ (x + y), both 5*1 and 5*2 have precisely ^ ^ elements. Since Fp has only p elements, there exist x^ G 5*1 and —y^^ — 1 G 5*2 so that x'^ = —y*^ — I. This proves the result. D 3.4 Proposition. Let L he a finite RA 2'loop and K a field such that char/I / 2. (i) / / L is of type /, then all simple components of AK(L^ L') are split algebras. (ii) If L is of type II and A is a simple component of AK{L, L') which is not splits then A = {Ii\ —1, —1, —1). Write Z{L) = {ti) x (^2) x ••• x (tr), where (ti) = C2n., i = 1,2,..., r a n d V C {ti). By Proposition 1.3 and Theorem X.2.1, PROOF.
onil
2 ( A K ( L , 1 0 ) = f 7 7 7  T  ^ ^ ^ ^ ( ^ 2 n i )[C2n. X . . . X C2ne] ^ e ^ ^ i
lU,
where A\ = A^(^2"^») ^^^ ^2"^» is a primitive 2"^»th root of unity, i = 1 , . . . , r. By Corollary VI.4.8, AK{L, L') = ®Ui ^^^ where each Ai is a CayleyDickson algebra over its centre Ki, Furthermore, because of Lemma VII.2.2, if the centre of Ai contains a 2^th root of unity with n > 1, then Ai is split. Thus, if m^ > 1, then Ai is split so, if L is of type I, rii > 1 and thus also each nii > I, Therefore each Ai is split. If L = {Z{L)^ X, y, u) is of type II, we only have to consider those simple components of A A ' ( ^ , L^) which have centre K (the others being split). By Lemma VII.2.2, the nonsplit simple components of AxiL^L^) are of the form A =
{{K[ZiL)])e,ixe)Mye)\iue)%
e a primitive central idempotent in AK{L^L'). The field {K[Z{L)\)e = K does not contain a fourth root of unity. Since a — (xe)'^^ /? = (ye)^, 7 = {ueY have orders which are powers of 2, it follows that a,/?,7 G { i l l It is easily seen that if one of these values is + 1 , then the equation x'^ —
284
XI. ISOMORPHISMS OF LOOP ALGEBRAS
OL'ip' — (iz^ + a/3w^ = 7/^ has a nontrivial solution. Hence, by Theorem 1.3.4, the algebra A is split, a contradiction.
Thus a = P = 'y = —l^soA
=
(A^1,1,1).
D
We now establish the analogue of Theorem X.4.9 for finite RA 2loops. 3.5 T h e o r e m . Let F and K be fields with characteristics
different from 2.
Let 7^2 be the class of finite RA 2loops of type IL Then F is equivalent
to
K on 7^2 if ^ ^ ^ only if
(a) 0{F) =
0{K),
(b) ind2(F) = ind2(A') and (c) c{F) = PROOF.
c{K).
Assume t h a t F and K are equivalent on 72^2• First, we prove
that F and K are equivalent on the class of all finite abelian 2groups. Assume t h a t A and B are finite abelian 2groups such t h a t FA = Set / =
M\Q{Q^),
FB.
the Cayley loop, and \ei L  I X A and M = I X B. Then
FL = FM^ thus KL = A'M, since F and K are equivalent on 712 Hence, by Theorem 1.2, K[LIV] M/M'
^ K[M/M'],
Since L/L'
^ C2 x C2 x C2 x yl and
^ C 2 X C 2 X C 2 X ^ , wehaveA'[C2xC2xC2X/l] ^
Ii[C2xC2xC2xB].
Using the fact t h a t KC2 = K ® A^ we conclude t h a t KA
= KB.
Thus
F and A' are equivalent on the class of all finite abelian 2groups so, by Theorem X.4.8, 0 ( F ) = 0{K)
and ind2(F) = ind2(A').
If c ( F ) 7^ c(A'), then, without loss of generality, c{F) = 1 and c{K) — 0, so, by CoroUary VII.2.3, F[M,e{Q)]
= F [ M i 6 ( Q , 2 ) ] , but K[Mie{Q)]
^
A^[Mi6(Q,2)]. Thus c ( F ) =r c(A'). Conversely, suppose t h a t conditions (a), (b) and (c) hold. For L and M in 7^2, assume FL ^ FM. AF(L,L')
^
AF(M,M').
By Theorem 1.2, F [ L / L ' ] ^ F[MIM']
Since both L/L'
and M/M'
2groups, conditions (a) and (b) imply t h a t Ii[L/L^]
and
are finite abelian
= A ' [ M / M ' ] , by The
orem X.4.8. Because of Theorem 1.2, it remains to show t h a t A x ( / / , L^) = A K ( M , M ' ) . We have Z{L)
^ V x 02^2 x • • • x C2nr and Z{M)
^ M' x
C2'n2 X • • • X C2m., where L' ^ C2 = M ' , n, > 0 for i = 2 , . . . , r and m^ > 0 for J = 2 , . . . , s. By Proposition 1.3, 2 ( A F ( i / , L')) = F[C2n2 X • . • X C2nr]
3. THE EQUIVALENCE PROBLEM
285
and Z{AF{M,
M'))
^ F[C22 X • • • X C2ms].
Hence F[C2r^2 X • • • X C2nr] ^ F[C22 X • • • x C2ms] ^ 0 ^ ^ F,, where Fi = F or Fi = F{^2^^) wi^h 6i > 2 for i = 1 , . . . , /:. Similarly,
and 2:(AK(M, M'))
= K[C2m2 X . . . X C2m.].
Conditions (a) and (b) imply t h a t F and K are equivalent on the class of finite abelian 2groups, by Theorem X.4.8. Therefore, K[C2n2 X • . . X C2nr] ^ A^[C22 X ' ' . X 0 2 ^ . ] ^ 0 ^ 1 A^', where K^ = K or A^ = A X 6  . ) with a, > 2, z = 1 , . . . , ^. Write AF(L, L') = 0 f = i A{F^) and
AF(M,M')
^ 0 ^ ^ B^{F^), where A ( F t ) and /5,(/s) are
CayleyDickson algebras over Fi. Since Ap'{L^L')
=. A/7(M, M ' ) , we have
Ai(F^) = Bt(Fi) for i = 1 , . . . , A;, reordering, if necessary. Write A/^'(L, L') = 0 t i C^(^'^) ^nd
AA'(M, M O
^ 0 t i A ( A ; ) , where C,( A\) and P ( A\) are
CayleyDickson algebras over AV For A\ = A'(^2^t) with a^ > 2, we have Ci{Ki)
= Vi{Ki)
since these algebras are split over the same field Ki. It
remains to analyze the case Ki — K. If c[K) = c ( F ) = 1, then all the algebras Ci[K) and Vi{K)
are split
and thus isomorphic. If c{K) = c{F) = 0, then some of the algebras Ci{K) and Vi[K) may be division algebras while, by Theorem 1.4.16, the others are all isomorphic to (A^ 1,1,1). Thus it remains only to show t h a t the numbers of nonsplit algebras in
AK{L^L')
and in A/v'(M, M ' ) are equal.
By Propositions 3.3 and 3.4, we may assume t h a t char F = char A' = 0, so we may assume t h a t Q C F and Q C K. Also, c{K) = c{F) = 0 implies 0{F)
= 0{K)
— 0. Let mK{L) denote the number of direct summands of
KL which are CayleyDickson division algebras. From KL = K ® Q Q L , KM mK{L)
^ A ®Q Q M , FL ^ F ®Q QL and FM = rriQ^L) — mp^L)
and mK{M)
= rriQ^M)
FL = FM, we conclude t h a t 771^(1)  mF{M), and hence A K ( L , L') ^ A K ( M , M ' ) .
^ F ®Q Q M , we obtain = mjr{M).
From
Thus, mK(L) =
mK{M) D
286
XL ISOMORPHISMS OF LOOP ALGEBRAS
Now it is easy to see which fields will yield a positive solution to the isomorphism problem for finite RA 2loops of type II. 3.6 Corollary. A field K of characteristic different from 2 determines the class 72.2 of all finite RA 2loops of type II if and only if ind2(A) = I, 0{K) = 0 and c{K) = 0. Because of Theorem 3.5, K is equivalent to Q on TZ2 if and only if ind2(A^) = 1, 0{K) = 0 and c(K) = 0. Since we have shown in Theorem 2.11 that Q determines the elements of 7^2, the result follows. D PROOF.
To obtain a more general result, we now study the RA loops of type I. 3.7 Theorem. Let F and K be fields of characteristic different from 2. Let 7l\ he the class of all finite RA 2loops of type I. Then F is equivalent to K on TZ\ if and only if (a) 0 ( F ) = 0[K) and (b) ind2(F)  ind2(A0. Suppose that F and K are equivalent on TZ\. To prove that conditions (a) and (b) are necessary, it is sufficient, by Theorem X.4.8, to show that F and K are equivalent on the class of all finite abelian 2groups. Assume that A and B are finite abelian 2groups such that FA =. FB. Let / be a finite indecomposable RA loop with Z{I) = C4 and / / / ' = C2 X C2 X C2 X C2. Let L = / X ^ and M = / X B. Since FL^ FA®FI ^ FB ® FI = FM and F and K are equivalent on 7^l, KL = KM, Hence, Ii[L/L'] ^ K[MIM'], by Theorem 1.2. Since LjV ^ C2 x C2 X C2 x C2 x A and MIM^ ^ C2 x C2 x C2 x C2 x 5 , it foUows that KA ^ KB, Thus K is indeed equivalent to F on the class of all finite abelian 2groups. Conversely, suppose that conditions (a) and (b) hold. Let L and M be RA loops in T^i. We have Z{L) = 02^1 x C2"2 X ••• X €2^1, where ni > 1 and n^ > 0 for i = 2 , 3 , . . . ,^, and Z{M) = C2"^i X C2"^2 X ••• x C2mr, where mi > 1 and m^ > 0 for i = 2, 3 , . . . , r. As usual, we assume that the first factors of each decomposition contain L' and M', respectively. If FL ^ FM, then F[L/L^] ^ F[M/M'] and ^F{L,V) ^ AF(M,MO, by Theorem 1.2. By Theorem X.4.8, conditions (a) and (b) imply that K[L/L^] = K[M/M^]. Again, because of Theorem 1.2, we need only prove that AK{L,V) ^ AA'(M,M'). PROOF.
3. THE EQUIVALENCE PROBLEM
287
By Proposition 1.3,
[A (f 2^^! j : A J OTni—1
Z{/^K{M,
M'))
^ _  —  _ _  A ^ ( ^ 2  i )[C22 X . . . X C2mrl [A(f2'^i j . ' A J onil
and o m i —1
Z{AF{M,
M')) ^ J—
^ ^ ^ ( 6  1 )[C22 X • • • X C2m.].
Writing 2 ' ^ i  V [ F ( 6 " i ) : F] = 2" and 2 ™ '  V [ ^ ( 6 ' " i ) : i^] = 2 ^ it foUows that Z{AFiL,
L')) ^ 2 " F ( 6 " i )[C2n2 X • • • X Ci",] S F ( 6 " : )[C'2X • • • X C 2 X C 2 " 2 X • • • X Ci",] a factors
and also t h a t Z(AF(M, M'))
= 2 ' ' F ( 6 " ' . )[C2m2 X • • • X C^mr] = F ( 6 ' " i )[C2 X ••• X C2XC2"'2 X • • • X C2mr]. 6 factors
Since >Z(A/r(L, L')) = Z ( A i ? ( M , M ' ) ) , comparing these equations gives Certainly conditions (a) and (b) imply t h a t 0 ( F ( ^ 2 " i ) ) = 0(A^(^2^i)) and t h a t ind2(F(^2"i)) = ind2( A'(^2"i ))• Therefore, A'(^2"i) = A^(^2"»i) stnd [ F ( 6 " i ) : F] = [ A X 6 " i ) : ^ 1  Applying Theorem X.4.8 to
Z(AF{L,L'))
we also obtain
Z{/S.F{M,M'))
K{^2^i )\C2 X . . . X C2 xC2"2 x . . . x C 2 n t ] V
a factors
= A(6"^i )[C2 X . . . X C2^XC2^2 X . . . X C2mr]. 6 factors
^
288
XI. ISOMORPHISMS OF LOOP ALGEBRAS
Thus ZiAKiL,
L')) ^ Z{AKiM,M'))
^ ®Ui
I^^
where A', = A'(^2<'i) f°r some d, > 0. Write A K ( I , L') ^ e i ^ i Ci{Ki) and A K ( M , M ' ) ^ @U^ Vi{Ki), each Ci{Ki)
and Vi{Ki)
is a CayleyDickson algebra.
m i > 1, Lemma VII.2.2 implies t h a t aD Ci{Ki) and Vi{Ki) hence isomorphic. Hence
AK{L,V)
Then KL ^ KM PROOF.
are split and
= A x ( M , M').
3.8 C o r o l l a r y . Let K be a field of characteristic ind2(/l') = 1 and 0{K)
where
Since ni > 1 and D
different from 2 such that
= 0. Let L and M be finite RA 2loops of type L
if and only if L/V
^ M/M'
and Z(QL)
^
Z(QM).
The assumptions and Theorem 3.7 imply t h a t K is equivalent
to Q on 721; thus the result follows from Theorem 2.2.
D
For finite RA 2loops in general, we have the following theorem. 3.9 T h e o r e m . Let F and K be fields of characteristics Let TZ be the class of all finite RA 2loops.
different from 2.
Then F is equivalent
to K on
TZ if and only if (a) 0{F)
=
0{K),
(b) ind2(F) = ind2(A') and (c) c{F) = c ( A ) . PROOF.
Suppose F and K are equivalent on TZ. From Theorem 3.5, it
follows t h a t conditions (a), (b) and (c) hold. Conversely, suppose t h a t conditions (a), (b) and (c) hold. Let L and M be loops in TZ and write Z{L)
= C2"i X C2"2 X • • • X €2^1, where rij > 1 and
Ui > 0 for i = 2 , 3 , . . . , / , and Z{M)
= C2^i X C2^2 x • • • x C2mr, where
mi > 1 and mi > 0 for i = 2, 3 , . . . , r. Assume t h a t L^ and M ' are contained in the respective first factors of these decompositions. If FL = FM^ then, by Theorem 1.2, F[L/U]
^ F[M/M']
and
AF{L,V)
^
Theorem X.4.8, conditions (a) and (b) imply t h a t K[L/L'] Therefore, again by Theorem 1.2, it is sufficient to prove t h a t A x ( M , M').
By
AF{M,M').
^
K[M/M'].
A A ' ( / / , L^)
=
Because of condition (c) and Proposition 3.4, this proof is
essentially the same as t h a t of Theorem 3.7.
D
Chapter XII
Loops of Units
In this chapter, we study the unit loop U{1L)
= {a e ZL \ a/3 = I = /3a for some f3 G IL}
of the integral loop ring of a finite RA loop L,
Recall from §11.5.3 t h a t
U{7.L) is a Moufang loop. In the first section, we deal with some technical results which enable us, in some instances, to reduce the study of unit loop of ZL to the unit loop of Z[T(L)], where T[L)
is the torsion subloop of L. In the second section, we
give necessary and sufficient conditions for the unit loop to satisfy a group identity. The third section deals with central units. A description of the centre of the unit loop is given and finitely many generators for a subloop of finite index in this loop are given. In Section 4, we exhibit subloops of finite index in the unit loop and, in Section 5, we apply our results in order to describe the full unit loop ZY(ZL) for two specific RA loops. 1. R e d u c t i o n t o t o r s i o n l o o p s Let L be an RA loop with torsion subloop T.
Recall t h a t T is indeed a
subloop which is locally finite, normal and finite if L is finitely generated (Lemma VIII.4.1). The lemma which follows enables us to show t h a t if QT is a direct sum of division rings, then, modulo the trivial units, the unit loop of ZL is determined by the units of 2.T. The main part of the proof is as in the associative situation [Seh78, Lemma VI.3.22]. 1.1 L e m m a . Let L he a finitely generated subloop T and let K be a field. Suppose 289
RA loop with normal
that KT
torsion
= L^i ® • • • © D^ is the
290
XII. LOOPS OF UNITS
direct sum of division rings and that every idempotent of KT is central in KL. Then every unit fi G KL can be written in the form fJ' = ^ diii^ with di e Di, ii e L. Write Di = (KT)ei^ where Ci is a primitive central idempotent of KT and Yl^i — 1 Since each Ci is central in KL and since T is normal in Z/, the division ring Di is normal in KL in the sense that PROOF.
Dia = aDi, (Aa)/? = A(«/?), {aD^)(3 = a{DiP) and a{f3Di) = {af3)Di for any a^P £ KL, To see why, these equations in the case that a tain (A^i)^2 = Di{iii2) for ^1,^2 d = {Y^dtt)ei = '^dt{tei)^ dt G K^t
note that it is sufficient to establish and j3 are elements of L. So, to obG L^ for example, let d G Di^ write G T and use
[(t€^)ii]i2 = [t{e^i^)]£2 = [tiiiei)]i2 = [iti,)e,]i2 = itii){ed2) = [^'(^i^2)]ei,
= {til){i2e^) = [{thy2]e^ for some t' G T, by normality of T,
= t'[{e,e2)ei] = t'[e,{i,e2)] = (<'e.)(^i^2) to conclude that {dt\)i2 — d\i\i2) for some d' G Di. Since T is normal in L, L is the disjoint union of cosets of T (see §11.1.3); that is, L — UQGQ ^^ ^^^ some transversal Q. If g G L\T^ then TqOTq'^ = 0 since g''^ has infinite order, thus we may assume that q~^ G Q whenever qeQ. Let /i be a unit in KL. Then /i can be written /^ = X^/Xj^j with fij G A T and the qj distinct elements of Q, and, for any i,
(1)
e,/i = J^€i(/i^^^) = J^rf^g^,
dj = e^^j G Dt, since 6^ is central. Since L/T is a finitely generated torsionfree abelian group, it can be ordered (see Lemma VIIL1.3) by an order relation <, say. Thus, in (1), we may assume that the qj are such that Tqi < Tq2 < • • • < Tq^ Similarly, we can write V
1. REDUCTION TO TORSION LOOPS
with d'j e Di and Tq[
291
Tq'^,
By normality of A , if ct,6 G A , there exist ai,a2,61,62,63 G A
such
that aqj • bq'i^ = ai{qj • 6^^) = ai(g^6i • q[) = ai{b2qj
' q'k) = ^1(63 • qjq'k) = ^263 • qjQk
Thus ei/j. ' eifi~^ can be written Y^djk • ^ j ^ ^ for djk G £^i. On the other hand, Artin's Theorem (Corollary 1.1.10) gives eni • e^/i"^ = e^; thus, e, r= ^djk
qjq'k
The left hand side of this expression has support in T , while the terms of the right hand side have supports in Tqiq'i, Tqiq^,...
.Tq^ql.
The first coset here is the unique smallest, the last the unique largest and, if either u > 1 or t; > 1, these are diflferent. Since Di is a division ring, it follows also t h a t d n 7^ 0 and d^v 7^ 0. Hence u = v = I so t h a t e^/x = diqi, di G Di, and n
n
/i = ( ^ e,)/i = ^ 1=1
d.^i, rf, G A , g't G L,
1=1
as required.
D
1.2 C o r o l l a r y . Let L be a finitely generated torsion
subloop.
Suppose
normal
that QT = Di ® • • • © Dn is a direct sum of
division rings and that every idempotent U{1L) PROOF.
RA loop and T its
of QT is central in QL,
= [U{ZT)]L =
Then
L[U{ZT)].
We use the same notation as in the statement and proof of
Lemma 1.1. So let // be a unit in ZL. Let < be an order relation on the finitely generated torsionfree abelian group L/T and let Q be a transversal for T in L with the property t h a t q~^ G Q whenever q £ Q. Because of Lemma 1.1, we can write // = ^ i=l
diqi,
di G A ' , Qi € Q
292
XII. LOOPS O F UNITS
and, similarly, n
(2)
/x^=5]gX,
d\eD,,q[eQ.
1
It is important to realize t h a t there is no reason here to expect the qi or q[ to be distinct. We can and do assume, however, t h a t Tqi
Tqn
and
Tq[
Tq'^.
Since for q^q' ^ L^ a ^ Di^ b G Dj and i / j , we have qa ' bq = q\ai
• bq) = q{a2b • q),
for some a i , a 2 G Di^ it follows t h a t q'a  bq — 0. Therefore, n 1
and each q^d^ • rfj^^ /
0. The left hand side has support in T while the
terms of the right hand side have supports in Tq[q\^... ordered Tq[q\
< • • • < Tq'^q^.
^Tq'^qn which are
If the first and the last of these cosets are
different, we have a contradiction. Thus each Tq[qi — T, so each q[qi G T and, by choice of the transversal Q, q[ — q~^ for each i. In the expression fi — Yl^i^ii
collect equal ^^ and write qeQ
where each /i^ is a sum of certain d^. Since /i has coefficients in Z, each ^q G ZT. Note t h a t ^q^q' = 0 \f q ^ q' since the division rings represented by the di in /i^ are all different from those represented in /i^/. Since q'^ = ^ ~ \ the q' in expression (2) for /i~^ will collect exactly as the qi in the expression for //, so t h a t we have
qeQ
with z/gZ^o/ = 0 if g' / ^'. Also, I = fifX ^ =
^ q.reQ
fiqq • r
^Ur.
1. REDUCTION TO TORSION LOOPS
293
The left side has support in T while figq • r~^i/r does not, unless r — q. So (3)
1=
^^iqqq^Uq,
Choose some fig^ ^ 0. We claim t h a t {li^q ' q^i/q)fig^qo = 0
if q yi qo.
Indeed, observe first t h a t the element in question can be expressed as a linear combination of terms of the form {aq • q~^b)cqo where b and c come from different components Di. Now (4)
aq • q~^b = (aq • q~^)bi — ab\^
where 6i and b come from the same Di] hence [aq • q~^b)cqo — ab\ • cq^ — [abi ' ci)qo — (a^ •biCi)qo^ with a and ai in the same division algebra, and c and ci in the same division algebra. Since b\Ci = 0, (fXqq • q~^i/q)iigQqo = 0 and the claim follows. From (3), we now obtain /i^o^o = if^qoQo'Qo^ i^qo)l^qoQo sind, since x = xyx implies that xy is an idempotent, we see that i^igQqo • %^i^qQ is a (nonzero) idempotent. A calculation similar to (4) (using normality of T ) shows t h a t f^qoQo'Qo^^qo ^ ZT. Since idempotents in ZL are trivial (Corollary VIII. 1.7), f^qoQo'Qo ^qo = 1 So q^ Ug^ and, therefore, Ug^ are invertible. Similarly, j^ig^ is invertible. Since ^gfig^ — Q iox q :^ q^/xt follows that /x^ = 0 for q ^ q^ and fi = fig^qo G [U{ZT)]L,
So we have shown that U{ZL)
C [ZY(ZT)]L.
The other inclusion is obvious. Finally, if i e L smd ji e U{ZT), U{ZT),
so [U{ZT)]L
= L[U{ZT)]
then both i'^lit
and £/i£^ are in
follows from //£ = £ ( £  V ^ ) and i^i =
1.3 P r o p o s i t i o n . Let L be an RA loop with torsion either an abelian group or a hamiltonian
subloop T.
If T is
2loop and if every subloop ofT
is
normal in L, then U{ZL) = [U{ZT)\L PROOF.
Clearly [ZY(ZT)]L C U{ZL),
=
L[U{ZT)].
To prove the other inclusion, we
have to show t h a t , if /x G ZY(ZL), then /i G [ZY(ZT)]L.
Replacing L by
the loop generated by the support of /i and three elements which do not associate, if necessary, we may assume t h a t L is finitely generated. Hence,
294
XII. LOOPS OF UNITS
by Lemma VIII.4.1, T is finite. We claim that QT is a direct sum of division rings. If T is an abelian group, this is obvious. If T is a hamiltonian 2loop, then T = Qs X E OT T = MieiQs) x E^ where £ is a (possibly trivial) elementary abelian 2group. So, by Corollaries VII.2.3 and VII.2.4, either QT = {QE)Qs = {®Q)Qs = ®QQs = e(4Q © H(Q)) or QT  {QE)[Mre{Qs)] = (®Q)[M^e{Q8)] = ©Q[Mi6(g8)] = ©(8Q ® (Q,  1 ,  1 ,  1 ) ) . By Theorem 1.3.4, both H(Q) = ( Q ,  l ,  l ) and ( Q ,  1 ,  1 ,  1 ) are division algebras. So QT is indeed the direct sum of division rings; in particular, every idempotent of QT is central in QT. Hence, by Theorem VII. 1.4 and Proposition VII.2.1, every idempotent of QT is a linear combination of idempotents of the type jff, where ^ is a subloop of T. Since every subloop of T is normal in L, it follows from Lemma VI. 1.2 that every idempotent of QT is central in QL, so the result follows from Corollary 1.2. D
2. Group identities Let X — {a:i,X2,...} be a countable set and let F be the free group on X. If G is a group and w = K;(XI,. .. ^Xn) is a nonempty reduced word of F in the variables x i , . . . ,Xn, we say that K; = 1 is a group identity for G if w{gi^... ^gn) = 1 for all ^ i , . . . ,^7^ G G, Without restriction, we may assume that 11; is a word in at least two variables, for if x^ = 1 is a group identity for the group G, then so is (0:1X2)^ = 1. Furthermore, since a free group of finite rank can be considered as a subgroup of a free group of rank 2 (see, for example, [Rob82, Theorem 6.1.1]), we may assume that it; is a word in precisely two variables. Since Moufang loops are diassociative, this notion can be extended. 2.1 Definition. A Moufang loop L satisfies a group identity if there exists a nonempty reduced word w = w{xi^X2) in the free group F on {xi^X2} such that w(£i,i2) = 1 for all ^1,^2 ^ L,
2. GROUP IDENTITIES
295
2.2 Examples. 1. An RA loop L satisfies the identity {xyx'^y'^y — 1 (see Theorem IV.1.8). 2. An nEngel loop is, by definition, a loop satisfying the identity (... ((x, y), y ) , . . . , y) = 1 (t/ repeated n times). 3. A Moufang loop L is said to be an FC loop if, for every a £ L^ the set [a] = {x~^ax \ x £ L} is finite. Suppose C = {^1,... , gk) is a finitely generated group such that U{ZG) is an FC group. For each i, 1 < i < k, let Ui = {u e U{ZG) \ (u^gi) = 1}, the centralizer of gi in ZY(ZG). Since U{7.G) is FC, each Ui has finite index in U{ZG). Hence U{ZG) is a finite extension of its centre Z{U{ZG)) = r\i
L2^'"2
in
Ln = {l}
of subloops Li, all normal in L, such that Lii/Li is contained in Z{L/Li) for z = 1 , . . . ,n. Just as in group theory, a Moufang loop has an upper central series {1} = Zo{L) C Zi{L) C Z2{L) C ... Zi^\[L)lZi[L) — Z{L/Zi{L))^ which terminates at L in a finite number of steps if and only if L is nilpotent. In this case, the smallest n with Zn{L) = L is called the nilpotency class. A Moufang loop L also has a lower central series L = ML)
2 7i{L) D ML)
2 •••
where 71 (L) = L' and, for i > 1, 7.+i(L) = {{L, j,iL)),
(L,L, f,iL)),{L,7.(1),
I),{7^iL), L, L))
is the subloop generated by all commutators and associators of the indicated types. The loop L is nilpotent if and only if the lower central series terminates in a finite number of steps. When this occurs, the nilpotency
296
XII. LOOPS OF UNITS
class is the smaDest n for which jn{L)
= {1} (and this definition agrees
with the previous). 2.3 L e m m a ( M a l ' c e v [Mal53]). A nilpotent tency class n satisfies defined inductively
the group identity
loop L of nilpo
x^ = yn, where Xn and yn are
by
xo = x,yo = y and, for m> PROOF.
Moufang
0, XTn\i = Xmym, 2/m+i ~ yrnXm
We prove the lemma by induction on n. If n < 1, then L is
commutative, so L satisfies xi = yi.
Now assume t h a t L has nilpotency
class n > 1 and t h a t the result holds for nilpotent loops of nilpotency class less than n. Since the loop L/Z(L) hypothesis yields t h a t Xn\Z[L)
is of nilpotency class n— 1, the induction — yniZ{L).
It follows easily t h a t
x^i
and yn'i commute, so Xn = yn
D
In [ M a l 5 3 ] , Mal'cev shows t h a t nilpotent groups can actually be characterized as those groups satisfying a certain group identity. To be more precise, let XQ, t/o, ^ i ^ ^2? • • • he a set of independent generators for a free group. Define words Xn and Y^ inductively as follows: XQ
=
XQ,
YQ
= yo
and, for n > 0,
It turns out that a group G is nilpotent of class n if and only if n is the least positive integer such t h a t Xn = Vn is a group identity for G. For a positive integer m, recall that a subset A^ of a ring A is said to be nil of bounded exponent
less than or equal to m if a ^ = 0 for all a G A^.
The following result is well known. Our proof, which appears in [ R o w 8 8 , Proposition 2.6.26], is due to A. Klein. 2.4 L e m m a . Let R be an associative
ring with I, let m be a positive
and let I be a nonzero nil right ideal of bounded exponent to m. Ifxel PROOF.
with x ^  ^ 7^ 0, then {Rx'^'^Rf
integer
less than or equal
= {0}.
Note t h a t m > 2 since / is nonzero. To prove the result, it is
suflScient to show t h a t x'^~^rx'^~^
— 0 for any r G i2. So assume r £ R and
2. GROUP IDENTITIES
let r' = x'^~^r.
297
Then xr^ = 0 and ( r ' ) ^ = 0, because r' G / . Hence 771 — 1
For any i, 1 < i < m — 2, we have (^r j
X = (x
rj
X = X
rx
ri,
for some r^ G Ra;. Thus
It follows t h a t 0 = ( r ' x ^  i ) ( l + r ' ' ) , where r'' = J2T=~i^ ^ G /2x. Since Rx is nil, the element r ' ' is nilpotent; therefore, 1 + r'' is invertible. So we obtain
0 = r'x^i = x^Vx^^ 2.5 P r o p o s i t i o n . [ G J V 9 4 ] an infinite identity.
commutative
n Let A be an associative
algebra with 1 over
domain R and suppose that LI (A) satisfies
a group
Then there exists a positive integer m such that, for all a^b^c ^ A
with a^ = be — 0, the right ideal bacA is nil of bounded exponent
less than
or equal to m. PROOF.
By assumption, 11(A) satisfies an identity in the variables x\
and X2 of the form
where A; > 1 and r^, 5^ are integers, all of which are nonzero except possibly r^^i.
If we make the substitution xi = xy^ X2 = yx (where x and y are two
variables), then we get a group identity for U{A) of the form (6)
{xyY^iyxY' '' •{xyY^iyxY^ixyY^^^ = 1. We first establish the proposition in the special case where b = c. So, let
a, 6 G ^ be such t h a t a^ = b^ = 0. We claim t h a t abaA is a nil right ideal of bounded exponent less than or equal t o m, where m is a number given by a formula in the Vi and Si. We consider separately the cases c h a r i ? = 2 and char/Z 7^ 2. Suppose t h a t char i2 ^ 2. We show t h a t k
m = 2 + 2r^+i + 2 ^ (  r ,  + 5,).
XII. LOOPS O F UNITS
298
For u G A, we have (aua)^ = (bauab)^ = 0, hence 1 + aua and 1 + bauab are in U{A). We evaluate the group identity in U{A) by replacing x and y with 1 + aua and 1 + bauab respectively. We get ((1 + aua){\ + bauab)y^{{\ + bauab){l + aua)y'' • • • • • •((! + bauab){\ + aua)y^{{l
+ aua){\ + bauab)y^'^'^ = 1,
for all u e A and, after computing all products, an identity of the form f{aua^ bauab) = 0, where /(xi,X2) is a nonzero polynomial in the noncommuting variables xi,a:2 with integral coefficients and zero constant term. Replacing u by A^^, with A G iZ, and expanding, this identity can be written in the form Api + X^P2 + '" +
= 0,
AV
where p i , . . . ,p^ are polynomial expressions in aua and bauab with integral coefficients. Since R is infinite, there exist i distinct elements A j , . . . , A^ in R. Then Pi
P2
0
Pi
0
0
where j Ai
A2
A?
^2
LM
A2
A2
• .
• A, 1 • A?
• ^i.
Since /Z is a domain, det A = ( Yli=\ ^i] Yliyji^i ~ Aj) 7^ 0. It follows that p^ z=: • •' = p^ = 0. Since each variable in (6) (when the left side is written as a reduced word) may appear with exponent either 1, —1, 2 or —2, pi = 0 implies 2'^{abauy = 0 for some integers r and 5, with 0 < r and 1 < 5 < m. To clarify this statement, suppose, for instance, that the r^ and the Si are all positive integers. Let aua = c and bauab = d. Since c ^ A with c^ = 0, we have (1 + c)^ = 1 + kc for any /: G Z, so we can write ((1 + Ac)(l + A d ) r ( ( l + Ad)(l + A c ) r • • • = 1
2. GROUP IDENTITIES
299
as ((1 + Ac)(l + Xd)y'\l
+ Ac)(l + 2Xd) . ((1 + Ac)(l + Xd)y'\l
+ 2 A c ) . . . = 1.
By a direct calculation, we get (A^cdp^2A^cd(A^cd)"i^2Ac + ( t e r m s of lower degree in A) = 0. Thus 2''X\cdyc^
+ (terms of lower degree in A) = 0,
where r > 0 , ^ > 1 , l < n <
rfc+i + E f = i ( k t l + \^i\) ^nd t G { 0 , 1 } . It
follows t h a t Q = p^ = 2^(cc/)^c^ Thus 2'^{auabauahY^^ t h a t 2''{abauy
= 0 and this implies
= 0, with 5 = 2n + 3 < m .
Since char i2 ^ 2, 2'^{abauy
— 0 implies {abauy
= 0 for all u ^ A. T h u s
abaA is a nil right ideal of bounded exponent less than or equal to m. Suppose now t h a t char/2 = 2. In the group identity (5), replace x^ by a:iX2 and X2 by 0:1X3. In this way, we obtain a group identity for U{A) of the form (7)
ix^X2y'{x^x^y^
• • • ( x i X 2 P ( a ^ i ^ 3 r ' ' ( ^ i ^ 2 r + ' = 1,
where each variable in the left hand side (written as a reduced word) may appear with exponent 1 or —1. Let u £ A.
Since chari? = 2, ((a + ba)u{a + o.b)y
1 + (a + ba)u{a + ab) is in U{A).
= 0, and thus
Evaluate the group identity (7) in U{A) by
replacing Xi with 1 + atxa, X2 with 1+6at/a6 and 0:3 with
\\[a\ba)u{a\ab).
We get ((1 + aua){\ + bauab)y^[{\
+ aua){\ + (a + ba)u{a + ab))y'
• • • == 1.
Now take X £ R and replace u by Xu. As above, we get an equality of the form Xpi + X^p2 + • • • + X^p£ = 0 and another Vandermonde determinant argument gives p£ = 0. By a direct calculation, it is easy to verify t h a t P£ = 0 implies {abauy
= 0 for some integer s with I < s < m and m again
given by a formula in the exponents r^ and Si. Thus abaA is a nil right ideal of bounded exponent less than or equal to m. This concludes the proof of the special case.
300
XII. LOOPS O F UNITS
T h e proposition in general now follows quickly. Let 6, c G A he such t h a t be — 0. Then (cn6)^ = 0 for any u E A and so, for a £ A with a^ = 0, cubacubA is a nil right ideal of bounded exponent less t h a n or equal to m. Hence bacA is a nil right ideal of bounded exponent less t h a n or equal to 3m+1.
D
Let A be an associative ring and S a multiplicatively closed subset of the centre of A such t h a t no element of 5 is a zero divisor in A (in particular 0 ^ 5).
The ring of fractions
localization
of A with respect to 5 , also called the
of A with respect to S (see, for example, [ C o h 7 7 ] ) , is the ring
s^A = {s\ \se s,ae A}, with addition and multiplication defined by s^^ai
+ S2^a2  {s\S2)~^{s2ai
{s~^ ai){s~^ a2) =
+ Sxa2),
{siS2)~^{aia2)
for a i , a2 G /I and 5i, ^2 G 5 . If 5 = / ? \ { 0 } for some subring R of the centre of 4 , then we write R~^A rather than the more cumbersome {R\ 2.6 L e m m a . Let A be an associative
algebra with unity and let R be a
subring of the centre of A whose nonzero elements A.
{0})~M.
are not zero divisors
Suppose that bac = 0 for any a^b^c ^ A with a^ — be — 0. Then
idempotent PROOF.
Let e be an idempotent in / 2 ~ M . Then, for some 0 ^ r £ R^ Hence f{r  f) = 0 and {fu{r
any u ^ A, From the hypothesis, we obtain f{fu{r
— f)){r
 f)^
any u ^ A. Similarly, from (r — / ) ( ( r — f)uf)f
= 0 for
— f) — 0. Since
(r — / ) ( r — / ) = r{r — / ) , we have r'^fu{r — / ) = 0, and so fur
— fuf
= 0, we obtain ruf
and consequently eu = ue for any u £ A. 2.7 C o r o l l a r y . [ G J V 9 4 ]
subring
PROOF.
of R~^A
associative
fuf
algebra with
of the centre of A whose
are not zero divisors in A, IfU(A)
every idempotent
=
for D
Let A be a semiprime
and let R be an infinite
elements
every
of R~^ A is central.
f = re e A dind f^ = rf.
unity
in
nonzero
satisfies a group identity,
then
is central.
Assume t h a t there exist a^b^c ^ A such t h a t a^ = be = 0. By
Proposition 2.5, bacA is a nil right ideal of bounded exponent.
Because
2. GROUP IDENTITIES
301
A is semiprime, Lemma 2.4 implies bac = 0. Now Lemma 2.6 gives the result.
n
2.8 C o r o l l a r y . Let L be an RA loop with torsion 2loop, then the following 1. U(TL)
conditions
satisfies a group
2. for any finitely group
are
identity;
generated group G contained
in L, K{ZG)
satisfies
a
identity;
subloop ofT
Moufang
loop and
every
= [U{ZT)]L
sat
is normal in L.
if any of these conditions
isfies the identity PROOF.
If T is a
equivalent:
3. T is an abelian group or a hamiltonian
Furthermore^
subloop T.
(x^^y^)
holds, thenU{ZL)
= 1.
Since Z does not have zero divisors, the rings ZG and ZL are
semiprime for any subgroup G of L, by Corollary VI.3.6 and Theorem VI.2.7. Clearly, if ZY(ZL) satisfies a group identity, then so does U(ZG) for any finitely generated subgroup G in L; hence (1) implies (2). Now assume (2). By Corollary 2.7, for any finitely generated group G contained in L, any idempotent in QG is central in QG.
Hence, if £ G L
and g ^ T has order n, then the idempotent n~^{\ + g + g^ +   • + g'^~^) is central in QG, where G is the group generated by g and i. It follows t h a t i{^g) = {g)i for any £ G ^ so, by Corollary IV. 1.11, the cyclic group (g) is normal in L. Since g is arbitrary in T , every subloop of T is normal in L, so T is an abelian group or a hamiltonian Moufang loop. Thus (3) holds. Finally assume (3). U{ZL) = [U{ZT)]L,
By Proposition 1.3 and because T is a 2loop,
In c a s e T is a hamiltonian Moufang 2loop, by Higman's
Theorem (Theorem VIII.3.2), U{ZT) Z{L)
= ± T , so U{ZL)
= ±L,
Since P e
for any £ G L, it follows in this case t h a t U{ZL) satisfies the identity
(x^,X2) = 1. So, for the rest of the proof, we assume T is an abelian group and show t h a t ji^ G U{ZT)Z{L) with V G U{ZT)
for any
JJL G U(ZL),
For this, write ji = ut
and ^ G //. By diassociativity, the loop generated by u and
^ is a group, thus
Because supp(i/) C T and because T is normal in L, iyt~^
G U{ZT).
It
follows t h a t /x^ G [ZY(ZT)]Z(iy), as asserted. Since \U(ZT)]Z{L)
is an abelian
302
XII. LOOPS OF UNITS
group, we again obtain t h a t U{ZL) satisfies the identity (3^1,2:2) = 1, hence proving (1). T h e result follows.
D
Although we now show t h a t Corollary 2.8 can be proved more directly, even without the restriction on the torsion subloop, we chose to offer the above proof primarily because of Proposition 2.5 and its possible applications. One such application is given in [ G J V 9 4 ] and [ G S V 9 1 ] , where it is shown t h a t , if the unit group of a group algebra KG of a torsion group G over an infinite field K satisfies a group identity, then KG satisfies a polynomial identity. Furthermore, in [ P a s 9 5 ] , necessary and sufficient conditions are given for U{KG)
to satisfy a group identity.
2.9 C o r o l l a r y . Let L be an RA loop with torsion following
conditions
are
1. ZY(ZL) satisfies a group
in L, U{7.G) satisfies
a
identity;
subloop ofT Furthermore,
Moufang
2loop and
every
is normal in L.
if any of these conditions
satisfies
PROOF.
the
identity;
3. T is an abelian group or a hamiltonian
U{ZL)
Then
equivalent:
2. for any finitely generated group G contained group
subloop T.
the identity
Assume U{ZL)
hold, then LI{ZL) — \JJ{2.T)]L and
[x'^^y'^) — 1. satisfies a group identity.
As in the proof of
Corollary 2.8, it follows t h a t (1) implies (2) and t h a t (2) implies t h a t T is either an abelian group or a hamiltonian Moufang loop with every subloop normal in L.
Since free groups do not satisfy group identities, it follows
from Corollary VIII.5.8 and Theorem VIII.3.2 t h a t T is a 2loop in the hamiltonian Moufang case. The remainder of the statement follows from Corollary 2.8.
•
We conclude this section by describing the RA loops L for which U{7.L) has various properties, nilpotence for example.
For this, we need three
technical lemmas. Recall t h a t a group G is said to be solvable if G has a series C ^ G o 3 G i D • • O G ,  {1} in which Gij^\ is a normal subgroup of Gi and Gi/Gi^\ 0,...,rl.
is abelian for i =
2. GROUP IDENTITIES
303
2.10 Lemma. Let G be a group such that one of the following is satisfied:
conditions
1. for every finitely generated solvable subgroup H ofU{ZG), is nilpotent or H' is torsion; 2. the torsion elements ofU{ZG) form a subgroup.
either H
Suppose the set T of torsion elements of G is an abelian group and every subgroup of T is normal in G. If G' — { l , ^ } , then, for any g ^ G and t e T, gtg^ = t ort~^. Let g^G^l^t^T and suppose t has order n. Since the cyclic group (t) is normal in G, we have g~^tg — f for some i, 1 < z < n and gcd(i,n) = 1. To prove the result, we assume that i / 1 (thus g and t do not commute) and show that i = n — I. Since g'Hgt'^ = f"\ it follows that f"^ = s. Hence g'^t^gt''^ = f2i2 _ 2 So n = 2i — 2. These observations imply, in particular, that n is even and i is odd. Consider the Bass cyclic unit PROOF.
6 = (1 + ^ + ... + f^ )^(^) +
f.
n Recalling the form of the inverse of such a unit (see §VIII.2) and noting that n I [1  i(l + f)], we have b^ ^(l+f
+ '" + f2)^(^)
+
1 n
= (1 + f + •. • + /^(^i))^W +
1, n Hence b~^ = g~^bg^ since t is central. Thus g~^bgb~^ = b~^. If the torsion elements oiU{TG) form a subgroup, then b~^ is of finite order (as a product of two torsion units). Otherwise, let H be the group generated by g and b. Then the preceding yields b~^ ^ H' — 71 (77). Since g'b^'^gb^'^
= 62^62^ = ft^"^^',
it follows by an induction argument that b'^"^ G 7m(^)? the mth group in the lower central series of H, for any m > 1. Since H is solvable, either H is nilpotent or H' is torsion and we see again that 6 has finite order. Since
304
XII. LOOPS OF UNITS
T is an abelian group, Theorem VIII. 1.4 yields t h a t 6 is a trivial unit. By Lemma VIII.2.7, i = n — 1 as required.
D
2 . 1 1 L e m m a . Let L be an RA loop with torsion subloop T. If the
torsion
units in the integral loop ring ILL form a suhloop, then every subloop of T is normal in L and T is either an abelian group or a hamiltonian
Moufang
2loop. P R O O F . Let t e T, i e L and consider a = ?i{l  t). Clearly a'^ = 0, so 1 + Q: is a unit in ZL. Since the loop generated by i and ^ is a group, the element /3 = {I + a)~^^(l + a ) is a torsion unit in the integral group ring Z{t,i).
Note t h a t
/? = (1  a ) ^ ( l + a) = t  at + a = t + a{l  t) = t + ?£{!  2t + t^). Since L/V
is abelian, Proposition VIII. 1.4 implies t h a t the torsion units
of Z[L/L'] are triviaL Hence the natural image of/3 in Z[L/L'] is a trivial torsion unit.
It is then not hard to see t h a t supp(/3) contains a torsion
element, say g ^T.
Since, by assumption, the torsion units ofU{ZL)
form
a subloop, the element l3g~^ is also a torsion unit and with 1 in its support. So, by Proposition VIII. 1.5, l3g~^ and hence also /3 are trivial units. Since it has coefficient 2 in 2tit^ it follows t h a t It = t or it = tH for some i > 0, or it = fit'^ for some i > 0. Hence, either ^ = 1, iti'^
= f or i'^Vi =
t'^;
thus C~^ti G (t). In particular, every subloop of T is normal in L, so T is either an abelian group or a hamiltonian Moufang loop. Let II = ^teT
l^tt be a torsion unit and suppose /x^ ^ 0 for some t G
T . Since the torsion units form a subloop, fit~^ is a torsion unit as well. Since T is locally finite (Lemma VIII.4.1), the support of /j.t~^ generates a finite subloop and therefore, by Proposition VIII. 1.1, fit~^ = ± 1 . Thus Consequently, for each
finitely
generated subgroup G of T , the (finite) group ± G is normal in
the torsion units of U{Z{L))
are trivial.
U{ZG)
so, for any /i G ZY(ZG), fi'^g/j,''^ = g for all g ^ G^ where n is the order of the automorphism group of ±G, Consequently U{ZG)
Thus /x^ is in the centre of
satisfies the group identity {x'^^x'2) — 1, so
U{ZG). U(ZG)
does not contain free subgroups of rank 2. By Theorem VIIL5.7, if G is not abelian, it is a 2group. Since G is an arbitrary finitely generated subgroup of T , we obtain t h a t T is either abelian or a hamiltonian Moufang 2loop.
D
2. GROUP IDENTITIES
305
We require a basic fact about divisible abelian groups. Recall that an abelian group D is said to be divisible if for each d E D and any positive integer m, there exists d\ £ D with d = d'^. Obviously, the additive groups Q and Q/Z are divisible. 2.12 Lemma. Let A be an abelian group with subgroup B and suppose f:B —> D is a group homomorphism into a divisible abelian group D, Then f extends to a homomorphism (fi: A ^ D. P R O O F . Let S be the set of all ordered pairs (H, Z) by setting (p\a) = 1. On the other hand, if a has finite order n modulo H ^ then a^ e H ^ and we can choose d £ D with d'^ = (p{d^). One then easily verifies that the map (p': (H^a) —> D given by (p'(ha^) = (p(h)d\ for h ^ H^ also extends (p. In either case, this contradicts the maximality of {H^ (p). D
The next lemma is standard in the study of dimension subgroups. Our proof is along the lines of the one given in [Seh78, Section III.l]. 2.13 Lemma. Let G be a group. 1. For any xi^... such that
^Xn E G^ and >2:i,... ,Zn ^ Z, there exists (3 £ Az(G)^
2. G n ( l + Az(G')2) P R O O F . TO
=G'.
prove 1, note that if m € Z, m > 0, and x £ G, then
m{x  1) = {x""  1) + (x  l ) (  x " '  ^  x'"2
x+
ml)
and m{x  1) = (x"'"  1) + (x  l ) (  m + x^ + • • • + x'"+^ + x""^).
306
XII. LOOPS OF UNITS
So m{x  1) € (x^  1) + A2(G)2 for any m € Z. Since (a  1) + (6  1) = {ab I){a
1)(6  1)
for any a^b £ G^ the first statement now follows by an induction argument. To prove 2, note that one inclusion follows immediately from the identity x~^y~^xy — 1 = x~^y~^{xy — yx) = x^y\{x
 \){y l){y
\){x  1)],
which holds for any x^y £ G, To prove the reverse inclusion, it is sufficient to show that \i g e G\G', then g I ^ Az{G)'^. For x e G,\et x = xG', Since 'g is not the identity element in the group GjG'^ there exists a nonzero homomorphism f^) ^ E — (^jZ. Because ^ is a divisible group, this map extends to a homomorphism / : G/G' —^ /T, by Lemma 2.12. Since f{'g) ^ 0, the map / : ZT —» H defined by a = ^ a^x 1^ ^ ckxf{x) xeG xeG is a homomorphism of additive groups. Since f({xl){yl))
=
f{xyxyl)
= 7(^)7(x)7(y)7(T)0, it follows that /(Az(G')2) = {0}. Since f{g  I) = J{g)  J{\) = Jig) ^ 0, clearly g\ ^ Az(G)^ D R. Baer has proven that a solvable nEngel finitely generated group is nilpotent (see, for example, [Rob82, 12.3.7, page 360]). The following result is based upon and unifies the main theorems of [GM95]. 2.14 Corollary. Let L be an RA loop with torsion subloop T. following are equivalent: 1. 2. 3. 4. 5. 6. 7.
U{ZL) is an RA loop. U{ZL) is nilpotent. U(ZL) is nilpotent of class 2. \U{ZL)\' is a group of order 2. U{ZL) is nEngel for some n > 2. U{ZL) is 2Engel. U{ZL) is FC.
Then the
2. GROUP IDENTITIES
307
8. [ll{ZL)Y is a torsion loop. 9. The torsion units in ZL form a subloop. 10. T is an abelian group or a hamiltonian Moufang 2loop and x~^tx = t"^^ for any t ^ T and any x £ L, Moreover^ ifT is an abelian group and i £ L is an element that does not centralize T, then £~^ti = t~^ for all t E T. PROOF.
Consider also the following condition.
10'. T is an abelian group or a hamiltonian Moufang 2loop and every subloop of T is normal in L. Any RA loop L whose unit loop satisfies any of (1) through (7) has the property that, for any finitely generated group G contained in L, U{ZG) satisfies a group identity. Hence, because of Corollary 2.9, any of these conditions implies (10'). The following implications are also clear: • • • • •
(1) (1) (1) (1) (1)
=> (3) (2); (4); ==> =^ (7); = ^ (6) = ^ (5); = > (8) => (9). = > •
Because of Lemma 2.11, (9) and, hence, also (8) imply (10'). Next, we show that any of (2), (4), (5), (7), (9) (hence any of the conditions (1) through (9)) implies (10). Note that in each case, we also have (10'). If (2) or (5) holds, then for any finitely generated solvable group H contained in ZY(ZL), the group H is nilpotent. In the case of (4) or (7), we know (see, for example, [Pas77, Lemma 4.1.6]) that H^ is torsion for any finitely generated solvable group H contained in U{ZL). So, in all cases under consideration, if T is an abelian group, applying Lemma 2.10 to 2generated subgroups, we have i~^tt = t^^ for any t e T and i ^ L. If T is a hamiltonian Moufang 2loop, this condition is obviously satisfied as well. So, to prove (10), we now assume that T is an abelian group and that ^ G T is not central. For some i e L,we have i'^ti = t'^ with t'^ ^ 1. We have to prove that i'^tii = t~^ for any ti G T. Suppose to the contrary that ^i € T, i'^tii = ti and tl ^ I. Since ^i and t commute, the three elements ^, /, ^i associate, by
308
XII. LOOPS OF UNITS
Corollary IV.1.3, so they generate a group. Thus
and so tit'^
= tit
or
tit'^
=
tl'^t'^,
contradicting t'^ ^ I ^ t\. It suffices finally to show that (10) implies (1). For this, we first claim that, if (10) holds, then U{ZL) = HL^ where 7i is a subgroup oi Z{U{ZL)). By Corollary 2.9, U{ZL) = [ZY(ZT)]Z/, so the claim is obvious if T is central. If T is a hamiltonian Moufang 2loop, then, by Higman's Theorem (Theorem VIII.3.2), U{ZT) = ±T and again the claim follows. In the remainder of the proof of the claim, we assume that T is an abelian group and not central. For any a = YjteT^ti € ZT, let a' = J2teT^tt~^' Note that (a/?)' = a'/3'. The assumptions in (10) imply that, if / is not central, then t~^ = f^ = st, where s is the unique nonidentity commutatorassociator of L. Therefore, if a' = a, then a is a linear combination of elements t G Z{L) and elements of the form t + t* = (1 + s)t^ which are also central, by Corollary III. 1.5. Hence Tl = {a E U{ZT) \ a = a'} is a central subloop of ZY(ZL). We prove that ZY(ZL) = HL. For this, let a = Xllli ^iU G U{ZT), with each z, e Z and ti G T. It is sufficient to show that a G HL in the case that a has augmentation 1. By Lemma 2.13, m
m
a= 1 + 5^^,(^,1)= (n^'O +/^ = ^o + /3 for some /3 G A(T)2, where ^o = UZi^TThus a = (I + 6)to, with 6 = f3t^ G A(T)2and l + SeU{ZT). Let j = (1 +Sy(I+ S)^ = Eterltt, each jt € Z. Then 7^7 = 1, so "^t^x^t ~ 1 Since 7 also has augmentation 1, 7 = ^ for some g £ T. Consequently, again by Lemma 2.13, ^ = (1 + sy{l + S)' G T n (1 + A{Tf)
= r ^ {1}.
Hence I + 6 = {I + Sy e H, Therefore a G HL, So U{ZT) C HL and U{ZL) = HL^ as required. This proves the claim. Since H. is central, HL has the same unique nonidentity commutatorassociator as L and, since L has the LC property, so does HL. Corollary IV.2.2 says that U{ZL) is an RA loop. D
3. THE CENTRE OF THE UNIT LOOP
309
3. T h e centre of t h e unit l o o p In this section, as a consequence of the famous Dirichlet Unit Theorem, we show that the unit group of the integral group ring of a finite abelian group is finitely generated. Then we exhibit a subloop of finite index in Z{U(T{L)))^ where L is an arbitrary finite RA loop. We begin with some terminology. Let K = Q(0) be an algebraic number field and let f(X) G Q[^] be the minimal polynomial of 9. Suppose that f{X) has ri real and r2 pairs of complex conjugate roots. The pair [ri,r2] is called the signature of K. If r2 = 0, then K is called totally real while, if ri = 0, K is said to be totally imaginary. 3.1 Theorem (Dirichlet Unit Theorem). Let R be the ring of integers of an algebraic number field K and let [ri,r2] be the signature of K. The unit group U{R) of R is the direct product of the cyclic group of roots of unity in K and a free abelian group of rank ri + r2 — 1. PROOF.
We refer the reader to [Kar88, Corollary IV.3.13].
D
Let ^ be a complex root of unity. Since Z[^] is a Zorder contained in the ring R of integers of Q(0? it follows from Lemma VIII.2.6 that the commutative groups U{R) and ZY(Z[^]) have the same torsionfree rank. Hence the Dirichlet Unit Theorem has at once the following application. 3.2 Corollary. Let n > 1 and let ^^ be a primitive nth root of unity. Then U{2[i^ri\) = (i^n) X F, a direct product, where F is a finitely generated free abelian group. Furthermore, if n > 2, the rank p{F) of F is ^(n) — 1; otherwise p{F) — 0. 3.3 Theorem (Higman). If A is a finite abelian group, then U{ZA) =
±AxF,
where F is a free abelian group of rank p{F)=\{\A\
+
n^{A)2c{A)^\),
where n2{A) is the number of elements of order 2 in A and c[A) is the number of cyclic subgroups of A. In particular, the only torsion units in ZA are the trivial units.
310
XII. LOOPS OF UNITS
Let n be the order of ^ . From Theorem VII.1.4 and the ensuing remark, we know that QA = 0 1 a^Qi^d) stnd PROOF.
ZAC^^^^adZ[U]
= R.
where each ^d is a primitive dth root of unity and ad is the number of cyclic subgroups of order d. Recall that (d) is the Qdimension of Q{^d)' By Lemma VIII.2.6, [1{(R): 1{(ZA)] < oo. Hence, because of Corollary 3.2, both these abelian groups are finitely generated with the same torsionfree rank. By Theorem VIII. 1.4, aU torsion units in ZA are trivial, hence U{ZA) = ±A X F , where F is a finitely generated free abelian group of finite rank p{F). Furthermore, Corollary 3.2 yields
p{F)= Yl a , ( ^  l ) d\n4>2
d\n4>2
c/n,d>2
= \ ( Z ) °<''^(^)  2 ^ ad + n2{A) + l) d\n
d\n
= \{n~\n2{A)2c{a)
+ l),
D
Higman's Theorem can be used to determine when the free abelian part is trivial (that is, when all units are trivial), but we have already established this result by more elementary means in Chapter VIII. We now give the structure of the loop of central units of the integral loop ring ZL of a finite RA loop L. 3.4 Proposition. Let L be a finite RA loop. Then Z{U(ZL)) = where F is a free abelian group of rank 1 (\L\ 2 \ 2
\Z{L)\ + n^iL/L') 2
 n2{Z{L)IL') + {c{LlL')
±Z(L)F,
+ n2{Z{L)) + l ) +
c{Z{L)IL')c{Z{L)))
Let G be a nonabelian group such that L — M(G,*,^o) ^ii^d, as usual, let s denote the nontrivial commutatorassociator of L. Write R — Z{ZL). Note that Z{U{ZL)) = U{Z{ZL))^ because both loops are equal to ZY(ZL) n Z{ZL). Clearly R C R^^ ® R ^ , and both R and R^^ 0 R ^ PROOF.
3. THE CENTRE OF THE UNIT LOOP
311
are Zorders in Z{QL). By Lemma VIII.2.6, U(R) is a subgroup of finite index in the direct product U{R^^) X U{R^^)^ so its torsionfree rank is the same as that of the direct product. We concentrate on each direct factor separately. First, since ZL^^ = Z[LIV] is commutative, we have R^^ = Z[L/V], Theorem 3.3 therefore yields that U{R^^) = ±LFi^^, where Fi is a free abelian group of rank
By CoroUary VI.1.3, R ^ = Z[Z{L)]^.
Because
Z[Z{L)]CZ[Z{L)y^®Z[Z{Lr^ and since the rings Z[Z(L)] and Z[Z{L)]^^®Z[Z(L)]^ are each Zorders in the ring Q[Z(L)], it follows from Lemma Vin.2.6 and Theorem 3.3 that U{R^^) = ±Z{L)F2^ where F2 is a finitely generated free abelian group. Furthermore, since Z[Z{L)]^^ =. Z[Z{L)IL']^ we also obtain the rank of
piF,) = i {\Z{L)\ + n2(2(L))  2c(Z(L)) + 1)  i {\Z{L)/L'\
+ n2iZ{L)/L')
 2c(Z(L)/L') + 1).
So U{R) is a subgroup of finite index in a finitely generated abelian group of torsionfree rank p{Fi) + p{F2). Hence U{R) has torsionfree rank p{Fi) \p{F2) and the result follows. D Theorem 3.3 gives a complete description of the structure of the unit group of the integral group ring of a finite abelian group. To obtain a presentation of this group, one would also like to have a description of the generators of the free abelian part. This, however, is a very difficult problem, as even the analogous problem forZY(Z[^]) is unsolved. Therefore, as a compromise, one hopes at least to find a description for generators of a subgroup of finite index in the unit group. In this direction, we have a fundamental theorem due to H. Bass and J. Milnor [Bas66]. For a detailed proof, we refer to [Seh93, Chapter 2].
312
XII. LOOPS OF UNITS
Recall that the Bass cyclic units in the integral loop ring of a finite RA loop are the units of the type 1 _ 7<^(^)
n where g £ L has finite order n, 1 < i < n, gcd(i,n) = 1 and ^ = 1 + ^ +
3.5 Theorem (Bass—Milnor). Let A be a finite abelian group. Then the group generated by the Bass cyclic units is of finite index in U{2.A), Next we give generators for a subloop of finite index in Z{U{2.L)). Recall (Corollary III.4.3 and preceding remarks) that r G ZL is central if and only if r = r* and, therefore, that elements of the form rr* are always central. In particular, if 6 is a Bass cyclic unit, then 66* is a central unit. 3.6 Theorem. Let L be a finite RA loop. Then the loop B = (66* I 6 a Bass cyclic unit in ZL) is of finite index in
Z(U{ZL)).
Since Z{L) is a finite abelian group, the (finitely many) Bass cyclic units, hence also the squares of the Bass cyclic units of Z[Z(Z/)], generate a subgroup of finite index in U{Z[Z{L)])^ by the BassMilnor Theorem. Call this subgroup J9i and note that B\ C B since 6^ = 66* for any central 6. Since PROOF.
Z[Z{L)]CZ[Z{L)Y±^®Z[Z{L)]
15
and because both rings are Zorders in Q[>Z(Z/)], Lemma VIII.2.6 yields that 5 i , and thus also 5 , contain subgroups of finite index in the group U{Z[Z{L)])^^, Note that we have identified the latter group with ^ ^ + U{Z[Z{L)]y^. As before, we have also 2(ZI)CZLl±^®Z[2(L)]lf^, each ring a Zorder in 2(QL), so, to prove the theorem, it is sufficient to show that B ^ = {v'^ \ v £ B] is oi finite index in U{ZL)^^ ^
3. THE CENTRE OF THE UNIT LOOP
313
U(Z[L/V]). We claim that if 6 is a Bass cyclic unit in Z[L/V]^ then there exists a Bass cyclic unit 6 G ZL with b^^ = 6 or 6 . Indeed, write _
1 — 7^("') 71
where ^ G L/V has finite order n, 1 < i < n, and gcd(i,n) = 1. Clearly g E L has order m, and m = n or 2n, Because of Proposition V. 1.1, if n is odd, we may assume that n — m. Consider then the Bass cyclic unit 1 — j4>{'^) m
If n = m (for example, if n is odd), then clearly b^^ hand, if m = 2n and n is even, then
= b. On the other
+ 2(i + 5 +  + r')*<"^^^^—9+^^—^—g. n
n
Since 'g^'g = 'g for any i > 0, it foUows that
V
n
n
= (1 + ^ + . . . + ^l)20(n) ^ (^2i'^(n) ^ J _ .^(n)^ ^ ~ ^
/ f
= (T+5 + • • •+r'')'*^"^ + H i + ^^^"^) (1  ^*^"^) t
= (T+5 + • • •+r')*^'"> + ^(1  i*^'"')?. Because ^ 5 * = ii^gr''^'" for any i > 0, it follows that 'g = ^ 5 . Hence 6' = (T + p + • • • + 5^1 )'^() + X ( l _ i't>(m)^^\±l = (1 + P + • • • + 5^)^()l±^ + 2^(1  i*('"))5i^ = 6if^, which establishes our claim. Since LjV is abelian, the claim and Theorem 3.5 once again imply that B^^ is indeed of finite index in U{ZL)^^ =
U(Z[LIV]),
D
314
XII. LOOPS OF UNITS
4. D e s c r i b i n g large s u b g r o u p s Let i be a finite RA loop. In this section, we exhibit a snbloop V of finite index in the unit loop of ZL. In many cases, it turns out that V is a normal complement of ±L in the sense that V < U(ZL), V H L = {1} and U{ZL) = ±LV. The results are based mainly on work by E. Jespers and G. Leal [JL91, JL93b]. We introduce two subloops of the unit loop of ZL. Let L be an RA loop with V = {1,5} and let Ui and U2 be the following subloops oiU{ZL): Ui = U{ZL) n {QL^^
+ ^ )  U{ZL) n {ZL^^
+
^ )
U2 = U{ZL) n ( ^ + Q L i ^ ) = U{ZL) n (i±^ + ZL^f^). We list a few properties of these loops. As usual, the augmentation map QL ^^ Q is denoted €. 4.1 Proposition. Let L be a finite RA loop with L' = {^f^}
Then the
following statements are true. 1. 2. 3. 4.
Hi = {fieU{ZL) \^= l + a(l + s),a € ZL}. ZY2 = {/i G ZY(ZL) I /i = 1 + a ( l  6),a G ZL}. Hi is a central subloop ofU{ZL) and Hi 01^2 = {I}V = V(ZL) = {/i G U{ZL) I /i = 1 + a ( l  5),a G ZL, 6(a) even} is a torsionfree normal subloop ofU{ZL) and V = H2/{l',s}. 5. UiV is a normal subloop of finite index in U{ZL). 6. / / LIV has exponent at most 4, then U{ZL) = ±LV; thus V is a torsionfree normal complement of ±L inU{ZL). P R O O F . For (1), let A = {^i e U{ZL)
 /x = 1 + a ( l + 5),a
G ZL}.
If
^  \ \ a ( l + 5) G A, then
/ ^ = ^ + ^ ( l + 2a), hence A CUi. Conversely, let ji — Ylg^LJ^gQ G ZYi, where each /x^ G Z. Then // = OL^^ + ^ ^ for some a = YlgeL^gd ^ ^ ^ ' ^^^^ ^9 ^ ^* Since 5(1 + 5) = 1 + 5, we may assume that supp(a) is in a transversal (containing 1) of L\ Consequently, for g G supp(a), it follows that asg = 0 and s ^ supp(a). Comparing coefficients, we see that if 1 ^ ^ G supp(a), then IXg = ^ag ^ Z, Therefore ag is even. Similarly, comparing coefficients.
4. DESCRIBING LARGE SUBGROUPS
315
/xi = i a i + i G Z. Hence a i is odd. Thus a = 1 + 2/3 with f3 G ZL. Therefore /i = (1 + 2 / ? ) ^ + ^ = 1 + /?(1 + 5); that is, fi e A, The proof of (2) is similar. For (3), since ZL^^ = Z[L/V] is commutative, Hi is a central subloop of U{ZL). Assume /i eHi nZY2. Since /x € ZYi, ( / /  1)(1  5 ) = 0. Furthermore, (/i  1)(1 + 5) = 0 since // G K2' It follows that /x  1 = 0; that is, // = 1. To prove (4), let /x = 1 f a ( l  s) e U2 with a = Y^g^L^gd ^ ^^• Then /X5 = 1 — (1 + a ) ( l — 5), so multiplication by s changes the parity of 6(a). Thus U2 — V{1,5}. Next we show that s ^V. To the contrary, if s — 1 + a ( l — 5) where a = Y^g^L^gd ^ ^^ ^^^ ^(^) ^^en, then a^ = ags for ^ ^ {1,5}, so a = a'(l + 5) + a i + a^^ for some a' G ZL. Consequently, 5 = 1 + (ai + a55)(l — 5) and thus 0 = 1 + a i — a5, contradicting the fact that e(a) is even. In order to prove that V is a loop, it is enough to note that V is closed under multiplication and inverses. The former is clear and the latter follows because, if //"^ ^ V, then, by the above, /x~^ = i/s for some i/ G V. However this implies 5 = //^' G V, a contradiction. To prove that V is torsionfree, note that each element /x G V has 1 or s in its support, and also that €(/i) = 1. So, by Corollary VIII.1.2, if // is a torsion unit in V, then fj, = I or s. But the latter is impossible, as we already have seen. Hence V is torsionfree. It is now also clear that U2 is the direct product V X { l , ^ } . To finish the proof of (4), we show that V is a normal subloop of ZY(ZL). Let ly = I + a(l — s) E V with e(a) even. If //i,//2 € U(ZL) and if 9 is any of the inner maps T(/xi), i?(/xi,//2), L(/ii,/X2) (see §11.1.9), then 19 = I and s9 = s^ so 1/9 = I + (Q:^)(1 — s) which is in V because €(a) even implies €{a9) even. Hence V is indeed a normal subloop. To prove (5), we first note that ZYiV is a normal subloop of U{ZL) because U\ is central and V is normal. Since the index of V in U2 is 2, to show that U\V has finite index in ZY(ZL), it is enough to prove that U1U2 is of finite index in U{ZL), For this, we claim first that U{Z + 2ZL) C ZY1ZY2. For if ^ =z m+2j G ZY(Z + 2ZL), then fx~^ = n+2<5, where m, n G Z. Consequently, 1 = rnn + 2(7 + 6 + 2j6)^ so mn (and hence m) are odd. Write m = 1 + 2m',
316
XII. LOOPS OF UNITS
m' G Z. Then // = 1 + 2(m' + 7). Since ji = M 2^ "^ /^ 2^' ^^ obtain /x = 1 + 2(m' + 7 ) i ^ + 2{m' + 7 ) ^ ^ = 1 + (m' + 7)(1 + 5) + (m' + 7)(1  s) = [1 + (m' + 7)(1 + s)][{l + (m' + 7)(1  s)l which is an element of ZYiZY2 This establishes the claim. Since Z+2ZL C ZL and since the rings Z + 2ZL and ZL are each Zorders in QL, it follows from Lemma VIIL2.6 that U{Z + ZL) is of finite index in ZY(ZL). By our earlier claim, U1U2 is of finite index in ZY(ZL), giving (5). By Theorem VIII.3.1, Z[LlV] has only trivial units. From this observation, it follows easily that U{ZL) = zbLV. Since V fl L = {1}, the result in (6) also follows. D In Theorem 3.6, we gave generators for a subgroup of finite index in the centre oiU{ZL), Since U\ is central, we therefore know this subgroup implicitly, up to finite index. We now show how to compute V when L has cyclic center. In so doing, we consider two separate cases. First we introduce some notation. Recall that GLL(2, R) denotes the general linear loop; that is, the loop of invertible 2 x 2 matrices in Zorn's matrix algebra 3 ( ^ ) over a commutative ring R. (See §11.5.5.) Let ^n be a primitive nth root of unity. Then Q(^n) is a Galois extension of Q and its Galois group is denoted Gal(Q(^n)/Q) The norm n{a) of an element a G Q(^n) is defined by
^(^) = n ^(^)^EGal(Q(e„)/Q)
It is well known (see, for example, [Kar88, Proposition 1.2.29]) that a G Z[^n] is invertible in Z[^ri] if and only if 71(a) = ± 1 . For a subloop S of GLL(2, Z[^n]), let 5n(det)=±i = { ^ G 5  n ( d e t A ) = ± l } and '^n(det)=±l = Sn{det)=±\l{I'>
I]
where / is the identity matrix of 3 ( ^ )  We also write 5'det=±i = { ^ G 5  d e t A . r ±1}
4. DESCRIBING LARGE SUBGROUPS
317
and *5'det=±l = Sdet=±l/{Ij
I}'
4.2 Theorem. Let L be a loop of type C\ with cyclic centre generated by an element ti of order 2'^^. Let ^ be a primitive 2'^^ th root of unity and let
V':QLi^3(Q(0) be the mapping defined in Proposition VIL2.7, Then V = V ^ ^ and the restriction of if) to V ^ ^ induces an isomorphism from V onto ' ^ ( V ^ Y ^ ) , where
1. V(vif^) = 1 + 2a 2v 2w 1 + 26
1 + 2Z[^] 2(Z[^])3
2(Z[^])3 1+2Z[^]
n(det) = l
a + 6 € 2Z[e], V + w G 2(Z[^])= if mi > 1, and
2. i^iV'^) = l + 2a 2w
2v 1 + 26
1 + 2Z 2Z^ 2Z^ 1 + 2Z
J det=l
a + 6 G 2Z, v + w G 2Z^ if mi = 1. PROOF.
Assume first that mi > 1. Given an element
a = l + 2[(7o + Jix + 722/ + 73xy) + (70 + 71^^ + 72^ + 73^2/)^] l  s
318
XII. LOOPS OF UNITS
in ZY2, it follows that V>(aif^) =
2(i7{ + l{, iJ
+ iiJ, iix^ + 7 ^ 0
1 + 2(7o^  ili)
and clearly 1 + 2Z[^] 2{Z\i\f 2(Z[^])3 1 + 2Z[^]
V^(aif^) e
n(det)=±l
Now we have to determine which matrices A =
1 + 2Z[^] 2(Z[^])3 2(Z\i\f 1 + 2Z[^]
1 + 2a 2v 2w 1 + 26
n(det)=±l
actually belong to '^{JA^^')Let v = (vi,i'2?^3) and w = \i Then A G V ' ( ^ 2 ^ T2 ^ ) if ^i^d only if the linear systems 1
i
1
I
0 0
0 0
\A li
0 0 0 0
{wx^w^^w^.
a
b
ij
V2
LT^^.
W2 j
t i 0 0
and
1 0 1 0 0 i 0 i
0 0 1 1
[7/1
Vi
^
= V3
W/\
 '^3 J
each have a solution in Z[^]. These systems are equivalent to 1 i 0 0' [70^ 0 2i 0 0 li 0 0 1 i %' 0 0 0 2i
y\
a 1 a—b V2
y2 + W2\
and
i 0 0 0
1 0 0 2 0 0 0 —i 1 0 0 2
\A
V\
1
'^3
wA
yz + ^^3]
4. DESCRIBING LARGE SUBGROUPS
319
Hence, A e il^ ( ^ 2 ^ ) if and only if a+6 e 2Z[^] and v+w G 2(Z[^])^. Notice that a + 6 G 2Z[^] implies that det A € 1 + 4Z[^]; therefore, n(det A) = 1. Since V = U2IL'^ and '^(^y^) = —/, the result follows. The case m\ = 1 follows in a similar way. D We remark in passing that, when mi = 1, the condition a + 6 G 2Z is actually equivalent to det A = 1. 4.3 T h e o r e m . Let L he a loop of type £2 ™^^ cyclic centre generated by an element ti of order 2'^^ ^ mi > 1. Let ^ be a primitive 2^^ th root of unity and let
be the mapping defined in Proposition VIL2.8. Then V = V ^ ^ and the restriction of ^ to V ^ ^ induces an isomorphism from V onto '^{V^^), where
V'(vif^) = 1 +2a 2{wx,W2,W3)
2{v\,V2.,V3) 1 + 26
l + 2Z[e] 2{Z[^]f
2{Z[i]f 1 + 2Z[^]
I n(det)=l
a + be 2Z[^], V2 + W2, vii + t/^i, t;3^ + ^3 G 2Z[.<] I . PROOF.
The proof is similar to that of Theorem 4.2.
D
To compute V{ZL) for an arbitrary indecomposable RA loop L, one proceeds as follows. Given a unit ji = l + a ( l  5 ) G V(ZZ/), write fi = ^ ^ + ^ ^ ^e, where e runs over all primitive central idempotents in A(JL, L ' ) . Since /ie G V((ZZ/)e) and Le is an indecomposable RA loop with cyclic center, it follows from Theorem VII.2.5, Proposition VII.2.7 or Proposition VII.2.8, and Theorem 4.2 or Theorem 4.3, that /x can be written as a sum of matrices (with entries in the ring of integers of a cyclotomic field extension of the rationals) and perhaps one element in a nonsplit CayleyDickson algebra. The problem is then to determine which of these sums are actually representations of units
320
XII. LOOPS OF UNITS
in V(ZL). This leads to systems of linear equations and then to other equations relating the components in the different matrices. Since the integral nonsplit CayleyDickson algebra has only trivial units—the only units are the elements in ±{l^i^j^k^£^i£^j£^ki} because the norm of a unit has to be 1—the appearance of such a component will imply some other equations relating the coefficients of /x. Note also that, to write fi G (ZZ/)e as a matrix using Proposition VIL2.7 or Proposition VII.2.8, one has possibly to replace /i by another element LJ such that (ct;e)^ = —e and to choose appropriately the group G of index 2 in L to ensure that {QG)e is a matrix algebra. The same method can also be applied to RA loops which are not indecomposable; that is, to loops of the form LxA where L is an indecomposable RA loop and A is an abelian group.
5. E x a m p l e s In this section we apply the method outlined at the end of the previous section to obtain the unit loops of the integral loop rings of two indecomposable 2loops. In [Goo92], a description of the unit loop for all RA loops of order less than 64 is given. T h e unit loop of Z[M32(^t, 16)] Let L = M32{Ei,l6) = M(16r26,*, 1) = D\J Du.v^here D = lGT2b = {x,y,ti \ x^ = y^ = l^t'l = l,(a:,?/) = / i , / i central ). The group D is of type Vi and L is an indecomposable RA loop of type £ i . (See Chapter V, Sections 2 and 3.) Because of Proposition VII.2.7, L = D U Dw with w = tiu; thus w'^ = tj = s and ( t i ; ^ ) ^ =  ^ . Furthermore, any element in V{ZL) can be written uniquely in the form fj. = I + {(ao + /Soh f a i x + /3itix + 0:2 + /?22/ + (^s^V + l^shxy) + («o + f^oh + a[x + fi[tix + a'2 + /322/ + «3^2/ + l^3hxy)w] (1  s) where all ai,l3i,a',fi[ G Z.
5. EXAMPLES
321
5.1 T h e o r e m . Let L = M32(Ei,l6), ThenU{ZL) = ±L[V{ZL)] V(Ziy) is isomorphic to the loop of matrices of the form l + 2Z[i] 2{Z[i\f 2{Z[i\f 1 + 21[i\
1 + 2a 2v 2w 1 + 26
where
det=l
with a + 6 e 2Z[j] andv + w 6 2(Z[t])^. Furthermore, the isomorphism maps /i = 1 + {(ao + /3o*i + oi\x + Afix + a2 + /322/ + otsxy + fi^tixy) + (ao + /3o^i + aia; + f3[t^x + a^ + /3^t/ + a'^xy + /9^iixt/)w} (1  5) fo
2 ( ( a 2  / ? i ) + (ai+/32)i, 1 + 2((ao  ;33) + (as + /?o)0 (/?;+a^) + (  a ' , + ; 3 ^ ) 0 ((A + 02) + (  a i + /32)«, (  a o  / 3 3 ) + (/5o + a3)«,
PROOF.
orem 4.2.
1 + 2(a[, + 13':,) + (;9^  a^)i
This follows from Proposition VII.2.7, Proposition 4.1 and TheD
The unit loop of ZfMieCQs, 2)] Let L = Mi6((58,2) = D4 U Z)4'« = M(£'4,*,a^), where we use the classical representation for the dihedral group £^4 of order 8: D^ = ( a , 6  a^ = 6^ = 1,6a = a^6}. As before, D^^ is a group of type V\. To use the results of the previous section, we should use for D^ the presentation {x,y \x'^ = y^ = l , ( x , y ) = ti,t\
= 1).
As a consequence, we cannot directly apply the isomorphism given in Section 4. Instead we go through the whole procedure again, as in [JL93b]. 5.2 Lemma. The mapping VP: V ( Z L ) 
2Z + 1 (2Z,2Z,4Z)
(4Z,2Z,2Z) 2Z+ 1
det=±l
XII. LOOPS OF UNITS
322
defined by
a^))
2s+1 w
V 2t+l
where a = ai \ a2Ci + a^b + a4ab f3 = (3i+fi2a
+ fisb + (i4ab
«s = a i + a2  a s + a4 V = (4(a2 + a 4 ) , 2(/3i + /32  /?3 + ^4). 2(/32 + /^a)) w = ( 2 (  a 2 + a a ) , 2(/3i  /32 + /J3  /?4),4(/?2 + /?4)) t = ai  a2 + Ois  a4^ each ai^Pi G Z, is a PROOF.
monomorphism.
Let e n , 622, ^12 and 621 be the following elements of the group
ring QD4: €11 — ^(1 — a — b + ab) ^21 = (  « + 0.b)
2
2
'
e\2 = Uo' + b)
2
'
^22 = U l + « + 6  a6) la^
'
One can easily verify t h a t e n +^22 =
2^'> ^^^ identity element of A ( L , L'),
and t h a t eiie22 = 0, e n = ei2e2i, e22 = ^21^12 ^ind ej2 = ^21 — 0. Thus {€ii?^i27C2i,e22} play the role of the elementary matrices in M2(Q). Note also t h a t .io^
.10^
= €11  ^22 + 2ei2  e2i, €21  ^11 + e22
and a6 l  g ^ = ^11  ^22 + 2ei2. For L = M ( G , * , ^ o ) , remember t h a t RL = RG + RGu and A ( L , L ' ) = A ( G , G ' ) + A{G,G^)u
(see Lemma V L l . l ) . Here, A{G,G')
^ M2(Q). Now
we represent an element it; G ZY2(ZZ/) as an element of A ( L , L') = M2(Q) + M2(Q)/7. For this, note t h a t a'^{la'^)
= {la'^),sow
= l + {a +
f3u){l
5. EXAMPLES
323
a^), where a = a i + 0:20 + 0:36 + 0406, f3 — Pi + /?2a + l^sb + ji^ab and each a,,/3j G Z. Hence t(; = l + (a + / 3 u ) ( l  a ^ ) = i ^ + [(2a+l) + 2/?«]i^ = i ^ + [1 + 2(ai + a2a + 03^ + a^ah) + 2(/Ji + /?2a + IS^h + / ? 4 a 6 ) u ] i ^ = ^
+ [(1 + 2(^1 + a2  "3 + a4))eii + 4(a2 + 04)612 + 2 (  a 2 + Q;3)e2i + (1 + 2(ai  02 + "3  Q;4))e22] + [2(/3i + /32  /33 + /?4)eii + 4(/32 + /34)ei2 + 2(/32 + /?3)e2i + 2(/?i  ^^2 + /33  /34)e22]«.
It follows that the mapping tp.HiiZL) ^ M2(Q) + M2(Q)f/ defined by 1^(1 + ((ai + 020 + 036 + a4ab) + (A + /J2a + /Sgi + /34a6)ti)(l  a^)) 1 + 2 ( 0 1 + 0 : 2  0 3 + 0:4)
4(02 + 04)
2 (  0 2 + 03)
1+2(0102 + 0304)
+
2(/3i+/32/93 4/34) 2(/32+/33)
4(/32 + /34) t/ 2(/?i/32+/33/?4)
is injective and preserves multiplication. The isomorphism from M2(Q) + M2{Q)U to 3(Q) given in §1.3.5 now allows us to obtain an injective, multiplication preserving mapping
ip': Ui{ZL)^ S =
2Z + 1 (21,21,41)
(41,21,21) 2Z+1
defined by ip'('w) = [^*J"^ 2t+i ] ' where s,t,\/,\N are as in the statement of the lemma. Since the determinant function is multiplicative on 3(Q), it follows that the image of rp' is contained in 5det=±i • Finally, i)'(a^) = V'(l  (1  a^)) = I Hence, by Proposition 4.1, rp' induces the required monomorphism 9 : V(1L)
2Z + 1 (21,21,41)
(41,21,21) 2Z+1
D det=±l
XII. LOOPS OF UNITS
324
5.3 T h e o r e m . The monomorphism (p defined in Lemma 5.2 yields the isomorphism V(ZL) ^ <^ A
A^ PROOF.
2s+1
(4x1,2X2,22:3)
(2i/i,22/2,4t/3)
2f+l
2Z+1 [(2Z,2Z,4Z)
(4Z,2Z,2Z) 2Z+1
I 2 I (X2 + 2/2) det=l
Because of Lemma 5.2, it is sufficient to show that A=
r
25+1 (2j/i, 2^2,42/3)
(4x1,2x2,2x3) e v(v(ZL)) 2<+l
if and only if 2  (x2 + 2/2) and det ^ = 1. Clearly A € i^(V(Zi/)) if and only if the linear systems 1 0 0 1
1 1 1 1
1 0
1 " ai 1 «2 0 «3 1 _ .^4.
s Xi
y\ _ t \
1 0 0 •1
and
1 1 1
1 0 1 1
1 1 0 1
>r
XT]
/?2
2/3
/33
3:3
A.
.^2]
have an integral solution. These systems are equivalent to 2 0 0 0 1 0 0 1 1 0 0 1 1 1 I 1
"ai"
' s + t "]
Ot2
Xi +t \
Ot3
xi + y\
_a4
t
and
\ 2 1 0 1
0 0 1 0 1 0 1 1
0 0 1 1
"/Ji"
X2  2/2I
P2
J/3  J/2
f33
ys + 2^3
A.
.
2/2 J
and hence have an integer solution if and only if 2  {^s\i) and 2  (x2 —1/2). Since det ^ — {2s\1)(2/+ 1) \{2x\y\ +3:2^/2+ 2x3^/3) = ± 1 , the condition 2 I (5 + ^) is equivalent to det A—\ and the result follows. D
5. EXAMPLES
325
We now show that V{ZL) is closely related to V(Zi?4). For this, we introduce a certain subgroup of PSL(2,Z), the integral projective special linear group; namely, 2Z + 1 2Z
4Z 2Z+1
= {Ae
25+1 U
M2(Z) I A =
det=l
5, t,k,ieZ,
4k 2« + 1
det yl = 1 i / { / ,  / }
and we also define three natural subgroups of V{ZL). 5.4 Definition. 25+1 (2t/i,0,0)
V^ = iA
(4x1,0,0) 2i+l s,t,xi,yi
25+1 (0,22/2,0)
Vo = lA
e Z, detA = 1 W { / ,  / } ;
(0,2x2,0) 2t+l
s, t, X2,2/2 € Z, 2 I (X2 + j/2), det A = n / { / ,  / } ;
V.
25 + 1
(0,0,2x3)
(0,0,4t/3)
2f+l
I
s,t,X3,y3e
Z, detyl = 1, W { / ,  / } .
5.5 Proposition. 1. Each V, is a group isomorphic to 2Z + 1 2Z
4Z 2Z + 1
^ V(Z£>4)det=l
2Z + 1 2Z
4Z 2Z+1
det=l
XII. LOOPS OF UNITS
326
3. V{ZD4) is a free group of rank 3. Moreover, vi = I + {1  a'^b)a^{l + a'^b) =
1 4 and 0 1
t;2 = 1 + (1  aH)a{l + aH)
V3 = l + ilb)a\l
1 0 2 1
3 8 2 5
+ b)^
are free generators, where the matrix representations are those given by the isomorphism of part (2). Note that v\, V2 and v^ are bicyclic units o/ Z£>4. 1. This statement is obvious for Vi. That V2 is a group is easily verified, and conjugation by [J }] gives an isomorphism between PROOF.
{and
25+1
2A; 1
2e
2Z + 1
4Z
2Z
2Z+ 1
2t + 11
I
2(A; + ^), det^ = l W { / ,  / }
; thus V2 = Vi. 7
det=l
Furthermore, conjugation by [^J] gives an isomorphism between 2Z + 1 2Z 4Z 2Z+1
and det=l
2Z+1 2Z
4Z 2Z + 1
det=l
Hence V3 = V2. 2. This follows at once from Proposition 4.1 and Theorem 5.3 by restricting (f to the set {it; = 1 + a ( l  a^) \ a = ai \ a2a + asb + a4ab^ e{a) G 2Z, w G U{ZD4)}, which is V(ZD4), and then applying the natural isomorphism between Vi
and [ i J S ^ i J s ^ . 3. The group G=
^ 2Z + 1 4Z 2Z 2Z + 1
det=l
5. EXAMPLES
327
is a subgroup of index 2 in the free group 2Z + 1 2Z 2Z 2Z + 1
det=l
By Corollary VIII.5.2, the latter is free of rank 2 and has free generators [II] and [21] The NielsenSchreier Theorem (see, for example, [Rob82, Chapter VI]) shows that G is a free group of rank 3 and, using the Schreier transversal { [ Q ? ] ? [ O ~ I ] } ' ^^^ obtains the generators
[1 0' 2 1
1 2 0 1
2
"1 4' 0 Ij
and
r1 2' 0
1
1 0' "1 2' 2 1 0 1
" 3
8
2
5
Let
(0,0,0) 1
VP(1 + (1  a^b)a{l + a^b)) = >f{l + (a + 6)(1  a^)) 1 (0,0,0)
(4,0,0) 1
and ip{l + (1  6)a^(l + b)) = ip{l + {a  ab){l  a^)) 3 (2,0,0)
(8,0,0) 5 D
The isomorphism in part 2 then yields the result.
5.6 Remark. Examining the proof of part 3 of Proposition 5.5 and using the generators of V(Z7?4) given there, it follows that the generators of Vi are 1 (2,0,0)
(0,0,0) 1
1 (0,0,0)
(4,0,0) 1
3 (2,0,0)
(8,0,0) 5
XII. LOOPS OF UNITS
328
that those of V2 are r
1
(0,2,0)
[(0,2,0)
3
1 (0,0,0)
(0,4,0) 1
5 (0,2,0)
(0,18,0) 7
1
(0,0,0)
5 (0,0,8)
(0,0,2) 3
and that those of V3 are 1
(0,0,2)'
(0,0,0)
1 J ' [(0,0,4)
1 J
5.7 Corollary. The Moufang loop V(ZL) is generated by the free groups V\j V2 and V3. Thus V{'LL) has generators vi
=
1 + (  a + a6)(l  a^),
V2 =
l + (a + 6 ) ( l  a 2 ) ,
V3 =
l +
V4 =
l + [{a +
v^
1 + [(1 _ a  6 +a6)i/](l  a^),
=
{aab){la^), bab)u]{la'^),
^6 zr
1 + [(3a + 36  Sab) + (  5 + 4a + 46  4a6)iz](l  a^),
vj
=
1 + [ (  a +a6)tx](l  a^),
vs
=
1 + [(a + 6 ) i i ] ( l  a ^ ) ,
vg =
1 + [(2a  26 + 2a6) + (  3 a  26 + ab)u]{l  a^).
Let S be the subloop of V(ZL) generated by the groups Vi, V2 and V3 and let PROOF.
25 + 1 [(2yi,2i/2,4t/3)
(4x1,2x2,2x3)1 2/+1 J
^
^'
in particular, 2  (X2 + 2/2) ^i^d det A = 1. Consider the following statements. 1. We may assume 2^ + 1 > 0. 2. If Xi = 0 for some i, 1 < i < 3, then we may assume yi = 0 too. Similarly, if some yi = 0, then we may assume Xi = 0. In particular, upper and lower triangular matrices of V(ZL) belong to S. 3. If gcd(25+ l,xi) = 1, then Ae
S,
4. If (25 + 1) I xi, then we may assume xi = y^ = 0.
5. EXAMPLES
329
5. If 0 < gcd(25 + 1, a:i) = 2^1 + 1 < 25 + 1, then we may assume A — \
2sifl
(0,2x^,24)1
6. In assertions 4 or 5, if {2si + 1)  2:3, then A ^ S. 7. In assertions 4 or 5, if 0 < gcd(25i + 1,0:3) = 2s2 + 1 < 2^1 + 1, then we may assume 252 + 1
A =
(4a:;',2a7^',0)
These assertions, if true and used a number of times, show t h a t A £
S.
The rest would then follow from Theorem 5.3 and Remark 5.6. Thus it is sufficient to verify the seven statements. 1. Since / equals —/ in V(ZL), this is clear. 2. We show t h a t if Xi = 0, then we may assume yi = 0. So suppose 2s + 1
(0,2x2,2x3)
(2t/i, 2^/2,4^/3)
2^ + 1
A = and let
2s\ 1
(0,2x2,2x3)
(0,22/2,4^/3)
2^ + 1
B = Clearly B G V(ZL).
While the following calculation is obvious, it is included
as motivation for further computation in the proof and to indicate how this proof can be adapted to show the last p a r t . Consider, therefore.
B+
0
(0,0,0)
(22/1,0,0)
0
B\I+B
0
(0,0,0)
(22/1,0,0)
0
1
1 (25+l)(22/i,0,0)
(0,82/12/3,4J/12/2) 1
1
(0,0,0)
(2s + l)(22/i,0,0)
1
(0,82/12/3,42/12/2) 1
B It follows t h a t A = B
1 (0,0,0)
XII. LOOPS OF UNITS
330
Let
C =
1
(0,0,0)
[(25+l)(2yi,0,0)
and
1 J
Then C , P G 5 and A = B{C[D{EF)]},
E =
1 (32j/2j/2y3,0,0)
(0,0,0) 1
1 (0,0,0)
D
(0,82/1^3,0) 1
where
and
F =
1 (0,0,0)
(0,0,4yij/2) 1
Note that both E and F are in S. Thus ^ G 5 if and only if fi 6 5 , so, if xi = 0, we may also assume j/i = 0. Similar arguments also work for X2 and Xz as well as for the rest. 3. Since gcd(25 + 1,^i) = 1, there exist x,y £ Z such that x{2s + 1) + ySx\ = I. Let B =
r
X (2j/,0,0)
(4x1,0,0)1 25+1
Then 5 e 5 and AB =
1 .1
x' 0*l
[y' 2t' + l
is in V(ZL). Hence, we may assume that A =
1
(4x1,2x2,2x3)
(22/1,22/2,41/3)
2t + \
in which case A
1 (0,0,0)
(4x1,0,2x3) 1
1 (2yJ,22/^,42/^)
(0,2x2,0) 2^'+ 1
By statement 2, we may assume 2/J = 2/3 = 0, so ^ G S. 4. In this case, Xi = x(2s + 1) for some x G Z. Consequently A =
1 25+1 (0,2X2,2X3)1 [(22/1,22/^, 4y^) 2t'+ I (0,0,0)
(4x,0,0) 1
is in V(ZL), where 2/2 = 3/2— 4xx3, 2/3 = 2/3 + 2xx2 and t' = t — 4x1/1. Thus, by statement 2, we may assume Xi = 2/1 = 0 .
5. EXAMPLES
331
5. Let 0 < 2^1 + 1 = gcd(25 + l,a;i). Then there exist x,y G Z such that x{2s + 1) + ySxi = 2^1 + 1. Multiplying A on the right by the matrix
(42^,0,0)] 2s+l 251+1
L(2y,0,0)
(which belongs to S) shows that we may assume A=
2^1 + 1 (2t/i,22/^,4t/^)
(0,2a:'2,2x9 2/' + l
and (2si + 1)  Xj. The result now follows from statement 2. 6. In this case X3 = a;(25i + 1). Let B =
1 (0,0,0)
(0,0,2a;) 1
be an element of S. Then 251 + 1 {2yl2y'^,4y'^)
AB =
(0,2x^,0)1 2t" + \
is in V(ZL). The result then follows from statement 2. 7. Let 0 < 2^2+1 = gcd(25i + l,x^) < 25i + l. Then there exist x,y G Z such that x{2si + 1) + 2/8x3 = 2s2 + 1. Let B
(0,0,2(^))l (0'0'4j/) If^
(which belongs to S). Then AB
252 + 1 i2y'{,2y'^,4y'^)
{4x'{,2x'^,0) 2t" + \
Again, the result follows from statement 2.
€ ViZL). D
Chapter XIII
Idempotents and Finite Conjugacy
In Chapter XII, we found exphcit conditions under which the unit loop of the integral loop ring of an RA loop is an FC loop. The main thrust of this final chapter is to investigate this problem for alternative loop algebras over fields of any characteristic. Crucial in tackling this problem is knowledge of when all idempotents are central. This is what will be explored in the first section. As an application, one can also determine when all nilpotent elements are trivial. In the final section, the FC problem is answered fully.
1. Central i d e m p o t e n t s In this section, we give necessary and suflScient conditions under which all idempotents of loop algebras of RA loops are central. Most of the main results and proofs are based on [GM]. In the case of a group algebra KG^ S. Coelho solved the analogous problem when K is a field of positive characteristic (provided the set of torsion elements of G is a locally finite group) [Coe87], and S. Coelho and C. Polcino Milies settled the issue for fields of characteristic zero [CM88]. (See also Example 1.11 in this section.) For loop algebras over fields of positive characteristic our proofs are almost verbatim those of the group ring case [Coe87]. A field K is said to be a splitting field for a finite abelian group G if the group algebra KG is the direct sum of fields, each isomorphic to K. Clearly, a field K is a splitting field for a finite abelian group G of order n if and only if KG is semisimple artinian with n orthogonal nonzero primitive idempotents. 333
334
XIII. IDEMPOTENTS AND FINITE CONJUGACY
1.1 L e m m a . Let K he a fields G = (x) a cyclic group of order n and ^n CL primitive
nth root of unity over K. Suppose char/if does not divide n. If j
and k are integers with 0 < j ,fc< n — 1, then 1 ^j — n ^7=o(^^xy 2. CjCk = 0 ifj ^ k. In particular, PROOF.
K{(n)
^^ ^^ idempotent
in K{^)G
and
i^ ct splitting field for G.
Let 0 < j ,fc< n  L Then
\i j ^ k^ then ^ = ^n~
is a nontrivial root of unity of order m with
m dividing n, hence also an n t h root of unity.
Thus Y^^=Q ^^ = 0 and
YllZo i^ — ^ ^^d thus CjCk — 0. On the other hand, if j = A;, then we obtain
1=0
t=0
The next result is a special case of a well known theorem of R. Brauer. (See, for example, [ C R 8 1 , §15.16 and §17.1].) 1.2 C o r o l l a r y . Let K be a field of characteristic abelian group of order m . If in i^ o, primitive
zero and let G be a finite
nth root of unity over K and
m I 71, then A'(^n) is a splitting field for G and the primitive K{in)G
are the elements
idempotents
of
of the form ,
{n/\H\)\
in E «!."")•. i=0
where H runs through the subgroups of G such that G/H G/H
= {xH) PROOF.
is cyclic, x £ G,
andO<j<j^l, Write F = A^^n) From Theorem X.2.1, we know t h a t
id a primitive dth root of unity and ad = nd/[F{id)'
^ ] , where nd is the
number of elements of order d in G. Since F contains a primitive m t h root
1. CENTRAL IDEMPOTENTS
335
of unity, each field F{^cl) is F and thus
So F is indeed a splitting field for G. Now let e be a primitive idempotent of FG. Then Ge is a finite subgroup in the component Fe = F and each of its elements is a root of unity whose order divides m. So Ge = {x) is a cyclic group of order i dividing m and Ge = G/H for some subgroup H of G. It follows that {FG)e ^ F is an epimorphic image of the group algebra F{x) = (FG)H, From the first part of the proof, we know that F is also a splitting field for (x). Furthermore, by Lemma 1.1, eo, e i , . . . , e^_i are the primitive idempotents of F{'x), The above epimorphism then shows that, for some J, 0 < j < ^ — 1,
where x is an element of G with xe = x. Hence n
^^ 1=0
as desired. Conversely, let ^ be a subgroup of G with GjE = (Hx) = (^), a cyclic group of order £, and let
t=0
Since {FG)H ^ (F{x))/, where
F[G/H],
by Lemma VL1.2, it foUows that {FG)e
^
1=0
Because of Lemma 1.1, the element / is a primitive idempotent of F{x). Hence e is a primitive idempotent of FG, D 1.3 Corollary. Let K he a field of characteristic p > 0 with prime subfield V. Assume A is a finite abelian group whose order is not divisible by p. If e is an idempotent in KA^ then e G VA^ where V denotes the algebraic closure ofV in K.
336
XIII. IDEMPOTENTS AND FINITE CONJUGACY
Recall that for an RA loop L with (locally finite) torsion subloop T, and for a prime p > 0, we denote by Tp/ the subloop of L consisting of elements whose order is not divisible by p and by Tp, the subloop of elements whose order is a power oi p. It was shown in Proposition V.1.1 that T = T2 X T21 and that T21 is a central subloop of L. 1.4 Lemma. Let L be an RA loop with torsion subloop T, and let K be a field of characteristic p > 0. If all the idempotents of KT are central in KL and t £ T has order not divisible by p, then the cyclic group (t) is normal in L. Let ^ € // be an element of order n with p )( n. By assumption, the idempotent e = ^ Y17=o ^^ i^ central in KL^ so, for every x G //, we have PROOF.
_i
e = xex
l ^ nn—\ 1
^ =n Lyxfx\ . t=0
Hence xtx ^ G supp(e); that is, xtx ^ = f for some 0 < i < n — 1. By Corollary IV.1.11, {t) is a normal subloop of L. D 1.5 Lemma. Let L be an RA loop with torsion subloop T. If K is a field of characteristic 2, then every idempotent of KT is central in KL. Since T21 is a central subloop of L, it is sufficient to show that each idempotent of KT belongs to KT21. So let e G KT be an idempotent. Since a subloop generated by finitely many elements has finite torsion subloop (Lemma VIII.4.1), the subloop generated by the support of e and three nonassociating elements of L is a finite RA loop. Hence, replacing Z/ by a finitely generated RA subloop containing the support of e, we may assume that T is finite. Because of Corollary VL4.3, KT2> = ®T=\ ^^t? ^^e direct sum of fields Ki. Hence PROOF.
KT = K[T2. X T2\ = (/1T2OT2 ^ ®Zi I<^T2^ Write e = Y^^=\ ^i^ where each e^ is an idempotent in A\T2. Then the augmentation of Ci is 0 or 1, so, either Ci or 1 + ^i belongs to A/v'.(T2), the augmentation ideal of KiT2. It follows from Theorem VL3.4 that AKX'^^) is a nil ideal. Consequently Ci = 0 or Ci = I. Thus e G 0 ^ i Ki = KT2'^ •
1. CENTRAL IDEMPOTENTS
337
1.6 Theorem. Let L = M{G^ *?5o) ^^ o,'^ R^ loop with torsion subloop T. Let K be a field of characteristic p > 0. Then every idempotent of KT is central in KL if and only if p = 2 or 1. Tp/ is an abelian group and 2. ifTpf is not central, then the algebraic closure Fp of Fp in K is finite and, for all t G Tpi and i £ L, there exists a positive multiple r of [F^: Fp] such that iti'^ = t^^ prove the necessity of the conditions, assume that every idempotent of KT is central in KL and that p > 2, By Lemma 1.4, every subloop of Tp/ is normal in L, so, Tp/ is either an abelian group or a hamiltonian Moufang loop. In the latter case, Tpt contains the quaternion group Qg^ hence, FpQg C KT. By Corollary VII.2.4, KT contains H{Fp), Since t(Fp) = 1, by Proposition XI.3.3, H(Fp) is split (and hence isomorphic to M2{Fp)). Thus KT contain noncentral idempotents, a contradiction. Therefore Tp> is an abelian group, proving (1). To prove (2), let t G Tp, and let K^ be a finite subfield of K. Then K^{t) is semisimple and thus, by Theorem X.2.1, A^'(^) = ®Ki, where each Ki is a cyclotomic field extension of K^. Furthermore, at least one of these fields, say A^i, is of the form A^i = K\^) with ^ a primitive root of unity of order o{t). Note that the natural projection K^{t) ^ Ki sends t to ^. Let i £ L, Since subloops of Tpf are normal in L, iti~^ = f for some i > 0. Since idempotents of KT are central, the inner automorphism a ^^ iai~^ of (the group algebra) K'{t^i) induces a field automorphism V^: A^i —> A'l. Since Ki is a finite field of characteristic p, 'tp is necessarily a power of the Frobenius homomorphism x \^ x'^, (See, for example, [Kar88, Proposition II.5.3]). Hence, ^^(0 = r for some r > 0. Since K' C A'l and the elements of K' are fixed under V^, we obtain also that fc^"^ = k for every k G K'. Hence, [A'': Fp]  r. Note that r depends on t and K'. P R O O F . TO
Now assume that Tpi is not central and let t G Tp/, i ^ L he such that it / tL We have shown that Ht~^ = t^"^ for some positive r. If [Fp: Fp] is infinite, there exists a finite subfield K' of Fp such that [A^^: Fp] — 2r. Hence, using again what we showed earlier, there exists a positive integer v such that
338
XIII. IDEMPOTENTS AND FINITE CONJUGACY
Since lU ^ = f^"^ ^ we have PU '^ = f^ "^ and so, because i'^ is central, t'P^'' = t. Hence
where v exponents are involved, and therefore M ^ i=: ^, a contradiction. Thus [Rp. F] is finite and (2) follows. We now show that the conditions are sufficient, lip = 2, then Lemma 1.5 implies that all idempotents of KT are central in KL, So assume p > 2. Thus both Tp and Tpt are abelian groups. (In fact, Tp is central.) Let e be an idempotent in KT, Write this element in the form
where kij E K^ gi ^ Tp and hj € Tpf. Let / be the image of e under the natural mapping exp' KT ^ KTpi. Then
/—
2L^
^^  ^ *
Clearly e f = 6 e AK(T,TP) = ker6Tp. It follows from Theorem VI.2.7 that 6'P'' = 0 for some n > 0. Since e and / commute, e = e^"" = / P " + 6^"" = /p" = / . So e G A'Tp/ and thus, by Corollary 1.3, e = Ylj ^j^j ^ FpA^ where A is a finite subgroup of the abelian group Tpi. If A is central, clearly e is central. Suppose A is not central and let i e L. Write ,4 = (^i) x • • • x {th) as the direct product of cyclic groups. Because of (2), there exist numbers Ui > 0 and ix > 0 such that Uii'^ = tf' ior I < i < h and e{Ut2 • "th)i~^ = (^1 • • 'thY • Since A is commutative, the subloop (A^i) of L generated by A and ^ is a group. (See Corollary IV.2.4.) Hence
Therefore, t^ * = t^ for any i = 1 , . . . , /i and so, for any a £ A^ iai~^ = a^"*. Since tx is a multiple of [Fpi F], for any k G Fp, k^"" — k. So we obtain,
M' = Y.^y; = J2kfhf = eP" = e, J
as desired.
3
D
1.7 Corollary. Let L = M^G^^^go) be an RA loop with torsion subloop T of finite order n. Let K be a field of characteristic p > 0. Assume K
1. CENTRAL IDEMPOTENTS
339
contains a primitive nth root of unity. Then all idempotents of KT are central in KL if and only if p = 2 or Tpf is central Because of Theorem 1.6, it is sufficient to show that, if all idempotents of KT are central in KL^ if p ^ 2 and if ^ G i/, ^ G Tp/ and iti'^ = tP\ then tP'' = t. For this, let (^rn be a primitive mth root of unity, where m = o{t). By Lemma 1.1, PROOF.
^ m—l i=0
is an idempotent of KT, Since this idempotent is central, 771 — 1
^
m
.
(Note that i and t generate a group, so conjugation by t defines an automorphism on K{i^t).) Comparing coefficients of ^P"^, we obtain Sm
sm
SO m I (p^ — 1) and t^"^ — t, as desired. Let K be a finite Galois extension of the field F with Galois group Q = Gal(AYF). If X G A, then the trace of x, denoted tr(a:), is defined by tr(x) X)a(x). Clearly tr(x) is fixed under any element of the Galois group G; therefore, tv{x) G F. For a group G and automorphism cr G ^, there is a natural extension of a to an Fautomorphism of KG defined by (^{Ylg^G^gd) — T.geG^MdClearly
creG
geG
geG
We shall require characterizations of split quaternion and split CayleyDickson algebras in addition to those given in Chapter I. 1.8 Proposition. For a field F, the following conditions are equivalent. (i) H(F) = (F, — 1, — 1) 25 a split quaternion algebra. (ii) H(F) has nonzero nilpotent
elements.
D
340
XIII. IDEMPOTENTS AND FINITE CONJUGACY
(iii) H(F) has nontrivial idempotents. (iv) The equation x^ \ y^ = —\ has a solution in F. Furthermore J if F — Q(^n)^ where ^n is o, primitive nth root of unity, then any of these conditions is also equivalent to (v). (v) The multiplicative order of 2 (mod n) is even. P R O O F . That (i), (ii) and (iii) are equivalent follows from Proposition 1.4.15. That (i) is equivalent to (iv) follows from Theorem 1.3.4. Now assume that F = Q(^n) with ^n ^ primitive nth root of unity. To prove that (iv) and (v) are equivalent for this field is much harder. We refer the reader to either [Mos73] or [FGS71]. D
In [FGS71], it is also shown that, for F = Q(^n)? (iv) is equivalent to the existence of a solution to a:^ + y^ = — 1 in Q(^p) for some prime divisor p of n, where ^p is a primitive pth root of unity. Furthermore, it is shown that the latter holds for all primes p = ±3 (mod 8) and for some primes p = 1 (mod 8). The condition fails, however, for any prime p = 7 (mod 8). 1.9 Proposition. For a field F, the following conditions are equivalent. (i) (ii) (iii) (iv)
(F, — 1, — 1, — 1) Z5 a split CayleyDickson algebra. (F, —1,—1,—1) has nonzero nilpotent elements. (F, —1,—1,—1) has nontrivial idempotents. The equation x^ + y'^ + z^ + w^ = —I has a solution in F.
Furthermore, if F = Q{^n), where ^n is ^ primitive nth root of unity, then any of these conditions is also equivalent to (v). (v) n > 2. in Proposition 1.8, the equivalence of (i), (ii) and (iii) follows from Proposition 1.4.15 and that of (i) and (iv) from Theorem 1.3.4. Suppose that F = Q(^n) foi* some ^n> ^ primitive nth root of unity. Again, to prove that (iv) and (v) are equivalent for K — Q(^n) is much harder. The result follows from the following fact, which is also shown by C. Moser in [MosTS] (see also [Pfi95, p. 41]): When n > 2, — 1 is a sum of four squares in P R O O F . AS
Q(^n).
•
1.10 Theorem. Let L — M{G^^.,go) be an RA loop with torsion subloop T. Let K be a field of characteristic zero. Then every idempotent of KT is central in KL if and only if the following conditions are satisfied.
1. CENTRAL I D E M P O T E N T S
1. Every subloop ofT 2. IfT
341
is normal in L.
contains a noncentral
element
of order n and if S,n i^ ^
primitive
nth root of unity over K, then the map a defined by (7(^n) = ^n is in G3l(K{^n)/K)' 3. T is either (a) an abelian group, or (b) a hamiltonian
group with the property
of odd order k, then the field K(^k)
solutions
of the equation x^ + y^ = —1, or
(c) a hamiltonian
Moufang
contain solutions PROOF.
that if it contains
element
RA 2'loop
does not
an
contain
and the field K does
not
of the equation X'^ + y'^ + z'^ + w^ = — 1 .
Assume t h a t every idempotent of KT
is central in KL,
By
Lemma 1.4, every subloop of T is normal in L. Hence T is either an abelian group or a hamiltonian Moufang loop. Suppose T is a hamiltonian Moufang loop. Let H he di finite (hamiltonian) subloop of T . either Qs or
M\Q{QS)',
By Theorem n . 4 . 8 , H E
S X E x A, where S is
is an elementary abelian 2group and ^ is a finite
abelian group of odd order. If a G A is an element of order A;, then, using Lemma VILO.l, we see t h a t KT contains the subring K[{a) X 5] ^ K{a)
®K
KS.
Because of Theorem X.2.1, K{a) contains the field K{^k) ^s a direct summand.
By Corollary V n . 2 . 3 and Corollary V n . 2 . 4 , KS
contains either
(A', —1, — 1, —1) or (A', —1, —1) as a simple component. Hence KT
contains
the simple subring /i ( 6 ) ® A  ( A ' ,  1 ,  1 ,  1 ) = ( A ' ( 6 ) ,  1 ,  1 ,  1 ) , in the case t h a t T is a hamiltonian Moufang RA loop, or the simple subring A'(a)®A(A',l,l) = (A'(6),l,l), in the case t h a t T is a hamiltonian group. Since all idempotents of
KH
are central, these simple rings contain only trivial idempotents, so they are division rings. By Propositions 1.8 and 1.9, the field K((k)
does not
contain solutions of the equations x'^ + y'^ + z"^ \w"^ = —1 and x^ + t/^ = —1 respectively. Furthermore, as
Q(^A:)
^ ^^ (^A:)^ we also get k = 1 when H is
a hamiltonian RA loop. Hence condition (3) follows.
342
XIII. IDEMPOTENTS AND FINITE CONJUGACY
We now show t h a t condition (2) must hold as weD. Suppose / G T is a noncentral element of (necessarily even) order n and let x be an element of L such t h a t xtx~^
= st^ where s is the unique nonidentity commutator
associator of L. Since (t) is normal in L, 5 G (^), so s is the unique element of order 2 in {t). Therefore s = W^^. Consider the group algebra A^(^) as a subring of the group algebra K{^ri){t)'
Since both rings are direct sums of
fields, the nonzero idempotents of K{t) are precisely those elements of A^(^) which are sums of primitive idempotents of A'(^n)(^) Let ei = and G = Gal(A'(^n)/A')
^J27=oi^rity
By Lemma 1.1, the set {cr(ei)  <J G ^ } is a
collection of orthogonal primitive idempotents in K{^n){t)'
Therefore,
creQ
is an idempotent in K{t)
and hence central in KL^ by assumption.
In
particular, 2^xa{ei)x~^ creQ Since each xa{€i)x~^
= xex~^
= e.
is also a primitive idempotent of
{xa{ei)x^
\aeG}
= {a{ei)
K{^ri){t)',
a G ^}.
Hence xe\X~^ = cr(ei) for some a £ Q. Now st is in the support of e i , with coefficient ^^n n^n
; thus t is in the support of xeix~^,
also with coefficient
• Since the coefficient oft in cT{e\) is <7(^^n)? we obtain t h a t
This finishes the proof of (2). Conversely, assume t h a t conditions (1), (2) and (3) hold. To prove t h a t every idempotent of KT is central in K L we may assume t h a t L is finitely generated; thus T is finite. We claim t h a t KT
is a direct sum of division rings and hence every
idempotent of KT is central in KT,
This is obvious if T is abelian, so assume
t h a t T is not abelian. Because of (3), T is hamiltonian, more specifically, T = QsxExA
or T = Mie{Qs)
x £ , where E is an elementary abelian
2group (possibly trivial) and A is an abelian group (possibly trivial) all of
1. CENTRAL IDEMPOTENTS
343
whose elements have odd order. Now, K[Mi6{Q8) >^E] = ^
{KE)[Mie{Qs)] (®K)MieiQs)
= (®AO®(e(A',l, 1,1))
by Theorem VII. 1.4 by CoroUary VII.2.3.
Hence, by assumption (3.c) and Proposition 1.9, K[Mie(Q8) X £^] is a direct sum of division rings, as desired. On the other hand, using again several times the Wedderburn decomposition of the rational group algebras of finite abelian groups, and because of Corollary VII.2.4, we obtain K[Qs xExA]
= iKE)[Qs x A] ^ i®K)[Qs X A)] ^ ®(KA)Qs
= ®(e^^adii'(^d)g8)
S T ®(®4^i«'^(^''(^<')'i'i))' for certain integers a^, primitive dth roots of unity ^d and some sum J^ of fields. By assumption (3.b) and Proposition 1.8, KT is a direct sum of division rings, again as desired. To prove the result, it is sufficient to show that every primitive idempotent e of KT is central in KL. Since s is central and of order 2, either e = e i ^ or e = e ^ . Since e G Z{KT) = KT^^ + Z{KT^) and since KT^^ is in the centre of KL^ (by Corollary VI.1.3), we may assume that e G Z{KT^^) = {K[Z{T)])^ (again by CoroUary VI.1.3); in particular, e G K[Z{T)\. Since {K[Z{T)])e is a field, we obtain that, for some subgroup H of Z ( T ) ,
Z(T)e^Z{T)IH={x), X G Z(T)^ X = xH and Z(T)/H a cyclic subgroup of this field, say of order m. Hence (K[Z{T)])e = {K[Z{T)])He\ where e' is the natural image of e in {K[Z{T)])H ^ K[Z{T)/H]. By CoroUary 1.2, we know the primitive idempotents of K(^rn)[Z{T)/H]. It foUows that e' is a sum of idempotents of the type ^ m—l t=0
344
XIII. IDEMPOTENTS AND FINITE CONJUGACY
where ^m is a primitive mth root of unity over K and 0 < j < m — 1. Hence e is a sum of idempotents of the type m —l
i=o
with 0 < i < m  1 and n = \2{T)\, Note that if j = 0, then Cj = Z(T) is a central idempotent in KL. Since ej is a direct summand of the primitive idempotent 6, it follows that e = e^; hence e is central. So, for the remainder of the proof, we may assume also that j 7^ 0. It is easily verified that {K[Z(T)])ej = K{^^)' Because there is a natural field epimorphism from {K[Z{T)])e onto {K[Z{T)])ej^ it follows that the fields {K[Z{T)])ej ^ K(CL) and {K[Z{T)])e ^ K{U) are isomorphic. Hence gcd(j, m) — 1. Replacing ^^ by ^^, if necessary, we may then assume that ei is a summand of e. Since any A'automorphism a of A'(^^) can be extended in a natural way to a A'automorphism of A^(^77i)[2^(T)], and since (7(e) = €, it follows that C7(ei) is also a summand of e. Clearly cr(ei) = e^ with gcd(j, m) = 1 since (jiXm) is also a primitive mth root of unity. Then
is a summand of e, where Q = Gal(A'(^r^)/A'). Since e is primitive,
e = 5^a(ei). To prove that e is central in A'^L, we have to show that yey~^ = e for any y £ L. li y and x commute, then, since H is central in K{^rn)L (by condition (1), H is normal in L), it is clear that ycr(ei)y~^ == <^(^i); hence yey~^ = €. So suppose that x and y do not commute. Then, since {x) is a normal subgroup of L and because s (which has order 2) is the only nonidentity commutator, it follows that s = x^/^, where k is the order of X, Because of assumption (2), there exists r £ Q with r(^fc) = ^^ ' ^ Without loss of generality, we may assume that ^m — ik • ^^ '^ restricts to a A'automorphism of A''(^^). It follows that
1. CENTRAL IDEMPOTENTS It remains to show t h a t r((j(6i)) = y(T{ei)y~^.
For this, write
i=0 where each ej(t) G Ki^m) / = y€jy~^
345
teZ(T)
^nd j is some integer with I < j < m — 1. Write
= ^t£Z(T)f(^)^'
^^
compute the coefficients
f{t).
Let i be an integer and let h ^ H, Write i = vm +1 for integers v and t with 0 < ^ < m — 1. From the definition of e^, it follows t h a t CjytlX ) = CjytlX
X j = n^'^ ~ n^'^'
because x ^ ^ G ^ . On the other hand, since x^ is central precisely when i is even and because yx^y~^ = sx^ = x^^'^x^ when i is odd.
/(/ix')= ; if i is even. It then easily foUows t h a t
/(/la;') = ^
i^((V2)+l)..
jf . i3 ^^^^
Hence
thus yej{t)y^ as required.
= ^i^jit)) for all / G 2:(T) = {H, x), so y(7(ei)?/i = r((7(ei)) D
1.11 E x a m p l e . Contrary to what is stated in [ C M 8 8 ] , the following example shows t h a t condition (2) in Theorem 1.10 cannot be weakened to "A' does not contain roots of unity of order n whenever T contains a noncentral element of order n " . Let G = {x,y
\ yxy'^
and it follows t h a t Z{G)
= x^,x^ = {x^,y^)
= 1).
Then x^ and y^ are central
and G/Z{G)
^ C2 x C2. Hence L =
M{G^ —l^y'^) is an RA loop with T{L) = {x) a cyclic group of order 8. Let K = Q(^8 + ^8^^) = Q ( \ / 2 ) . As a real field, K has only ± 1 as roots of unity.
346
XIII. IDEMPOTENTS AND FINITE CONJUGACY
Write 7
ei^lY^iM' i=0
7
and
e, =
lJ2i^;'xr. i=0
These two elements are orthogonal primitive idempotents in K(^s){^) and ^1 + ^2 ^ J^{^) is an idempotent, with coefficient of x equal to y/2 and coefficient of x^ equal to —\/2. It follows that the coefficient of x in y{ei + e2)y~^ is — \/2. Hence this idempotent is not central in KG and thus not central in KL^ although all the other conditions of Theorem 1.10 are satisfied. If n is even and ^n is a primitive nth root of unity over Q, then the map in ^ ^(''/2)+^ = ^n defines an element of Gal(Q(^n)/Q). Thus the next theorem is an immediate consequence of Proposition 1.8 and Theorem 1.10. 1.12 T h e o r e m . Let L = M{G^^^go) be an RA loop with torsion suhloop T. The following conditions are equivalent. 1. Every idempotent of QT is central in QL. 2. Every suhloop of T is normal in L and T is either an abelian group, a hamiltonian Moufang RA 2'loop or a hamiltonian group with the property that ifT contains an element of odd order n, then the multiplicative order of 2 (mod n) is odd,
2. Nilpotent elements We apply the results of the previous section in order to decide whether or not an alternative loop algebra contains nontrivial nilpotent elements. 2.1 Theoremi. Let K he a field of characteristic p > 0 and let L = ^{G^ *,5'o) be an RA loop with torsion suhloop T. Then KL has no nonzero nilpotent elements if and only if the following conditions are satisfied. 1. L does not contain elements of order p (in particular, p / 2). 2. Every suhloop ofT is normal in L. 3. If p z=z 0 and T contains a noncentral element of order n, then, for ^^y in) 0. primitive nth root of unity over K, the map a defined by cT(^n) = ^n'^^""^ is in Gal{K{(n)/K)Furthermore, (a) T is an abelian group, or
2. NILPOTENT ELEMENTS
(b) T is a hamiltonian
347
group and, if it contains
an element
of odd
order k, then the field K{Ck) does not contain solutions
of the
equation x^ + y^ =: —I, or (c) T is a hamiltonian solutions
Moufang RA 2loop and K does not
contain
of the equation x'^ + y'^ + z^ + w'^ = —1.
4. Ifp > 2, then (a) Tpi is an abelian group and (b) if Tpt is not central, then the algebraic closure Fp of Fp in K is finite
and, for all t G Tpi and i £ L, there exists a positive
multiple r of[Fp: Fp] such that M~^ PROOF.
= t^"^.
Assume t h a t KL has no nonzero nilpotent elements.
Then,
from the description of the prime radical in Theorem VI.3.4, we see t h a t L does not contain elements of order p. We claim, furthermore, t h a t all idempotents of KT are central in KL. KT and let x G KL.
Indeed, let e be an idempotent in
Since the ring generated by x and e is associative,
(ex(l  €))2 = ((1  e)xey
= 0. Hence ex{l  e) = {\  e)xe = 0. So
ex = exe = xe and it follows from Corollary III.4.2 t h a t e is central in
KL.
T h a t the mentioned conditions are necessary now follows from Theorems 1.6 and 1.10. We now show t h a t the conditions are also sufficient. By Theorems 1.6 and 1.10, all idempotents of KT are central in KL.
To prove t h a t any nilpo
tent element of KL is trivial, we may assume that L is finitely generated and hence t h a t T is finite. Since T does not contain elements of order p , the algebra KT is Jacobson semisimple (Corollary VI.4.3). It follows t h a t KT = 0 g ( A'^T)e is a direct sum of division rings, where e runs through the finitely many primitive idempotents of KT.
Since each such e is central in
KL^ we obtain KL = 0 g ( A ' L ) e , a direct sum of twosided ideals, and so it is suflScient to show t h a t each algebra {KL)e
has only trivial nilpotent
elements. As in the proof of Lemma XII. 1.1 leading to equation (XII. 1), any nonzero element x in {KL)e
can be written uniquely as a finite sum
X = diqi + ••• + dkqk, where each qi belongs to a transversal of T in L, and each di is a nonzero element of {KT)e.
Moreover, (diqi){djqj)
= d(qiqj) for some nonzero d G
348
XIII. IDEMPOTENTS AND FINITE CONJUGACY
{KT)e. Now a "leading term" argument as in the proof of Lemma XII.1.1 shows that {KL)e does not contain nonzero nilpotent elements. D 2.2 Corollary. Let L — M(G, *,^o) be an RA loop with torsion subloop T. Then QL has no nonzero nilpotent elements if and only if 1. every subloop ofT is normal in L and 2. T is either (a) an abelian group, or (b) a hamiltonian group and, ifT contains an element of odd order n, then the multiplicative order of 2 (mod n) is odd, or (c) a hamiltonian Moufang RA 2loop.
3. Finite conjugacy In this section, we find KL of an RA loop to depend on whether or torsion, we distinguish is based upon a paper
necessary and sufficient conditions for a loop algebra have a finite conjugate (FC) unit loop. Our results not L is a torsion loop and, for loops which are not the semisimple and nonsemisimple cases. This work by E. G. Goodaire and C. Polcino Milies [GM96a].
3.1 Lemma. Let L be an RA loop with unique nonidentity commutatorassociator s. Assume x and y are noncommuting elements of L such that y has order a power of 2 and the group (x,y) is finite. If K is a field of characteristic 2 and z is a central element of L with z ^ {x^y), then y has a conjugate y(z) inU{KL) with 1. zv G supp(y(2:)), for some v G (x^y), and 2. supp(y(z)) C Jt=o ^^{^iV)^ where k is the smallest positive integer such that z^ G {x^y) if such an integer exists, and otherwise k = oo. If Z{L) is infinite, y has infinitely many conjugates of the type y{z). Since char K = 2 and y has order a power of 2, it follows from Theorem VI.3.4 that I + y £ V{KL)^ the prime radical of KL, Hence, zx{l + y) is nilpotent, say of index n, and therefore /i r= 1 + zx(l + y) is an PROOF.
3. FINITE CONJUGACY
349
invertible element. Then fiyfJ^~^ is a conjugate of y and we note that My/^~^ = (1 + ^^(1 + y))y . (1 + zx{l + y) + {zx{l + y)f + • • • + {zx{\ +
yW'^)
= y + zx{l + y)y + yzx{\ + y) + • • • + y{zx{l + y))""^ + zx{l + y)yza:(l + y) + • • • + 2:x(l + y)y{zx{l + y ) r ~ ^ If no positive power of z belongs to (a:, y), then the terms involving different powers of z have no elements of their supports in common. The sum of the terms containing only the first power of z is zx{\ + y)y + yzx{\ + y) = z{xy + yx){l + y)  z{xy + sxy + xy'^ + sxy'^), which has four distinct elements in its support; in particular, fJ^yf^'^ has zxy in its support and, furthermore, supp(/x?//x~^) C [j^i^Q z^{x^ y). On the other hand, suppose there exists a positive integer k such that z^ G (a:,y). Let k be the smallest such number. Since z ^ (a:,y), by assumption, fc > 1. It follows that
= y + z{xy + yx){l + y)[l + z\x{y + 1))^ + z''\x{y + 1))^^ + • • •] +z^xy + yx){l + y)[x{l + y) + z^{x{y + 1))^+^ + . . . ] + . . . +z^'{xy + yx){l + y)[{x{l + y))'^ + z^{x{l + y))^^'^ + • • • ] . By the choice of A:, the terms involving z^^ have no elements in their support in common with terms involving z^'^^^ for i = 2 , . . . ,fc— 1. Since
is an invertible element and because z{xy + yx){l + y) ^ 0 (as above), it follows again that z{xy + yx){l + y)[l + z\x{y + I))' + z^'{x{y + l)f'
+ • • •] ^ 0
and so zv € supp(//7/^~^) for some v G {x^y). Furthermore, it is clear that supp(/xy/x^) C UfrJ z'{x,y). Finally, if Z{L) is infinite and {zi  i = 1,2,...} is an infinite set of central elements such that ZizJ^ ^ (x, y) for i ^ j , then {y{zi)  i = 1,2,...} is an infinite set of conjugates of y. D
350
XIII. IDEMPOTENTS AND FINITE CONJUGACY
3.2 L e m m a . The unit group of the classical quaternion
algebra H(Q) is
not FC. PROOF.
Write H(Q) = Q H  Qi + Qj + Qij with (i, j ) = Qs Then, for
any nonzero rational g, the element 1 + gi is invertible in H(Q) with inverse YjrrO — qi) Furthermore 1
(1 + qi)j{l + qi)^ = ^hiiJ + W ) ( l  ^0 1 + ^2' 1
r(i + qij + qij + g^t^j)
2'
1+ 9 ^ Since ^ i ^ 7^
((1^2)^.^2^.^').
? ;^2 for g'' 7^ ^ 9 ^ ~ \ it follows t h a t j has infinitely many
conjugates.
D
Lemma 3.2 is a consequence of a more general result of A. I. Lichtman concerning the existence of noncyclic free subgroups in noncommutative division rings [Lic82]. In particular, it is known t h a t the unit group of H(Q) contains a noncyclic free group. 3.3 L e m m a . Let K be an infinite field of characteristic
2. Then KQs
is
not FC. PROOF.
Write Qs = ( a , 6  a^ = l , a ^ = 6^, 6a = a^b).
Then, for any
k G A^ the element 1 + A;(l + a) is invertible with inverse (1 \ k{\ + a))^, since (1 + aY = 0. Now (1 + k{\ + a))b{\ + k[\ + a))^
= (1 + k{\ + a))b{\ + k{\ +
a)f
= 6(1 + k{l + a^))(l + A;(l + a) + k^{l + a)^ + k^(l +
af)
= 6(1 + k{a + a^) + k'^{l + a + a'^ + a^)), so, if A: 7^ 0 and k y^ 1, it is clear t h a t ba appears in the support of this element with coefficient k. Hence 6 has infinitely many conjugates.
D
3 . 4 L e m m a . Let K be a field and L a loop such that KL is alternative.
If
U{KL) are
is FC and the set K U Z{L)
central.
is infinite,
then all idempotents
of KL
3. FINITE CONJUGACY PROOF.
351
Assume e is a noncentral idempotent of KL,
Let t ^ Lh^
such
t h a t either et{\ — e) ^ Q or {I — e)te ^ 0. Suppose n — ei{\ — e) / 0. (The other case is proved similarly.) Then n^ = 0, so ^ = 1 + n is a unit in
KL.
For 0 7^ z G A^ U Z{L)^ the element 1 + zn \s s. conjugate of jj, because [ze + (1  eMze
+ (1  e)]"^ = [ze + (1  e)]{\ + n)[z^e = [ze + zn+{\
+ (1  e)]
e)][z^e + (1  e)]
= 1 + zn. Since the support of n is finite and because, by assumption, there exist infinitely many units z G A U 2 ( L ) , it follows t h a t /i has infinitely many conjugates, a contradiction.
D
For torsion RA loops L and for any field K^ it is particularly simple to determine when U{KL)
is F C .
3.5 T h e o r e m . Let L he a torsion U{KL)
is FC if and only if K and L are
PROOF.
U{KL)
RA loop and let K he a
field.
Then
finite.
If A' and L are finite, clearly U{KL)
is F C . Conversely, assume
is F C .
We first show t h a t the characteristic of K is not zero. contrary. Then ZL C KL and thus U{ZL)
Assume the
is FC as well. Since L is not
an abelian group, it follows from Corollary XII.2.14 t h a t L is a hamiltonian Moufang loop. In particular, QQs = 4Q © H(Q) is a subring of KL.
This is
impossible, however, because of Lemma 3.2. Thus char A' = p > 0 and the prime subfield of K is Fp, the field oi p elements. Assume t h a t p ^ 2. Because L2, the 2torsion part of L, is an RA loop as well and because L is locally finite (Lemma VIII.4.1), L contains a finite RA 2subloop LQ. Because p j( Lo, the loop algebra KLQ is a direct sum of fields and Cay leyDickson algebras, by Corollary VI.4.8. Furthermore, by Proposition XI.3.4, these CayleyDickson algebras are either split or of the type (A^, —1, —1, —1). The latter are also split because (Fp, —1, —1, —1) is split (Proposition 1.9 and Proposition XI.3.3). Consequently, KL
con
tains noncentral idempotents. By Lemma 3.4, K U Z{L) is finite and, since \L/Z{L)\
= 8, KL is also finite.
Finally we consider the case p = 2.
Since L contains an RA 2loop
and the torsion subloop of L is locally finite, certainly L contains a finite
352
XIII. I D E M P O T E N T S AND FINITE CONJUGACY
noncommutative 2group (x^y).
Lemma 3.1 yields t h a t 2{L)
is finite; t h u s ,
L is finite. So it remains to show t h a t K is finite. Assume the contrary. Then, by Lemma 3.3, L does not contain Qg Since L is not an abelian group, not every subloop of L is normal and so, since elements of odd order are central, there exist y^t £ L^ o(t) a power of 2, such t h a t y~^ty — st ^
(t).
(Here, as always, s denotes the unique nonidentity commutatorassociator of L.) Let k G A' and let /j. = I + k X^^ig o(t)\
^^ Then fi~^ = ji and
o(t)\
i=0
2=0
o(t)l
o(t)l
o(t)l
= y + ky Y,it' + y~'t'y) + f''y{i2 y~'^'y){Il ^02= 0
2=0
2= 0
We compute the coefficient of yt in this sum. Suppose y{y~^fy)
— yt. If i
is even, then f is central, so f~^ = 1, contradicting the fact t h a t t has even order. On the other hand, if i is odd, then t = y'^t^y the fact that s ^ (t). Thus y{y~^t^y) with i,j
contradicting
/ yt for any i. Suppose y~^VyP
—t
G { 0 , . . . ,o(^) — 1}. If i is even, then f"*"^ = ^, so j is uniquely
determined by i.
If i is odd, then y~^Vyt^
again contradicts s ^ (t). k + ^^k^
— st\
= t implies sV^^
— t^ which
It follows that the coefficient of yt in ^y^~^
is
which is fc or A: + ^^ since p = 2 and t has order a power of t.
In any case, yt G supp(/it/^~^) for infinitely many k. Hence y has infinitely many conjugates, a contradiction. So, indeed. A' is
finite.
D
We now investigate the situation with loops L which are not torsion and, again, we begin with a series of lemmas. 3.6 L e m m a . LeA K he a field of characteristic loop with unique nonidentity
commutatorassociator
E = {e ^ KT \e IfU{KL)
p > 0 and let L be an RA
= e,e central in
s. Let KL}.
is FC, then there exist only finitely many nonzero elements
form e(l — s) with e ^ E. In particular, with s ^ H and if H contains PROOF.
of the
if H is a torsion subgroup of
no elements
of order p, then H is
Z{L)
finite.
Let x and y be noncommuting elements of L. Suppose e is an
idempotent in KT which is central in KL.
Then (1 — e) + xe is a unit in
3. FINITE CONJUGACY
353
KL with inverse (1 — 6) + x~^e and [(1  e) + xe]y[{l  e) + x~^e\ = (1  e)y + esy = y  ey{\  s). li f ^ E and e(l — 5) 7^ / ( I — 5), then these idempotents clearly yield different conjugates of y. Hence the first part follows. For the second part, assume H is an infinite torsion subgroup of Z{L) without elements of order p and with s ^ H. Then H contains an infinite strictly ascending chain of finite subgroups
Each idempotent Hi — Hj is central in KL. Furthermore, since s ^ Hi^ it follows that
for i 7^ J, in contradiction with the first part.
D
3.7 Lemma. Let K he a field of characteristic p > 0. Let L be an RA loop with torsion subloop T ^ L. IfU{KL) is FC, then T is an abelian group. Since squares of elements in L are always central, the centre Z{L) of L contains a nontorsion element, so Z(L) is infinite. Because of Lemma 3.4, all idempotents of KL are central. Suppose p > 2. Then it follows from Theorem 1.6 that the loop Tp/ is an abelian group and so elements of even order commute. Since elements of odd order are always central, this yields that T is commutative and hence an abelian group. (See Corollary IV. 1.3.) If p = 2 and the loop T2 is not central in T, then, since T is locally finite, T contains a finite noncommutative subgroup. By Lemma 3.1, ll{KT) is not FC, a contradiction. Thus T ^ T2 X T2' (Proposition V.1.1) is an abelian group. Finally, if p — 0, since all idempotents of KL are central, either T is an abelian group or T contains Qs as a subgroup, by Theorem 1.10. However, the latter is not possible since H(Q) C QQg Q KT and ZY(H(Q)) is not an FC group, by Lemma 3.2. D PROOF.
3.8 Lemma. Let K be a field of positive characteristic p and let L be an RA loop with torsion subloop T ^ L. IfU(KL) is FC and L contains an element of order p, then p — 2 and T = L' x A for some finite abelian group A of odd order.
354
XIII. IDEMPOTENTS AND FINITE CONJUGACY PROOF.
Let x and y be noncommuting elements of L, let z be a non
torsion element in Z{L)
and let ^ G i^ have order p. By L e m m a 3.7, the
loop T is an abelian group. Suppose p /
2. Then, since elements of odd order are central, ^ =
Yl^Zo 9^ ^^ ^ central element of KL with ^
= 0 = (z^x^^
for any i > 0, so
1 + z^x^ is invertible and (1 + z'xg)'y{l
+ z'xg)
= {1  z'xg)y{\ = y + z'giy^
+
z'xg)
 xy) = y + z'g{\

s)yx,
where V — { l , ^ } . Since s ^ (^), it is clear t h a t yx G supp(^(l — s)yx)
and
thus z'yx e s u p p ( l + z'x^y^y{l
+
z'xg).
Because z has infinite order, y has infinitely many conjugates, a contradiction. Thus p — 2 and so, by Lemma 3.6, T21 is finite. Next we show that if t is an element of order 2, then t is central in L. Assuming the contrary, \et y ^ L be such t h a t yt 7^ ty. Thus yt — sty.
Let
i > 1 be an integer and consider the following conjugate of y:
[1 + z\l + t)]'y[\ + z\\ + t)] = [1 + ^^(1 + t)]y[\ + z\l + t)] = [y+z'{y
+ syt)][l +
z'{l+t)]
= y[l + z\l
+ st)][l + z'{l + t)]
= y[l + z\t
+ St) + z'^\\
+st + t + s)].
Since z has infinite order, z'^yt is in the support of this element. It follows t h a t y has infinitely many conjugates, a contradiction. So elements of order 2 are central. Next we show that if t is an element of order 2, then t — s. For this, let i > 1 be an integer and consider the following conjugate of y: [1 + z'x{\
+ t)]^y[l
f z'x{\
+ t)] = [1 + z'x{\ ^y
+ z'xy{\
+ t)]y[\ + z'x{l
+ t)]
+ t){\ + s)
= y + z'xy{\\t
+ s + ts).
If 5 / ^, then z'^xy belongs to the support of this element, again yielding a contradiction. So indeed t = s. Thus every subgroup of the abelian group T2 contains s.
3. FINITE CONJUGACY
355
Remember that T = T2 X T21 (Proposition V.1.1). We prove that T2 is cyclic of order 2 and hence that T2 = {1? 'S}. If this is not true, there exists g E T2 with g'^ = s. Assume g is central and let n = 1 + ^. Then n"* = 0 and, for any integer i > 0, (1 + z'xn)y{l
+ z'xny^
= (1 + z'xn)y{l
+ z'xn + {z'xnf
+
{z'xnf)
= y + z^n{l + s)yx (l + z'^nx + (z*n)^a;^) = y + z'n{\ + s)yx = y + z'{l + g + g^ + g^)yx. Since z^yx is in the support of this conjugate of y, it follows that y has infinitely many conjugates, a contradiction. So g is not central. Let h £ L be such that g and h do not commute. Then [h(l + g)f = hih + gh){l + g) = h{h + shg){l +g) = h\l
+ sg)il +g) = h''g{\ + s).
Since (1 + s)^ = 0, we obtain [h{\ + g)Y = 0. Consequently, for any integer i > 0, [\ + z'h{\+g)]g[l
+ z'h{\ + g)]'
= [1 + z'h{l + g)]g[l + z'/i(l + 5) + (z'/i(l + g)f + (z'/i(l + p ) f ] = 5 + [z'/i(l + 5)ff + gz%\
+ 5) + 3(z'/i(l + 5)]2
+z'/i(l + 5)P^'/^(l + ff) +ff[^'Ml+ 9)f +z'/i(l + 5)5[^'Ml + 9)? + ^'Ml + 9)9Wh{'^ + ^)]^ = 5 + z'(/i5 + gh){\ + ff) + ^2^(^/1 + hg){\ + p)/i(l + 5) +z^\gh + /ip)(l + 5)/i(l + g)h{l + 5)
= ff + z^h{\ + 5)(5 + 52)[i + z^h{\ + 5) + ^ " ' ( ^ 1 + 9)?]Since (1 + 5)(p + g'^) is central in KL (Corollary III.1.5) and because (1 + ^){g + g^){^ + 5^) = 0, it follows that [1 + z^h{\ + ^)]^[1 + z^h{\ + g)]^ =g + z'h{\ + s){g + y^). So z'^h is in the support of this conjugate, again a contradiction.
356
XIII. IDEMPOTENTS AND FINITE CONJUGACY
We have shown t h a t p = 2 and T2 = L\
Thus T = V X A, where
A = T21 is a finite abelian group of odd order.
D
3.9 L e m m a . Let K be a field of characteristic loop with torsion subloop T = L' = { l , ' ^ } .
2 and let L be an
Then, for any /J^^i^ E:
RA
U{KL),
fx~^i/fx = 1/ or si/. It is well known t h a t the units of the Laurent polynomial ring
PROOF.
K[X,X~^]
are trivial; t h a t is, of the form kX'^ with 0 ^ k e K and i G Z.
(This can be shown easily, as in the proof of Lemma VL2.1.) It then follows by an induction argument t h a t all the units in Laurent polynomial rings in several commuting variables are also trivial. So, if yl is a finitely generated torsionfree abelian group, then all units in KA are trivial as well, because KA = K[Xi^X^^^,..
, X r i , X ~ ^ ] . Obviously this implies t h a t all units in
KA are trivial for an arbitrary torsionfree abelian group A. By assumption, L/L^ is a torsionfree abelian group, so every unit of K[L/L^] is of the form ka for some k G K\ A; 7^ 0, and a = aV G LIL'. KL{\
f s) is the kernel of the natural epimorphism eii: KL
every unit /i G U{KL)
^
Since
K[L/L^]^
can be written in the form fi — ka + a ( l + 5), where
k £ Ii\ k ^ 0^ a ^ L and a G K L. As both a^ and a{l + s) are central in KL (Corollary III. 1.5), it is clear t h a t fx~^ — k~^a~^ + k~'^a~^a{l
+ s).
Consequently, for any g £ L^ figfi~^ = [ka + a ( l  s)]g[k~^a~^ = [kag + a ( l + s)g][k~^a~^ = {ka)g(k^a^)
+ A:"^a"^a(l + s)] + A:"^a~^a(l + s)]
+ kag[{k'^a^
a(l + s)] + a ( l +
s)g{k^a^)
= aga~^ + {k~^ag)a~'^ a{l + s) + [fc"^a(l + s)g]a~'^ = aga~^ f k~^[{aga~^
+ g)a~^]a{\
+ 5),
using diassociativity to write aga~'^ as {aga~^)a~^,
Since (1 + 5)^ = 0 and
aga~^ \ g \s 2g or (1 + 5)^, we obtain /J^gfJ^'^ = aga~^ G {gtSg}. Thus the lemma holds in the special case i/ = g £ L. In general, if 1/ G U{KL)^
then
u = kiai + a i ( l + 5), 0 7^ fci G K^ ai G I^, a i G KL^ and fii/fj,"^ = li[kiai + a i ( l + s)]ii~^ — k\\ia\[i~^ Since \ia\\i~^
f a i ( l + s).
— a\ or sa\^ as already seen, and since s{\ \ s) — \ \ s., the
lemma follows.
D
3. FINITE CONJUGACY
3 . 1 0 T h e o r e m . Let K he a field of positive
357
characteristic
p and let L be
an RA loop with torsion suhloop T ^ L. Suppose T contains order p. Then U{KL)
an element
of
is an FC loop if and only ifp = 2 and T = V x A^
where A is a finite abelian group of odd order. PROOF.
T h a t the conditions are necessary follows from L e m m a 3.8. For
sufficiency, first note t h a t as a finite group of odd order, A is in t h e centre of L. Since \A\ is relatively prime to p., the commutative group algebra KA is semisimple artinian and thus a direct sum of m fields Ki — {KA)ei., Ci is a primitive idempotent of
where
KA.
Suppose // and u are units of KL,
Let 5* be a finitely generated RA
subloop of L containing supp(/^) U supp(i/) U A.
Since SjS'
is a finitely
generated abelian group, we can write SIS'
^ BJS'
X
B2/S\
where B\ and B2 are subloops of S containing S\
B2/S'
consists of those
elements of odd order in S/S'
and B\IS'
has no elements of odd order.
Hence the finite group B2/S'
is isomorphic to v4 = T2', so there is an
epimorphism (^: S ^ A with (p{a) = a for all a G A. Therefore, S — SiA^ S\ = ker(/?, and, since A is central and Anker Thus KS — {KA^S\
— 0 ^  KiS\.
(f — {1}, we have S — S\X
A.
Since T = L^ x A^ the torsion subloop
of 5*1 is 5J = L' = {1,5}. Also, S is an RA loop, so, by Lemma 3.9, for i — 1,... ,m, {liei){uei){fiei)~^ Hence ^vji'^
— Y^^=\^i{^)^^i^
= uci
or
svci.
with ^i{s) G { l , ^ } . Consequently, v has at
most 2 ^ conjugates in KL and U{KL)
is indeed an FC loop.
D
3.11 L e m m a . Let K be a field of characteristic
different from 2 and let L
he an RA loop with torsion suhloop T. IfU{KL)
is an FC loop and T is an
infinite abelian group, then any subloop of T is normal in L. PROOF.
Let s be the nonidentity commutatorassociator of L.
Since
noncentral subloops H of L are normal in L if and only ii s ^ H (Corollary IV.1.11), it is sufficient to show t h a t if t G T is not central in L, then s ^ {t).
Suppose the contrary and let ^ G T and y e L he such t h a t
yty'^ = st ^ (t). Then [{t  l)yT]'^ = 0, so 1 + (^  l ) ^ ? i s a unit in
KL.
358
XIII. IDEMPOTENTS AND FINITE CONJUGACY
Because tt = t^ we obtain [l + {t
l)yt]h[l
+ (t
l)yt] = [1  (^  l)yt]t[l = t{tl)yt
+ (t
+ t(t
l)yt]
\)yt
= t ~ 2tyt + yt + t^yt = t  2sy?+ 2yt because t'^ is central. Since yt G supp(2t/?) but s ^ {t)^ we have y in the support of the conjugate [I + (t  l)yt]h[l
+ {t 
l)yt].
Since T is commutative, (tiy^t) = {y,t) / 1 for any t\ ^ T (see Theorem IV.1.14 and Corollary IV.1.3), so, since T is infinite, t has infinitely many conjugates, a contradiction. D For a prime p, we denote by Z(p^) the ;?Priifer group, that is, the additive group Z(p^) = Q ( P ) / Z
where
Q^^^ = { ^  a,n G Z}.
Clearly Z(p^) is a divisible abelian group. Less clear, but a standard fact, is that every proper subgroup of Z(p°^) is finite, a property which characterizes the pPriifer group within the class of infinite abelian pgroups. (See [Rot95, Chapter 10] for details.) For an abelian group A and a positive integer n, let >1^ = {a^ I aG A], Obviously, A^ is a subgroup of A. 3.12 Lemma. [Pas77, Lemma 14.3.2] Let A be an abelian pgroup.
Then
1. any divisible subgroup of A is a direct factor of A; 2. A has a unique maximal divisible subgroup dA; 3. if dA ^ {I}, then dA is a direct product of Z(p^)groups. Let D be a divisible subgroup of A. By Lemma XII.2.12, the identity map I: D ^ D extends to a homomorphism (p: A ^^ D and it is easily verified that A = D x ker(/?, giving (1). PROOF.
3. FINITE CONJUGACY
359
To prove (2), let dA be the subgroup generated by all divisible subgroups Di of A, i G X, some index set. Then {dAf
= {D\ \iei)
= {Di\ieJ)
= dA.
Thus dA is also divisible and therefore the unique maximal divisible subgroup. For (3), consider the nontrivial subgroup E = {d ^ dA \ d^ = 1}. Since every nontrivial element of E has order p^ E = Ylieli^'^)
^^ ^^^ direct product
of cyclic groups of order p. Since dA is divisible, for each z G X we can choose a sequence of elements of dA defined by di^i = di and d^^^j^ = di^n Then Di = {di^n I ^ > 1) =
Let F = {Di\i
AP"^)'
e I).
Clearly F is a divisible
subgroup of dA^ so dA = F X S for some subgroup S of dA^ by p a r t (1). Since E C F^ the group S must be trivial, so F = dA. It remains now to show that dA is in fact the direct product of the subgroups Di. To see this, suppose t h a t a\a2'  at — 1 with aj G Dj. If p'^ is the largest order of the elements aj and \i p^ > 1, then 1 = (aia2 •• atY^''
= a{^
oF^
"'O^r
^ (^i) X (^2) X ••• X (d^),
a contradiction. Thus p ^ = 1 and the product is indeed direct.
D
The following lemma is essentially proved in [SZ77]. 3 . 1 3 L e m m a . Let F C K be fields of characteristic positive
integer such that k'^ ^ F for each k ^ K.
p = 0 or gcd{m^p) PROOF.
p > 0 and let m be a If \F\ > m and
either
— I, then K = F.
Let a e K. Then
(1 + far
 1 = m / a + ( ^ ) ( / a ) 2 + • . . + ( / a )  = cj e F
for each f £ F. Since  F  > m, there exist distinct elements / i , . •. , / m in F. Then fj^i
+ fj^2
+ '•' + f'j'xm
= c/^,
\ <3
<m,
360
XIII. IDEMPOTENTS AND FINITE CONJUGACY
is a system of linear equations in the variables x i , . . . , Xm with determinant
n • • /r
/l
/2
/I
fi
Jm
J m
m fm
So its (unique) solution is ma^i^^a^^...
7^0.
,a^.
Therefore a ^ F.
F ^ K.
Thus D
The following lemma is taken from [Mil78]. 3 . 1 4 L e m m a . Let K be an infinite field of characteristic be a group which does not contain elements group, then each torsion element PROOF.
p > 0 and let G
of order p. IfU{KG)
is an FC
of G is central.
By Lemma 3.4, each idempotent of KG is central, a fact we
will use several times. Let ^ G G be a torsion element of order r and let g E G. The unit group U[K{t^g))
is clearly an FC group and hence a finite extension of its centre.
(See §XII.2.2.) Therefore, there exists a positive integer n such t h a t //^ is central in K{t^g)
for any fi G U{K{t^g)).
Since r is invertible in A', the
idempotent e = r ~ ^ ( l + t +    \ T " ^ ) is central in KG.
Hence (t) is a
normal subgroup of G. First we deal with the case p = 0. Because the group algebra K{t) is semisimple, we can write K{t) = ®Ki as the direct sum of field extensions Ki of K. Because (t) is normal in G, conjugation by g defines an automorphism (f of A'(^). As each Ki = K{t)ei
for some idempotent e^ and because
Ci is central in KG^ ^i^^'i) = ^^f Let Kf
be the fixed subfield of A'^; t h a t
is, K^ consists precisely of those elements fixed by (f. Then, by the above, k'^ G A'^^, for any ki G Ki.
Since Q C Kf^ the field K^ is infinite, so, by
Lemma 3.13, Ki = K'f. Thus ip is the identity m a p , so tg = gt. Secondly, we consider the case p > 0 and write n = p^m with z > 0 and g c d ( p , m ) = 1. Assume first t h a t K is an algebraic extension of its prime subfield Fp. Since K is infinite, there exists a finite subfield E of K with A^ > m. By Theorem X.2.1, E{t)
= © £ i , where each Ei = £^(6)? ^i ^ primitive root
of unity over E. Since (t) is a normal subgroup of (^,^), conjugation by g
3. FINITE CONJUGACY
defines an automorphism (/?: E{t) ^ E{t),
361
Since each Ei = E{t)ei for some
primitive idempotent €{ G E{t) and because €{ is central, we obtain t h a t ^{E^) = E„ Let Ef denote the fixed subfield of Ei. ^p m ^ ^^^. Since all the fields Ef
Given a^ € £^t, we know t h a t
are finite and because the Frobenius
homomorphism on a finite field is an isomorphism, there exists bi G Ef such t h a t oJ^^
— b^ ; thus a ^ = bi G E^ for all z. It now follows from
Lemma 3.13 t h a t Ei = Ef.
Hence ip is the identity and so tg — gt.
Now assume t h a t K is not an algebraic extension of Ep. Let 6 G K be an element which is transcendental over Ep. Since U{Fp{0)G)
C U{KG)
also an FC group, we may assume t h a t K = Fp{6). Set F — Fp{6^).
is
Then
E — {/jP^ I k G A'} and, furthermore, if e is an idempotent of A (^), then e = e^^ G F{t'P'').
So both group algebras K{t)
and E{t) have the same
primitive idempotents. Since both algebras are also semisimple, we have A'(^) = e A ' , and
F(^P')
= e F , where F, = F{t'P')e,,
primitive idempotent in F{t).
R\ = K{t)e,,
e, a
Also, F^ = A'f .
Again, we consider the automorphism (/?: A'(/) —> K{i) which is conjugation by ^ G C and denote by K'f and Ff the fixed subfields of A\ and F^, respectively. For an element fi G F^, we have //^ ^ = a^ G Ff. a^ = 6^ for so ne b^ G A ^ Since ^{ai)
— ai^ we obtain b\
Write
— ^{biY''
and
thus b, G A f . Hex.:e //^ = 6, G A f H F, = /;^. So Lemma 3.13 yields t h a t F^ = F'f \ thus the restriction of (f to F(^'^ ) is the identity. So gt''^ = f^ g. Since the order of t is relatively prime to p, we again see t h a t / and g commute.
D
3.15 L e m m a . [ P a s 7 7 , Lemma 14.4.3] Let p be a positive G — T{p^).
Suppose K is a field of characteristic
does not contain
a root of unity of order p^ for some positive
Then any idempotent idempotents extension
prime
e G AK{G)
e^ Furthermore, of K generated
is a finite sum of orthogonal
for each i, {KG)ei
integer
k.
primitive
= F where F is the field
by all p^th roots of unity for all n.
finite abelian group whose order is invertible
and let
different from p which
in K, then for any
If B is a idempotent
e G A/v'(G'), the algebra {K[G X B\)e is a finite direct sum of fields. PROOF.
Let e G A x ( G ) be an idempotent and let E — {KG)e.
Suppose
M is a maximal ideal of the algebra E. Denote by A the natural homomorphism E ^ E/M.
Because G is commutative, A(F) is a field extension of
362
XIII. IDEMPOTENTS AND FINITE CONJUGACY
K generated by the multiplicative group X{Ge). As a homomorphic image of Z(p°°), we have A(Ge) = {1} or X{Ge) ^ l{p°°).
If t h e former, however,
we would have A(Ai^(G)e) = 0, a contradiction, because e G A(e) ^ 0. Thus \{Ge) ^ Z(p°°), so \(E)
A.K{G)
and
^ F. Clearly, we may assume t h a t
XiE) = F. Since, by assumption, K does not contain a root of unity of order p^ for some positive integer k^ it follows from Theorem X.4.4 t h a t there exist roots of unity ^j and ^2 of orders p^^ j > 2, and p^^ respectively, such t h a t [K{^j):
/i^(^2)] > P Since ^j € A(Ge), we can choose g E G with
X(ge) = ^j. Let H be the finite subgroup of G generated by supp(e) U {g}. Then H = (h) is cyclic with, say, X{he) = ^ ^ , a root of unity of order p ^ ; clearly, m > j . Consider the epimorphism a: KH by (T{a) = A(ae).
Because KH
^ K(^m)
given
is semisimple, there exists a primitive
idempotent / G KH with cT{f) = 1. Clearly fe = f,so
E = Ef®
E{e  / )
and X(E(e — / ) ) = 0. We claim t h a t A: Ef > F is an isomorphism. T h e mapping is obviously onto, so we need to compute its kernel. Let a G Ef = {KG)f
be contained in the kernel of A and let Hi — ( / f , s u p p ( a ) ) .
Then
a
H\ is cyclic, generated by /ii, say, and we may assume t h a t h\ [H\: H] = p^ and \{hie) Because a G {KH\)f,
— h. Hence
= ^m+a for some suitable p'^'^^th root of unity. we can write a — X^^_o (^i^\ with a^ G
{KH)f.
Thus
0 = A(a) = Y. M«.)C+a and A(a,) G AX^m) = ^  But, by Theorem X.4.4, [F{Uia):
F] = [AX^m+a): A^^m)]  P ^
so ^m\a cannot satisfy a nontrivial polynomial over F of degree less than p^. Therefore X{ai) = 0 for all i. Furthermore, because A is an isomorphism on {KH)f^
we conclude t h a t ai — 0 and hence that a = 0.
It follows from the foregoing t h a t
E = Ef®E{e
f) =
Ef®M.
So any maximal ideal Af of £ is complemented in E by a minimal ideal and EjM
= F. Hence E is artinian and semisimple. This proves the first part
of the lemma.
3. FINITE CONJUGACY
363
Assume now t h a t 5 is a finite abelian group whose order is invertible in K.
Let e G A K ( G ) be an idempotent.
By the first p a r t , {KG)e
=
A \ © • • • © Kr is the direct sum of fields, each isomorphic to F. Then {K[G X B])e = {KG)eB
= {K\ © • • • © Kr)B
= 0^^^
lUB.
Since each group algebra KiB is semisimple artinian, it follows t h a t (K[G X B])e is the direct sum of
fields.
3 . 1 6 T h e o r e m . Let K he a field of characteristic
D p > 0 and let L be an
RA loop with torsion subloop T ^ L. Suppose T does not contain of order p. Then U{KL)
is an FC loop if and only ifT
and one of the following
conditions
1. KT
is an abelian group
holds.
is finite and, for all t E T and x e L, we have xtx~^
some nonnegative 2. T is finite and
elements
multiple r of[K:
= t^"^ for
Fp].
central.
3. T is central and of the form
Z ( 2 ^ ) X A with A a finite group and
V C Z ( 2 ^ ) , and there exists an integer k > 0 such that K does not contain a root of unity of order 2^. PROOF.
Note t h a t p / 2 since s £ T has order 2, where L' = {l?'^}.
Suppose U{KL) Z{L)
is an FC loop. Since squares in L are central and L ^ T^
is infinite. Hence, by Lemma 3.4, all idempotents in KL are central.
Furthermore, from Lemma 3.7, we see t h a t T is an abelian group. Assume t h a t T is infinite.
From Lemma 3.11, each subloop of T is
normal in L, so, if 7/ is a subloop of T with s ^ H^ then H is central in L. Because T does not contain elements of order p . Lemma 3.6 yields t h a t T2' is finite. Also, letting j^^ be a maximal subgroup of T2 not containing 5, then H is also finite. Hence T2/H is an infinite abelian 2group with the property t h a t 5 = sH belongs to all its nontrivial subgroups. Thus T2/H no direct products and so, by Lemma 3.12, T2/H divisible, it follows t h a t T2 = TfH
contains
^ Z ( 2 ^ ) . Since Z ( 2 ^ ) is
for all n > 1. Hence [T2: Tf]
< \H\.
Choose n so t h a t [T2: Tl""] is as large as possible. Then
so T2" is divisible. Thus, if D is the maximal divisible subgroup of T2, then [T2: D] is finite and, by Lemma 3.12, T2 = D x B with B
finite.
Since
r = T2 X T2' we have T = D x A for some finite subgroup A. Since D is
364
XIII. IDEMPOTENTS AND FINITE CONJUGACY
the direct product of Z(2^)groups and because T2IH = Z ( 2 ^ ) , we obtain t h a t D ^ Z(2°°). Thus T = Z ( 2 ^ ) x A for some finite group A.
Because
squares of elements of L are central, the group Z ( 2 ^ ) is central. Because of Lemma 3.6, we also have U = {1,'S} C Z ( 2 ^ ) . Thus s ^ A^ so A and hence T as well are central. We now show t h a t the field condition in (3) is satisfied. Suppose t h a t K contains a primitive 2^th root of unity (^ k > 1. Then, by Lemma 1.1, for any x G Z(2'^) of order 2^,
is an idempotent in KL^ and central as earlier noted. Furthermore, e ( l — s) ^ 0 since ^^
^ = —^ ^ ^. Because of Lemma 3.6, we know t h a t there
are only finitely many idempotents with these properties. Therefore, there exists A: > 0 such t h a t K does not contain a root of unity of order 2^. Suppose now that T is finite but not central. Let t £ T and g ^ Lhe such that the group G = (t^g) is not commutative. Since U{KG) group U{KG)
C U[KL)^
the
is an FC group. Furthermore, G does not contain elements
of order p. Hence Lemma 3.14 yields t h a t A' is finite. Since all idempotents of KL are central, we also obtain t h a t for a l H G T and x G Z/, xtx~^
= fP""
for some integer r > 0, a multiple of [A^: Fp] (Theorem 1.6). This finishes the proof of necessity. To prove the converse, assume t h a t T is an abelian group and suppose first t h a t (2) is the case; that is, T is finite and central. Since \T\ is not divisible by p = char A' (T contains no elements of order p), the group algebra KT is semisimple artinian and, because it is commutative, KT — A^i ® • • • ® Kn is the direct sum of fields AV Certainly every idempotent of KT is central in KL^ so we can apply Lemma X I L l . l and write any ji G U{KL)
in the form // — Y^^=\ ^i^i'> ^i ^ ^^^ ^^^ Qi ^ ^ Since KT is central
in A'L, we have {kiqi){kjqj)
= (kikj){qiqj)
so, in particular, [kiqi)(kjqj)
—0
\ii ^ j . Since kik~^ = e^, the identity of A^^, and Yll=\ ^i — 1? i^ follows t h a t /i~^ = Y17=i K^^7^'
If ^ = Z)r=i ^i'^i is another unit, hi G Ki, r, G L, then,
remembering t h a t hi and ki are central, we have ^i~^u^ — Y^^=\ Since L' — { 1 , ^ } , each q~^riqi is either r^ or svi. t h a t the set {/I'^u/i
\ /j, G U{KL)}
is finite, so U{KL)
^mi^'^i^i
It follows immediately is F C .
3. FINITE CONJUGACY
365
Now assume t h a t (1) is the case but t h a t T is not central (otherwise, we are in case (2) which has just been settled). As before, KT = Ki®is the direct sum of fields !{{.
 Q Kn
Since p is odd and elements of odd order
are central, not all 2elements can be central; thus, Tpi is not central. Since Tpf C T , Tpf is an abelian group. Also, K is finite. Thus both assertions (1) and (2) of Theorem 1.6 hold and again we conclude t h a t every idempotent in KT
is central in KL,
Once again, we have t h a t any unit /x G KL
can
be written in the form /x = X^^Li ^i^if where ki G Ki and qi G L. At this point, we require the fact t h a t each field Ki is normal t h a t for all a.P e
in KL in the sense
KL, A > = aK^,
{aKi)f3 = a{KiP)
{Kia)(5 =
K^{af3),
and a(PKi)
=
(af3)K^,
This follows from normality of T in L, centrality of idempotents of KT in KL (which implies t h a t the identity €{ of each Ki is central) and routine calculation (see the proof of Lemma XII. 1.1). In particular, q~^ Ki — Kiq~^, so we have (kiqi)~^
— q~^k~^ G KiL, thus /i~^ = Y17=\i^i^i)~^
^^ before,
if zy = X^^_ h^Ti, hi G A\, n G // is another unit in A'L, then n
ji'^u^
= ^{kiq^)~^{h,ri){k,q,)
where, for x G U{FL), by T{x)
n
— R{x)L{x)~'^.
^ ( ^ )  U{FL)
= ^ U{FL)
Recall t h a t T{x)
^{Kri)T{k^q^) is the inner m a p defined
is a pseudoautomorphism with
companion x~^ (Theorem II.3.3) and t h a t we apply this map to the right of its argument. Thus (*)
[{h,r,)Tik,q,)]c
with c = (kiqi)'"^.
= [h,T{k^q,)][r,T(k,q,)
Now hiT{kiqi)
• c]
G A\, by normality of A'^, and similarly,
for certain A:i,... , /^s G Ki, we have rtT{k,q^)
^ {q~^k^)ri{kiqi)
=
[{kiq^)r^]{qik2)
= [{f^3{Q^^'^i)]{Qif^2) = k4[{q~^ri){qik2)]
=
k4[{q~^r^qi)ks].
Since Vi and qi are in L and since L' = {l^'^}, the element q~^riqi is either ri or svi. Thus riT(kiqi)
G Ki{6riKi),
where 6 = 1 or S = s. From (*), it is
366
XIII. IDEMPOTENTS AND FINITE CONJUGACY
apparent that [ihiri)T(kiqi)]c € Ki{[Ki{6r,Ki)]c}
C {Ki[Ki{6TilQ]}c
C
[Ki{6rilQ]c
SO that {hiT,)T{kiq,) e lUiSnlQ
C {KT)[6r,{KT)]
=
{KT)[r,{KT)]
because 6 eT, Thus //"^r/// G T,7=iiI^T)[ri{KT)], showing that {//"^^M I /JL € U{KT)} is finite, because KT is finite. Once again, we have that U{KL) is an FC loop. Finally, assume that (3) holds. Let // G U{KL). Let ^ be a finitely generated RA subloop of L containing supp(/x) U A U L'. Denote by T{H) the torsion subloop of H, Since H is finitely generated, T{H) is finite. Since also A C T ( ^ ) C T =: Z(2^) x yl, there exists t e Z(2^) such that T{H) = (t) X ^ . Because there exists an integer A: > 0 such that K does not contain a root of unity of order 2^, we know from Lemma 3.15 that 1  ? = ei +
\ en,
the sum of the orthogonal primitive idempotents in {I — t )A [(^) X A]. Let 1/ G U{KL) and, as above, let H\ be a finitely generated RA subloop of L containing H U supp(i/) U supp(ei) U • • • U supp(en). Then A'[r(^i)] = A^® • e AV, the direct sum of fields A\ = (A^[T(/i'i )])/,• and each fi a primitive idempotent in K[T[H\)]. As each {K[T{H\)\)ei is a domain which is algebraic over A', the ring {K\T{H\)])ei is a field as well. So each e^ is a primitive idempotent in K\T[H\)\ and we may assume that r > n and /i = e i , . . . ,/n = en Write ii = X)Li M/^ ^^^ u = J2Ui ^fiBecause of Lemma XILl.l, ij.fi = aiQi and vfi = fiihi^ where a^,/?, G Ki and Qi^hi ^ L^ i = l^... ^r. Because the fields A\ are central in KL^ r
Again, since V = {1,'S}, higih~^ = Sigi, Si = 1 or Si = s. So aihiQih'^ ^ aiQi if and only liSi  s and ji{\s) ^ 0. Since 15 G {lt)K[T{H)xA],
3. FINITE CONJUGACY
367
the latter implies fi{ei +  —\ e^) ^ 0; hence, ji G { 6 i , . . . , e^}. So n
v^iu~^ = ^
r
aigi6i + ^
t=l
t=n(l
n
/x/, = ^fieiSi
+ (1  (ei + • • • + e^))//
t=l
So the conjugates of /x belong to the finite set n
{Y^fMeiSi\(l{ei Hence ll{KL) is an FC loop.
+ "' + €n))ti\6i
G {l,^}, 1
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Index
algebra
semi, see s e m i  a u t o m o r p h i s m a u t o t o p i s m , 58
CayleyDickson, 15, 17 composition, 27, 169 generalized q u a t e r n i o n , 17
Baer radical, 148
nonassociative, 5
Baer, R., 306
q u a d r a t i c , 27, 103, 104
Bass cyclic unit, 201
q u a t e r n i o n , 17
Bass, H., 311
real quaternion, 13
BassMilnor T h e o r e m , 312 B a u t i s t a , R., 256 B e r m a n , S. D., 195, 209 bicyclic unit, 200 bilinear
simple CayleyDickson, 18 q u a t e r n i o n , 18 split, 19 composition, 36
form, 23 m a p , 38 Brauer, R., 3, 334 Bruck, R. H., 3, 49
Zorn's vector m a t r i x , 21 algebraic field extension, 248 integer, 202 n u m b e r field, 202, 309
Cayley loop, 68, 81
a l t e r n a t i n g function, 8 a l t e r n a t i v e ring, 7
n u m b e r s , 14 Cayley, A r t h u r , 3 CayleyDickson
Artin Generalized T h e o r e m of, 12 T h e o r e m of, 13
algebra, 15 process, 16 central series lower, 295 upper, 295 centrally nilpotent, 295 centre
associative bilinear form, 166 associator loop, 52 ring, 5 a u g m e n t a t i o n , 149 ideal, 149
of a loop, 53 of a loop ring, 87, 88, 101
m a p , 149 A u t ( a u t o m o r p h i s m conjecture), 234, 237
of a ring, 6 characteristic, 88 Chein, O., 69, 80, 144 Chinese Remainder T h e o r e m , 249, 253 class conjugacy, 86
automorphism F, 230 inner, 228 of a loop, 58 pseudo, see p s e u d o  a u t o m o r p h i s m 377
378 nilpotency, 295 s u m , 86 Classification T h e o r e m , 140 Coelho, S., 333 Cohen, D. E., 256 C o h n , J. A., 195 commutator loop, 52 ring, 5 unique nonidentity, 87 commutatorassociator subloop, 53 unique nonidentity, 87 c o m p a n i o n , 58 component simple, 167 composition algebra, 27 split, 36 p e r m i t , 27 conjugacy class, 86 conjugate, 86 c o m m o n , 239 of a q u a t e r n i o n , 14 rationally, 236 conjugation, 65 C u r t i s , C. W . , 2, 36 cyclotomic extension, 249 field, 175 polynomial, 175, 249 Dade, E., 226 de Barros, L. G. X., 19, 271, 282 Deskins, W . E., 226, 258 d e t e r m i n a n t , 22 diassociative, 52, 68, 125 Dieudonne, J., 166 T h e o r e m of, 166 direct p r o d u c t , 109 s u m , 25 Dirichlet Unit T h e o r e m , 208, 213, 309 divisible g r o u p , 305 division ring, 7 normal, 290, 365
INDEX dual space, 25 Engel group, 306 loop, 295 equivalent fields, 261, 282 Euler (^ function, 175 exponent b o u n d e d , 296 of a Moufang loop, 132 extension, see field e x t r a loop, 109 F C , 295 Fernandes, N. A., 237 field algebraic extension, 45, 248 cyclotomic extension, 249 finite extension, 248 Galois extension, 248 normal extension, 248 separable extension, 248 fixed subfield, 360 flexible, 8, 63 form associative, 166 bilinear, 23 m a t r i x of, 23 nondegenerate, 23 q u a d r a t i c , 26 q u a d r a t i c , 26, 104 s y m m e t r i c , 23 free group, 215 s u b g r o u p , 214 Frobenius h o m o m o r p h i s m , 337, 361 Frobenius, G., 3 function alternating, 8 multiplicative, 14 Cloop, 116 Galois field extension, 248 group, 248, 316 general linear
INDEX group, 83 loop, 83, 316 Giambruno, A., 234, 297, 300 Glauberman, G., 125 group divisible, 305 Engel, 306 Galois, 248, 316 general linear, 82 hamiltonian, 68 identity, 294 inner mapping, 57, 86 multiplication, 50 ordered, 196 Priifer, 358 principle congruence, 215 projective special linear, 215, 325 solvable, 302 special linear, 82, 214 group algebra modular, 247 Hall, M., Jr., 69, 99, 138, 144 hamiltonian group, 68, 93 Moufang loop, 68 Hartley, B., 195, 220 Higman, G., 195, 204, 208, 225 Theorem of, 208, 309 Hurwitz, A., 2, 35 Theorem of, 30 ideal left modular, 148 nil, 147 nilpotent, 147 prime, 147 idempotent central, 300, 333 primitive, 174 trivial, 199 identity flexible, 8, 63 group, 294, 301, 302 left alternative, 7, 63 Moufang, 9, 63
379 right alternative, 7, 63 TeichmuUer, 6 index, 67 finite, 202 of a subloop, 62, 202 inner automorphism of a ring, 228 map, 57, 65, 86 integral element, 155 extension, 155 loop ring, 223 inverse left, 50 property, see right or left quasi, 148 right, 50 invertible, 81 involution of a loop, 80 of an algebra, 14 isomorphism normalized, 226, 259 problem, 223, 225, 247 isotope, 116 isotopic, 116 isotropic, 24 totally, 24 Jacobson radical, 148 Jacobson, N., 223, 230 Juriaans, S. O., 237 Kothe radical, 148 Karpilovsky, G., 158 Klein, A., 296 Kleinfeld function, 8 radical, 148 Klinger, L., 237 lack of commutativity, see LC Lagrange's Theorem, 51, 67, 114, 115 Laurent polynomial ring, 153, 356 LC, 9496, 98, 119 Leal, G., 125, 271, 314
380
left inverse property, 60 Lichtman, A., 350 linear function, 104 linearization, 7 Livingstone, D., 195 localization, 300 locally finite loop, 212 loop, 50 2, 125 Cayley, 68, 81 e x t r a , 109 F C , 295 G, 116 general linear, 83, 316 h a m i l t o n i a n , 301, 302, 307 Moufang, 68 indecomposable, 126 inverse property, 60, 63 locally finite, 212 Moufang, 64 nEngel, 295 nilpotent, 295 opposite, 55 quotient, 57 RA, 107 ring, 85 ring alternative, 107 special linear, 83 torsion, 125 t y p e 1, 271 t y p e 11, 271 unit, 82 L u t h a r , I. S., 237 Mfclaw, 115 Mal'cev, A.I., 296 Marciniak, Z. S., 218 Maschke, H., 3, 4, 147, 164 T h e o r e m of, 167 May, W . , 158 m e t a b e l i a n , 225 metacyclic, 226 Milnor, J., 311 modular algebra, 247 case, 267
INDEX ideal, 148 Molien, T . , 3 Moser, C , 340 Moufang identities, 9, 63 Moufang, R u t h , 2, 63 T h e o r e m of, 68, 82 multiplicative function, 14 nEngel, 295 nsquares problem, 35 NielsenSchreier T h e o r e m , 327 nil, 147 nilpotency class, 295 nilpotent, 147, 295 centrally, 125, 295 element, 147 Noether, E., 3 nonassociative, 5 ring, 5 nondegenerate norm, 104 norm, 27, 103 quaternion, 14 normal complement, 314 division ring, 290, 365 field extension, 248 subloop, 55 normalized form, 189 isomorphism, 226, 259 unit, 236 Norton, D. A., 68 T h e o r e m of, 75 nucleus left, 53 middle, 53 of a loop, 53 of a loop ring, 101 of a ring, 6 right, 53 order (in an algebra), 201 O s b o r n , J. M., 88 psequence, 262
INDEX Paige, Lowell, 4, 88 T h e o r e m of, 88 P a r m e n t e r , M. M., 208, 209, 212, 234 Passi, I. B. S., 237 P a s s m a n , D. S., 197, 198 Perils and Walker T h e o r e m of, 177, 252 Perils, S., 226, 252 Peterson, G., 234 Pflugfelder, H. O., 49 Pickel, P. F., 195, 220 Pickert, G., 49 polynomial ring, 153 power SLSSociative, 64, 88 Priifer group, 358 prime radical, 148 ring, 147 primitive i d e m p o t e n t , 174 root of unity, 175, 248 principal congruence group, 215 product direct, 109 projective special linear group, 215, 325 p s e u d o  a u t o m o r p h i s m , 57, 65 quadratic algebra, 27, 103, 104 form, 26, 104 quasiinverse, 148 quasiregular, 148 quasigroup, 49 quaternion, 1 algebra, 13 R A loop, 107 radical, 148 Baer, 148 hereditary, 149 Jacobson, 148 K o t h e , 148 Kleinfeld, 148 prime, 148 Smiley, 148 u p p e r nil, 148
381 Zhevlakov, 148 rank, 132 rationally conjugate, 236 reflection, 230 representation right regular, 167 right inverse property, 60 ring Laurent polynomial, 153, 356 nonassociative, 5 of algebraic integers, 202 of fractions, 300 polynomial, 153 prime, 147 simple, 7 ring alternative, 7 Ritter, J., 212, 213, 237 Robinson, D. A., 229 Roggenkamp, K. W., 226, 237 root of unity, 175 primitive, 175, 248 Sandling, R., 226 Schur, 1., 3 Scott, L., 226, 237 Sehgal, S. K., 212, 213, 218, 234, 237, 238 s e m i  a u t o m o r p h i s m , 65 semiprime ring, 147 semisimple algebra, 148 case, 267 Senior, J. K., 99, 138, 144 separable field extension, 248 polynomial, 248 signature, 309 simple c o m p o n e n t , 167 ring, 7 Smiley radical, 148 Smith, P. F., 208 special linear group, 214 projective, 325 special linear loop, 83
INDEX
382 Spiegel, E., 261, 266 split, 19 composition algebra, 36 splitting field, 333 subloop, 51 c o m m u t a t o r  a s s o c i a t o r , 53 index of, 62 n o r m a l , 55 subring unital, 201 s u p p o r t , 86 TeichmuUer identity, 6 tensor p r o d u c t , 39 T h r a l l , T . M., 252 T i t s , J., 218 torsion element, 125, 196 loop, 125 subloop, 212 unit, 196, 198 totally imaginary, 309 real, 309 t r a c e , 27, 103 associative, 104 c o m m u t a t i v e , 104 T r a m a , P., 237 t r a n s l a t i o n m a p , 50 transversal, 52, 290 trivial, see unit T r o j a n , A., 261, 266 unit, 81 Bass cyclic, 201, 312 bicyclic, 200 central, 198 torsion, 199 loop, 82 Engel, 306 F C , 306 nilpotent, 306 normalized, 236 torsion, 196, 198 trivial, 195, 196, 204 Bass cyclic, 204
bicyclic, 200 central, 212 unital, 5 unity, 5 Valenti, A., 234, 237, 297, 300 Walker, G. L., 226, 252 Weiss, A., 226, 237 W h i t c o m b , A., 224, 225 L e m m a of, 224 Williamson, A., 209 Wright, C. R. B., 125 Zorder, 201 Zassenhaus, H. J., 236 conjectures of, 236 ZCl (first Zassenhaus conjecture), 236, 239 ZC2 (second Zassenhaus conjecture), 237, 241 ZC3 (third Zassenhaus conjecture), 237, 241 Zhevlakov radical, 148 Zorn's vector m a t r i x algebra, 21 Zorn, M., 22
Notation
What follows is a list of symbols and other notation used in this book. Page references guide the reader to the place where the synnbol is defined or first mentioned.
Symbol
Meaning
P
the prime numbers
N
the natural numbers, 1,2,3,...
Z
the integers
Q
the rational numbers
R
the real numbers
C
the complex numbers
Fp
the field of p elements, p a prime
(j)
the Euler function, p. 175
^„
unless specifically stated to the contrary, this always denotes a primitive nth root of unity, p. 175, p. 248
(F, Qf,/^)
a generalized quaternion algebra over a field F, p. 17
»(F)
( F ,  l ,  l ) , p . 17
H(R)
(R,—1,—1), the classical division algebra of real quaternions, p. 17
(F, Of,/?, 7)
a CayleyDickson algebra over a field F , p. 17
C
the Cayley numbers, p. 15
Mn{R)
the ring of n x n matrices over a ring R
383
384
NOTATION
R[Xi,
X 2 , . . . , Xn]
the ring of polynomials in c o m m u t i n g indeterminates X\^. . . , Xn with coefficients in a ring R
R[Xi, X f \ . . . , Xn, X~^]
the ring of Laurent polynomials in c o m m u t i n g indeterminates X i , . . . , X n , p. 153
R^
the set of ordered ntuples over a ring R
RL
the loop ring of a loop L over a ring R, p. 85
M 04) A^
the tensor product of modules M and A'' over a c o m m u t a t i v e ring
Cn
the cyclic group of order n
Dn
the dihedral group of order 2n
Qg
the quaternion group of order 8
Mi6{Qs)
the Cayley loop, p. 68
Z{p^)
the pPriifer group, p. 358
M(G,*,^o)
the Moufang loop determined by a nonabelian group G with involution * and central element (70, p. 80
M{G,  1 ,
.9(1
the special case of M(G,^,
go) where * is the in
verse m a p on G, p. 81
M(G, 2)
M ( r ; ,  l , l ) , p. 80
In
where / is an ideal of a ring R, and n G N, this is {r e R\ rire
Lp
/ } , p. 157
where L is a loop (or group) and p is a prime, this is the set of elements of L whose order is finite and a power of p, p. 157
^p'
where L is a loop (or group) and p is a {)rime, this is the set of elements of L of finite order relatively prime to p, p. 157
[E: F]
where E/F
is a field extension, this is the dimen
sion of E" as a vector space over F , p. 175 G?i\{E/F)
the Galois group of a Galois field extension
E/F,
p. 248 F{a)
where F is a field and a is an element contained in some extension field of K of F , this is the smallest subfield of K containing F and cv, p. 45
NOTATION
Z[a]
385
the subring of Q(of) generated by an elennent a E C, p. 202
J\fil{R)
the upper nil radical of a ring R, p. 148
V{R)
the prime radical of a ring R, p. 148
J[R)
the Jacobson radical of a ring R, p. 148
GL(2, R)
the general linear group of degree 2 over a ring i^, p. 82
SL(2, R)
the special linear group of degree two over a ring R, p. 82
PSL(2, R)
the projective special linear group of degree two over a ring R^ p. 215
3(/^)
Zorn's vector m a t r i x algebra over a ring R, p. 21
GLL(2, R)
the 2 X 2 invertible elements of 3(7?.), p. 83
SLL(2, R)
the elements of GLL(2, R) of determinant 1, p. 83
Sdet=±\
the matrices of determinant 1 in a subloop S of GLL(2,Z[<en]),p. 316
'^det=±i
^^^ quotient loop 5'det=±i/{/, — ^}) where / is the identity m a t r i x of Zorn's vector matrix algebra, p. 317
5'^i(det)=±i
those matrices in a subloop S of GLL(2, Z[<^n]) with the property t h a t the norm of dei A is ibl, p. 316
^nidet^±\
^^^ quotient loop 5'n(det)=±i/{^, — 0 , whcre / is the identity matrix of Zorn's vector m a t r i x algebra, p. 316
x^
the right inverse of an element x in a loop, p. 50
x^
the left inverse of an element x in a loop, p. 50
R[x)
right translation by x, p. 50
L[x)
left translation by x, p. 50
T{x)
R{x)L{x)\p. 57
R{x,y)
R{x)R{y)R{xy)\p.b7
L[x,y)
L{x)L{y)L{yx)\p.b7
(a, 6)
the c o m m u t a t o r of elements x and y in a loop, p. 52
386
{X,Y)
NOTATION
for subsets X and F of a loop, this is the set of all c o m m u t a t o r s (x, y), x ^ X^ y G V, p . 52
{a,b,c)
the associator of elements a, 6 and c in a loop, p. 52
(X,Y,Z)
for subsets X , y , Z of a loop, this is the set of all associators (ic, y^z)^ x ^ X ^ y ^Y
^ z E. Z ^ p. 52
[a,b]
the c o m m u t a t o r of elements a and 6 in a ring, p. 6
[X,Y]
for subsets X and F of a ring, this is the set of all c o m m u t a t o r s [x, y], x E ^, y G F , p. 6
[a, b, c]
the associator of elements a, 6, c in a ring, p. 6
[X, Y, Z]
for subsets X , F , Z of a ring, this is the set of all associators [x, y, z], x G ^ , y G F , z £ Z^ p. 6
L'
the commutatorassociator subloop of a loop L, p. 53
Z{X)
the centre of a loop or ring X , p. 6 and 53
M{X)
the nucleus of a loop or ring X , p. 6 and 53
M{L)
the multiplication group of a loop L, p. 50
Inn(L)
the inner mapping group of a loop L, p. 57
[L: / / ]
where // is a subloop of an inverse property loop L, this is the index of H in L, p. 62
Aut(L)
the group of automorphisms of a loop L
CM
where A^ is a normal subloop of a loop L, this is the kernel of the m a p RL —> R[L/N],
which is
the linear extension of the natural m a p L —^ L/N, p. 149 6
ei^ where L is a loop, this denotes the augmentation m a p RL ^ R, p. 149
A ( L , N)
where L is a loop with normal subloop TV, this is kere^v, p. 149
A(L)
for a loop L, this is A ( L , L ) , the augmentation ideal of a loop ring RL, p. 149
N
where N is a finite loop contained in a loop ring, ^his is E n G N ^ ' P 1^1
NOTATION
387
N
where N is a. finite loop contained in a loop ring RL with \N\ invertible in i?, this is AnN^ p. 151
^
{9) — Yl7=o 5'*' where g is an element of finite order n in a loop ring, p. 200
p
(^) rr 1 Yl^=o 9^' where g is an element of finite order n in a loop ring RL^ with n invertible in R^ p. 218
U{R)
the set of units in a ring 7?, p. 81
U{RL)
the set of units in the loop ring RL
U\{RL)
the set of normalized units in the loop ring RL^ p. 236
TU{RL)
the set of torsion units in the loop ring RL, p. 236
TU\ [RL)
the set of normalized torsion units in the loop ring RL, p. 236
char R,
the characteristic of a ring R, p.88
p[A)
the rank of an abelian group A] that is, the minimal number of generators for A, p. 132
o(x)
the order of an element x in a loop, p. 128
ind2(A)
an invariant of a field K related to its 2sequence, p. 265 this is 1 if x^ 4 1 == 0 has a solution in the field K and 0 otherwise, p. 265
0[K) i{K)
this is 1 if x^ fy^ = —1 has a solution in the field K and 0 otherwise, p. 265
c[K)
this is 1 if x^ h y^ \ P f tf;^ = _ i has a solution in the field K and 0 otherwise, p. 282