Preface
T h e importance of group algebras has been clear since t h e work of T . Molien, G. Frobenius, I. Schur and H...
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Preface
T h e importance of group algebras has been clear since t h e work of T . Molien, G. Frobenius, I. Schur and H. Maschke in the early 1900s. Often studied because of their applications to group theory, they have also long since a t t r a c t e d attention as objects of interest in their own right. To t h e obvious interplay between group theory and ring theory which is offered through the theory of group rings, in this book, we incorporate a new element, lack of associativity. General quasigroup algebras were defined by R. H. Bruck in 1944, but it was not until the 1980s, with the discovery of a class of loop algebras satisfying important identities, the right and left alternative laws, t h a t the subject was seen to be substantive. For the past ten years, alternative loop rings have intrigued m a t h e m a t i cians from a wide cross-section of modern algebra, including loop theory, group theory and ring theory, both associative and nonassociative. As a consequence, the theory of alternative loop rings has grown tremendously. One of the main developments is the complete characterization of loops which have an alternative, but not associative, loop ring. Such loops are closely related to a class of 2-groups. As an application, a full characterization of the structure of alternative loop algebras has been obtained. In particular, one has complete descriptions of the loop representations in Zorn's vector matrix algebra. Furthermore, there is a very close relationship between the algebraic structures of loop rings and of group rings over 2-groups.
An-
other major topic of research has been the study of the unit loop of the integral loop ring. Here the interaction between loop rings and group rings is extremely fascinating. This is the first survey of the theory of alternative loop rings and related issues. Because of the strong interaction between loop rings and certain ix
X
PREFACE
group rings, we have included many results on group rings, some of which are new and some of which are known though, in the latter case, we often have a new viewpoint and novel, elementary proofs. This work has been written with minimal expectations on the part of the reader. We have assumed that the reader has had the equivalent of a standard first-year graduate course and hence that he or she is familiar with basic ring-theoretic and group-theoretic concepts. We have made an earnest effort to present a work which is self-contained, in the hope that it will prove to be a useful reference to the student as well as the research mathematician. There is an extensive bibliography of references which are either directly relevant to the text or which offer supplementary material of interest. The first three chapters are intended to serve as "short courses" on the fundamental algebraic structures with which this book is primarily concerned—alternative rings, Moufang loops and loop rings. While much of what is presented is undoubtedly in print elsewhere, we are unaware of a single source. The pace is leisurely, all terminology is carefully defined, all theorems are proven in detail. This book is mainly concerned with loop rings which are not associative, but which are alternative and hence nearly associative. An alternative ring is a ring in which the subring generated by any two elements is associative. Associative rings are alternative, though much of the interest in this work derives from the study of alternative loop rings which are not associative. Chapters IV and V discuss the properties of and classify those loops which, in characteristic different from two, have alternative loop rings which are not associative. We call these loops RA ("ring alternative") loops. The remaining chapters are devoted, in the main, to an analysis of the loop rings themselves and to the group rings of certain groups which are closely related to RA loops. In Chapter VI, we describe the Jacobson and prime radicals of an alternative loop ring and establish a Maschke type theorem for alternative loop algebras: over many fields, alternative loop algebras are the direct sums of simple algebras. The structure of alternative loop algebras over a field is further investigated in Chapter VII. We fully describe the simple components of
PREFACE
xi
a (semisimple) alternative loop algebra over the rationals by establishing concrete isomorphisms with Zorn's vector matrix algebras. Furthermore, we give a concrete description of the primitive central idempotents which, as it turns out, are "group determined". This work requires knowledge of the primitive idempotents of the group algebra of a finite abelian group and so, consistent with our goal to write a self-contained work, there is a thorough treatment of this subject in this chapter. In Chapter VIII, we begin a study of the units in an integral alternative loop ring. For example, we determine by elementary methods when all the units are trivial and when all the torsion units are trivial and, as a consequence, also obtain new proofs of the well known theorems of G. Higman and S. D. Berman for group rings. As with group rings, the central units of finite order in the integral alternative loop ring of a finite loop are trivial. Moreover, most such loop rings contain a free subgroup. A finite RA loop L is determined by its integral loop ring; that is, if G is another RA loop and ZG = ZL, then G = L, Thus, the well known "isomorphism question" has a positive answer within the context of integral alternative loop rings (which are not associative), as is shown in Chapter IX. After establishing for composition algebras, an appropriate analogue, due to N. Jacobson, of the Skolem-Noether Theorem for simple associative rings, we also show in Chapter IX that every "normalized automorphism" of ZL, L a finite RA loop, is the composition of an automorphism of L and an inner automorphism of the rational loop algebra QL. We also prove variations of three conjectures which H . J . Zassenhaus has formulated for group rings; for instance, we show that every torsion unit in ZL is conjugate to a conjugate of an element of ±L. We next turn our attention to the loop algebras of RA loops over arbitrary fields where our primary interest is the isomorphism problem. Since abelian groups play such an important role in the structure of RA loops, we first study this problem for group algebras of finite abelian groups and devote an entire chapter. Chapter X, to this topic. In Chapter XI, we apply our results to loop algebras. The structure of the unit loop and, indeed, finding all the units of an integral alternative loop ring, both in the associative and not associative cases, are hard and exciting areas of current research. Even for many loops
xii
PREFACE
of small order, the structure of the unit loop is not known. As a compromise, one aims for the structure of a subloop of finite index in the unit loop or, sometimes, just for a finite set of generators for such a subloop. These are the themes of Chapter XII. Under what conditions does each element of a unit loop possess only finitely many "conjugates" (in the usual associative sense of the word)? This is the central theme of our final chapter, for alternative loop algebras over fields. In the process, we also describe conditions under which all idempotents of a loop algebra are central and determine when aU the nilpotent elements are trivial. At the end of a preface, it is customary to thank family members, colleagues, typists and proofreaders for various things. Certainly, the three of us owe enormous debts of gratitude to our wives, Linda, Griet and Sandra, who have displayed remarkable patience and understanding during our years of work on this project. Many people have read various parts of our manuscript and made helpful comments, including Luiz de Barros, Guilherme Leal, Yuanlin Li, Daniel Robinson, Yiqiang Zhou and Yongxin Zhou. We would, however, like especially to identify and thank Michael Parmenter, our long time friend and colleague, for his extraordinary contributions. As for typists and proofreaders, we have only ourselves to thank, and to blame. The typesetting and proofreading were shared by the three of us, who assume the entire responsibility for errors.
Introduction
The formula (a^ + b'^)(c'^ + d'^) = (ac - bdf + {be + adf, which shows that the product of two sums of two squares is again the sum of two squares, is very familiar and most elegantly established with reference to the field of complex numbers. The first noncommutative "field" (more commonly called "division algebra") was discovered in 1843 by Sir William Rowan Hamilton and is denoted H(R) in Hamilton's honour. This algebra is the four-dimensional vector space over the real numbers, R, with basis {1, i,i, k}. Its elements, called quaternions^ are thus formal sums of the form (1)
q — a-\- bi + ej + dk
where a, b^e^d £ R. Two such sums are added "coordinatewise" (ai + bii + eij + d^k) + (a2 + 62^ + C2J + d2k) = {ai + a2) + (61 + b2)i + (cj + 02)7 + (di + d2)k and multiplied using distributivity and Table 1, which shows how to multiply basis elements. As in the complex numbers, there is in H(R) a notion of TABLE
1. Multiplication table for l , i , j . A: in the quaternions.
I1
~T1nr
i
i i i -1 J 3 -k k \k 3
3
k
k 3 k -3 i -1 —i - 1
2
INTRODUCTION
norm; for q = a-\- bi + cj + dk £ H(R),
\q\ = a' + b' + c' + d\ This norm is multiplicative: for any ^1,^2 € H(R), we have \qiq2\ = This fact corresponds to the formula
\QI\\Q2\'
(aj + al + al + al)(bl + bl + bl + bj) = (ai6i - 0262 - cisbs - 0464)^ + (ai62 + ^^2^1 + ^364 - 0463)^ + («1^3 + ^3^1 - Ct2&4 + ^^4^2)^ + (^164 + ^^461 + a263 - ^^362)^, which shows that the product of two sums of four squares is another sum of four squares. This particular formula plays an important role in number theory; for example, it shows that to prove that every natural number is the sum of four squares, it is sufficient to establish the result just for primes. For what integers n are there identities of the form n
n
n
t=l
t=l
t=l
with each Ci = X)j jt=i ^ijkO^ibj-, Zij^ G C, a biUnear form in the variables a^, bjl The answer, which was given by A. Hurwitz in 1898, is n = 1,2,4,8 [Hur98]. We refer the reader to a beautiful article on the history and solution of the n-squares problem by Charles Curtis [Cur63]. That there exist n-squares formulas for n = 1, 2, 4 and 8 follows from the existence of the real numbers, complex numbers, quaternions and a certain nonassociative algebra called the Cayley numbers. That these are the only integers for which formulas exist is a consequence of the fact that the reals, the complex numbers, Hamilton's quaternions and the Cayley numbers are the only alternative division algebras of finite dimension over the real numbers, a theorem which we prove in Chapter I. Alternative rings arose out of the work of Ruth Moufang in the 1930s [Mou33, Mou35]. Given a projective plane, one can label the points and the lines with elements from a set R and then define the addition and multiplication of elements of R in terms of incidence relations in the plane. (See [Hal59, Chapter 20] for an introduction to projective planes and their coordinatization.) One can then relate various geometrical properties of
INTRODUCTION
3
the plane to algebraic properties of the "ring", (i?, +, •). Two of the nicest theorems in this regard concern planes in which certain theorems due to Desargues and Pappus hold. A plane is desarguesian (pappian) if and only if it can be coordinatized by a planar alternative division ring (field). Since a field is a particular kind of alternative division ring (one which is commutative), a pappian plane is always desarguesian. Since a finite alternative division ring must be a field, a finite desarguesian plane is necessarily pappian. Much of Moufang's attention was directed to the m^ultiplicative structure of an alternative division ring. Just as the nonzero elements of a field form a group under multiplication, so the nonzero elements of an alternative division ring form a Moufang loop under multiplication. Much of this book concerns Moufang loops, which are introduced in Chapter II. Group rings were implicitly introduced in a paper by Arthur Cayley in 1854 [Cay54]. Given Hamilton's definition of a quaternion several years earlier, it was natural to consider more general algebraic expressions c^i^i
+ 0262 +
h
ajiCn,
where the a^ belong to some field F (originally the real or complex numbers) and { e i , . . . ,6^} is a basis for a vector space over F. Multiplication of two elements with the above form could be defined quite naturally by first showing how the basis elements multiply, as with the quaternions, and then extending this multiplication to linear combinations of basis elements by means of the left and right distributive laws. As a specific example, Cayley considered the case where the €{ were the six elements of the symmetric group 53, thereby giving the first instance of the now familiar group algebra 0^3. The importance of group algebras became clear in the early 1900s after the work of T. Molien, G. Frobenius, I. Schur, H. Maschke, and later R. Brauer and E. Noether, on group representation theory. Since then, group algebras have taken on a life of their own. The appearance of two large books on the subject [Seh78, Pas77] at almost the same time made it clear how the subject had grown by the late 1970s to an important field within its own right. The idea of relaxing the requirement of associativity and considering general loop rings or algebras is due to R. H. Bruck, who introduced the idea of a quasigroup algebra in a paper in 1944 [Bru44]. In this work.
4
INTRODUCTION
Bruck proved that over a nonmodular field, the loop algebra of a finite loop is the direct sum of simple algebras; thus, the well known theorem of H. Maschke for group algebras [CR88, Theorems 10.8 and 15.6] holds in the more general nonassociative context. Two years later, Bruck determined the centre of a loop algebra [Bru46], but the subject of loop algebras then seems to have laid dormant until 1955. In that year, Lowell Paige proved that if a commutative loop algebra over a field of characteristic diflFerent from 2 satisfies even the very weak identity, x'^x'^ = x^x, then the loop is an abelian group [Pai55]. In other words, there are no "interesting" nonassociative commutative loop algebras which are not already group algebras. This result strongly suggested that it was fruitless to expect that a loop algebra could satisfy any important identity without, in fact, being a group algebra. Nevertheless, in 1983, E. G. Goodaire showed that there do exist alternative loop algebras (which are not group algebras) [Goo83]. It is this paper which gave birth to a subject which has grown rapidly, as we hope to illustrate with this monograph.
Chapter I
Alternative Rings
1. Fundamentals A (nonassociative) ring is a triple (i?,-h,-)? where (i2,+) is an abehan group, (i?, •) is a groupoid (that is, • is a binary operation) and both distributive laws hold: a(b + c) = ab + ac^ (a + b)c = ac+ bc^ for all a^b^c ^ R. If, in addition, (i2,+) is a module over a commutative, associative ring $ such that a{ab) = {aa)b = a{ab) for all a G $ and all a, 6 G R^ then (i2, +, •) is said to be a (nonassociative) algebra. Except in Chapter VI, we assume that any algebra or ring R is unital; that is, we assume that R possesses an element 1 / 0 , called a unity, with the property that al = la = a for all a € ^ and, in the case of an algebra over a ring $, that J? is a unital ^-module. Unless we indicate specifically to the contrary, we also assume that all fields have characteristic different from 2. The term "nonassociative" means "not necessarily associative". If we intend that a ring (or some other structure) be not associative, we say so. Also, we usually suppress the operations when specifying a ring, referring to the ring "J?", instead of the technically correct "(i?, +,•)"• The concepts of subring, subalgebra, homomorphism and ideal are the same in general as they are in associative algebra and the basic isomorphism theorems hold as well. Nevertheless, certain well known "facts" in associative ring theory are not true in general. For instance, in a ring which is not associative, the square of an ideal / (the linear span of all products of the form xt/, x,y G / ) is not necessarily an ideal. Two important functions in nonassociative ring theory are the commutator and associator which, for elements a^b^c in a ring, are defined 5
6
I. ALTERNATIVE RINGS
respectively by (1)
[a, 6] = ab — ba
and (2)
[a,b,c] = (ab)c-
a(bc).
Each of these is a linear function of each of its arguments. For example, if ai, a2, 6 and c are elements of a ring R, we have [ai + a2,6,c] = [ai,6,c]+ [a2,&,c]; furthermore, if R is an algebra over a ring $ and a ^ ^, then [aa, 6, c] = a[a^ 6, c] for any a, 6, c in R. If X ^Y and Z are subsets of a ring, we often find it convenient to denote by XY the set of products xy, x £ X^ y ^ Y and to denote by [X, Y] and [X^Y^Z]^ respectively, the sets of all commutators [x^y] and all associators [x,y,z], X e X,y eY, z e Z, In any ring, there holds the Teichmuller identity: (3) g{w,x,y,z)
=
[wx, t/, z] - [w, xy, z] + [w, X, yz] - w[x, y, z] - [w, x, y]z = 0, which can be readily verified. 1.1 Definition. The nucleus of a ring R is J\f{R) = {a e R \ [a^ x, y] = [x, a, y] = [x, y, a] = 0 for all x, y G R} and the centre of i? is Z{R) = {ae N{R) I [a, x] = 0 for aU x € i?}. Note that in nonassociative algebra, elements of the centre are required also to associate with all pairs of elements. 1.2 Proposition. In any ring R, the nucleus and centre are subrings.
1. FUNDAMENTALS
7
Since the commutator and associator are linear functions of their arguments, both the nucleus and centre are additive subgroups of (i2,+). li w^x € M{R)^ it follows from the Teichmuller identity that [wx^y^z] = 0 for dX[ y^z ^ i?, so M{R) is also closed under multiplication and hence a subring. Finally, let w^x E Z{R), Then w, x and wx are all in Af{R) and hence in the centre since the calculation PROOF.
{wx)y = w{xy) = w{yx) = {wy)x = {yw)x = y{wx) shows that wx commutes with any y ^ R.
D
Just as in associative ring theory, a nonassociative ring R is simple if the only ideals of R are 0 and R itself. It is not hard to see that the centre of a simple ring with 1 is necessarily a field. 1.3 Definition. A ring R is alternative
if it satisfies
[x,x,y] = 0,
the left alternative
[2/,x,x] = 0,
the right alternative
identity,
and identity.
An alternative ring R with 1 is a division ring if every nonzero element of R is invertible; that is, given any x e R^ there exist y and z in R such that xy = I and zx = I. Any associative ring is alternative, but the converse is not true. In Sections 2 and 3 we shall meet some important examples of alternative rings which are not associative. Linearization is a fundamental and extremely useful tool in nonassociative algebra. By this, we mean the replacement of a repeated variable in an identity by the sum of two variables in order to obtain another identity. For example, linearizing the identity [x,x,y] = 0 involves replacing x hy x + z^ thereby obtaining 0=[x + z,x + z,y] = [x,x,y] + [z,x,y]+
[x,z,y]
+[z,z,y],
from which we conclude [z,x,t/]-|-[x,^,y] = 0 because [x^x^y] = [z^z^y] = 0. Thus, in an alternative ring, [x,z,y] = —[z,x,y] is also an identity. Similarly, linearization of [x^y^y] = 0 gives [x,y,z] = —[x,z,y]. In particular.
8
I. ALTERNATIVE RINGS
alternative rings satisfy (4)
[x, y, x] = 0,
since [x,y,x] = —[x^x^y] = 0. If i2 is a ring, a function f:R^-^Ris permutation o" of 1,2,.. .n,
the flexible identity,
said to be alternating if, for any
f(^ $, TUnear} of linear functionals on F . If / : V X F —> $ is a bilinear form on V and V ^ V^ the m a p (^y'. V —^ ^ defined by (^v{x) = f{v^x)
for x G V^ is an
element of V* (because / is linear in its second argument) and t h e m a p (f: V -^ V
defined by (p{v) = (fy is linear (because / is linear in its first
argument). 4 . 4 P r o p o s i t i o n . Let f:VxV—^Fbea a finite dimensional
nondegenerate
vector space V over a field F,
(i) The linear map (f:V—^
bilinear form on
Then
V*, defined above, is an
isomorphism.
(ii) / / U is a subspace of V, then dim V = dim U + dim f/-^. (iii) / / U is a subspace of V such that U nU^ PROOF,
= {0}, then V = U ® U-^.
(i) Let { e i , e 2 , . . . , e ^ } be a basis of V.
For i = 1 , 2 , . . . , n ,
define T^ G V* by
{
1
if i = J
0
otherwise.
The set { T i , . . . ,Tn} is linearly independent since if ^aiTi
— 0, evaluation
at Cj gives aj — 0. It spans V* because if T G K*, then T = ^ c^zT'z, where ai = T(ei),
Hence dim V = dim K*.
Since / is nondegenerate, we have kev(f = V-^ = {0}. Since dim K = dim y * < 00 and (f is one-to-one, it is necessarily surjective. (ii) For v eV,
define V^: V ^ (7* by {i^{v)){u)
= f{v,u).
T h e kernel of
V^ is precisely f/-^ so, by a theorem of elementary linear algebra, dim V = dim [/-'- -h dimim^/?. Since dim [/* = dim ?7, t h e proof will be complete if we establish i m ' 0 := U*. For this, suppose T G f/*. Thus T is a linear m a p U -^ F which can be extended to a linear m a p T': V -^ F^ t h a t is, to an element of V*.
By part (i), T ' = (fi(v) for some 1; G V , so T , being the
26
I. ALTERNATIVE RINGS
restriction of T' to C/, is the restriction of (p{v) to f/, which is ip(v). So ^^ is surjective, as required. (iii) For any subspaces U^ W oi V, we have dim(f/ + W) = dimU + dim Pr - dim{U D W). Since [/ n t/^ = {0}, dim(U + [/^) = dim [/ + dim U^. Thus f/ + C/-^ = F by part (ii).
D
4.5 Corollary. / / / is a nondegenerate bilinear form on a finite dimensional vector space V and if U is any subspace ofV^ then (U-^)^ = U. Clearly U C (U^)-^ and these subspaces have the same dimension since dim U + dim [/-'- = dim V = dim f/-^ + diin{U^)^. D PROOF.
4.6 Corollary. Let f be a nondegenerate bilinear form on a finite dimensional vector space V and let U be a subspace of V, Then the following statements are equivalent. (i) The restriction of f to U is nondegenerate.
(ii) t/nf/^ = {0}. (iii) The restriction of f to U-^ is nondegenerate. If / is nondegenerate on a subspace U and u ^ U f) U-^^ then f{u^v) = 0 for all t; G f/, so ^ = 0 and (i) implies (ii). Conversely, assume (ii), let u e U and suppose f{u^v) — 0 for all v ^ U. By Proposition 4.4, V = U + f/-"-, so /(ii, v) = 0 for all v £ V. Since / is nondegenerate, u = 0. Thus (i) and (ii) are equivalent. Noting that (ii) is also the statement f/-^ r\{U-^)^ = {0}, we see also that (ii) and (iii) are equivalent by the same reasoning which established the equivalence of (i) and (ii). D PROOF.
4.7 Definition. A function q: V -^ ^ is di quadratic form on the 4>-module V if 1. q{av) = a^q{v) for all a € $ and all v ^ V., and 2. the function / : (u^v) \-^ q{u + v) — q{u) — q(v) is bilinear on V. We refer to / as the bilinear form associated with q and caD q nondegenerate if the associated bilinear form is nondegenerate.
4. COMPOSITION ALGEBRAS
27
4 . 8 D e f i n i t i o n . A unital algebra A over a ring $ is quadratic
if each
element of A satisfies an equation of the form (22)
x^ -t{x)x
where t{x)^n(x)
+ n{x)l
= 0
G $ . The elements t{x) and n{x) are known, respectively,
as the trace and norm of x. The trace and norm of an element x ^ $ 1 are clearly unique.
Upon
defining (23)
t{al)
= 2a
and
n ( a l ) = a^
for Q; G ^ , we are assured t h a t t{x) and n ( x ) are uniquely defined for any X e A. Any algebra which arises from the Cayley-Dickson process is quadratic, with t(x) = X +'x and n{x) = x'x. 4 . 9 D e f i n i t i o n . An algebra A with unity over a field 7^ is a
composition
algebra if there is a nondegenerate quadratic form q on A such t h a t q{ab) — q{a)q{b) for all a^b £ A. A form with this multiplicative property is said to permit
composition.
The complex numbers form a composition algebra with respect to the form q defined by q{a + bi) = a'^ + 6^. As we showed in Section 2, the real quaternion algebra and the Cayley numbers are composition algebras. These algebras are division algebras whereas Zorn's vector matrix algebra is a composition algebra with respective to the determinant function (21), which is not a division algebra. Quadratic alternative algebras are almost composition algebras, as the following theorem shows. 4 . 1 0 T h e o r e m . Let A be a quadratic algebra over a field F of tic different from 2. Then the trace t is linear and the norm n is If in addition, PROOF.
A is alternative,
then n{xy) = n(x)n{y)
+ n{ax)l
Multiplying (22) by a^ gives a^x^ - a'^t{x)x + a ^ n ( x ) l = 0, so — n(ax)l
= a^t{x)x
—
quadratic.
for all x^y £ A.
Replacing x by ax in (22) gives a^x'^ — at(ax)x
at{ax)x
characteris-
a^n{x)l.
= 0.
28
I. ALTERNATIVE RINGS
Thus, if 1 and x are linearly independent and a / 0, we get t{ax)
=
at(x)
and n(ax)
= a^n(x).
=
at{x)
and n(ax)
= a^n(x)
Our assumptions in (23) show t h a t t{ax)
if x G F l , so these equations hold for all a G F and
X E A. To prove t h a t t is linear, it remains to show t h a t (24)
t{x + y) = t{x) + t{y)
for all x^y ^ A. This is clearly the case if both x and y are in Fl.
Suppose
X ^ F l . Let a,/3 G F , then, in (22), replace x hy ax + (31 and then x^ by t(x)x
— n(x)l.
a'^{t{x)x
We obtain
- n(x)l)
+ 2af3x + /J^l - t(ax + pi){ax
+ pi) + n{ax + /?1)1 = 0.
Since x is not in F l , the coefficient of x on the left of this equation must be 0; t h a t is, a'^t{x) + 2a(3 - at{ax
+ (31) = 0.
If a ^ 0, we obtain t{ax -\- (31) = at{x) + 2(3, Since this equation is also true if Q; = 0, it holds for all a ^ F. Thus it suffices to estabfish (24) in the case x ^ Fl^ y ^ Fl.
Moreover, since what we have just shown impfies t h a t
(24) holds if, for example, y = ax -\- (3^ we may also assume t h a t , whenever we have an equation of the form a x + /?i/ + 7 I = 0, necessarily a = /3 = 0. Replacing x by a: + 2/ in (22) gives x'^ + xy + yx -\- y'^ - t{x + y){x + y) + n{x + y)l - 0, and replacing x hy x — y \n (22) gives x'^ - xy - yx -\- y'^ - t{x - y){x - y) + n{x - y)l = 0. Adding these equations and replacing x"^ by t{x)x — n(x)l n{y)l
and y'^ by t{y)y —
gives
2{t{x)x
- n{x)l)
+ 2{t{y)y -
- t(x + y)(x + y)-
n{y)l)
t{x - y){x - y) + n{x + y)l + n{x - y)\ - 0.
Using our assumption to set to 0 the coefficients of x and y, we obtain 2t{x) - t{x + y)-t{x
- y) = 0
4. COMPOSITION ALGEBRAS
29
and 2t{y) - t{x + y) + t{x - y) - 0. Addition and division by 2 gives the desired result. To prove that n is quadratic, it remains to prove that the function / defined by / ( x , y) = n{x + y)-
n{x) - n{y)
is bilinear. This follows immediately from {x + y^ — t(x + y){x -\-y) + n{x + y)l = 0 and linearity of /, which shows that (25)
/ ( x , y) = t{x)y + t{y)x - {xy + yx).
Now suppose that A is alternative (and quadratic). We use Artin's Theorem (Corollary 1.10) freely, constantly employing the fact that the subalgebra generated by any two elements of A is associative. Multiplying (25) on the right by xy gives / ( x , y)xy = t{x)yxy + t{y)x'^y - (xyf
- yx'^y
= t{x)yxy + t{y){t{x)x - n{x)l)y - {xyf - y{t{x)x - n{x)l)y = t{x)yxy + t{x)t{y)xy - t{y)n{x)y - {xy^ - t{x)yxy +n{x){t{y)y - n{y)l) = t(x)t(y)xy - (xy)'^ - n(x)n{y)l and hence {xyf
+ / ( x , y)xy - t{x)t(y)xy
+ n{x)n{y)l
= 0.
Comparison with {xyY — t{xy)xy + n{xy)\ = 0 gives n{xy) = n{x)n{y) unless xy E Fl. To establish that n is also multiplicative in this case, we may assume that x (and hence also y) are not in F l , since (23) gives the result for elements of F l . So let xy = a l , a £ F^ with x ^ Fl and y ^ Fl, Multiplying x'^—t{x)x+ n{x)l = 0 on the right by y^ and noting that xy^ — ay and {xy^" = a-^l, we have a^l — at{x)y + n[x)y^ = 0.
30
I. ALTERNATIVE RINGS
If n{x) = 0, then a = 0 too; otherwise, a l — t{x)y = 0, so t(x) ^ 0 and y G Fl, Thus, when n{x) = 0, we have n(xy) = n{al) = 0 = n{x)n{y)^ as desired. In the case n{x) / 0, we have a^ ^ n[x)
at(x) n[x)
9
which, by comparison with y^ — t{y)y + n(y)l = 0, gives n{y) = and hence n{x)n{y) — o? — n ( a l ) = n{xy).
a'^/n(x) O
4.11 Theorem (Hurwitz). Let A be a composition algebra over a field F of characteristic different from 2. Then A is an alternative algebra, specifically, an algebra of one of the four types—F, (F^a), (F, a,/3), (F, a,/3,7) — which arise from the Cayley-Dickson process. In particular, any composition algebra which is not associative is a Cay ley-Dickson algebra. Conversely, any of the algebras F, (F^a), {F^a^/3), (F, a,/?,7) is a composition algebra. Let / be the bilinear form associated with the given quadratic form q on A; that is, PROOF.
/ ( x , y) = q{x + y)~ q{x) - q{y) for x^y E A. Note that / is symmetric. Since f(x^x) — 2q(x) and q is multiplicative, we have f{xy^xy) — 2q{xy^xy) = 2q{x)q{y) = f{x^x)q(y). Linearizing (x ^-^ x + z) gives f{xy, zy) + f{zy, xy) = f{x, z)q{y) + f{z, x)q{y) for all a:, y and z m A and hence (26)
f{xy,zy)
=
f{x,z)q(y)
since / is symmetric and char F 7^ 2. Setting y = 1, we see that g(l) = 1; otherwise f(x^z) — 0 for all x and z in ^ , contradicting nondegeneracy. Linearizing (26) {y y~^ y + w) gives f{xy, zy) + f{xy, zw) + f{xw, zy) + f{xw,
xw)
= / ( x , z)q{y +w) = f{x, z)[q{y) + q{w) + / ( y , w)] and thus (27)
f{xw, zy) + f{xy, zw) = f{x, z)f(y,
w),
4. COMPOSITION ALGEBRAS
31
SO, setting z = x, we get (28)
f{xy,
for all x,y,w
e A.
0 for all u eU}, (29)
xw) = f{y,
w)q(x)
Now let U = Fl
and U-^ = {a e A \ f{a,u)
=
With z = 1 and x e U-^ in (27), we get f{xw,y)
+ f{xy,w)
= 0
for an y,w e A and x G U-^. Since [/ n f/-^ = {0} (for example, / ( 1 , 1 ) = 2 ^ 0), we have A = U ® JJ-^ by Proposition 4.4. Thus any a G A can be written uniquely as a = a l + a' with a' G U-^. Then / ( a y , ti;) = / ( a y , ii;) + f{a'y,
w)
= f{ay,w)-f{a'w,y) = f{otw,y)-
by (29) f{a'w,y)
= /((al -
a)w,y).
Defining (30)
a =
a\-a'
we obtain (31)
fio^y-,^)
= f(G^'w,y)
for all a^y^w ^ A. Similarly, setting w = I and y G U-^ in (27), we get f(xy,z)
+ f(x,zy)
= 0
for all x^z ^ A and y G f/"*" and hence, again writing a = al + a', f{xa, z) - f{oLX, z) + f{xa', = f{x,az)
- f{x,za')
z) = f{x,z{al
Therefore (32) for all a^XyZ^
f(xa,z)
=
f{za,x)
A. So, for any a^y^w £ A^ f{ay,w)
= f{aw,y)
by (31)
= f{yw,a)
by (32)
= /(I/^,^)
by (31).
- a')).
32
I. ALTERNATIVE RINGS
In particular, with a = 1, we get f{y^w) fiWy^w)
— f{ay^w)
= f(ya,w).
= /(VyW) for any y^w ^ A and so
By nondegeneracy, we obtain ay = ya
for any a^y ^ A; thus, a i-^ a is an involution on A, Then aa = aa and (30) show t h a t aa € Fl for all a G ^ . Thus, for a £ A^ 2q{a)l = / ( a , a ) l = / ( l a , a ) l = / ( a a , 1)1
by (32)
= a a / ( l , 1) = 2aa and so (33)
^(^)1 = oM-
Similarly, ^ ( a ) l = aa^ so a commutes with a. Now (31), (28) and the fact that q(x) G F imply t h a t for any x^y^w f{x{xw),y)
= f{xy,xw)
£ A^
= f{y,w)q{x)
= f{y^q{x)w)
=
f{y,{xx)w).
Using again the nondegeneracy of / , we obtain (34)
x{xw)
=
{'xx)w.
Writing ^ = Ic + x — x and noting t h a t 'x + x G Fl^ we have also x{xw)
=
x^w.
Similarly, we can show t h a t {wx)x = wx^. Thus A is an alternative algebra. Now let B be any finite dimensional subalgebra of A which contains U = Fl and on which the restriction of / is nondegenerate (for example, B = U). Note t h a t B is closed under the involution since 6 = (6 + 6 ) — 6 and b + be Fl C B. Suppose B y^ A. By Proposition 4.4, we have A = If f(x^x)
= 0 for all x G B^^ linearization would give f(x^y)
B®B-^.
= 0 for all
x^y £ B^ and hence / ( x , a ) — Q for all x G B^ and all a G ^4, contradicting nondegeneracy of / . t£
Thus / ( x , x ) ^ 0 for some x G 5"*", so there exists
B^ with q{t) r= - a / 0. Now 1 G 5 implies t h a t / ( 1 , £ ) = 0, so £ G C/-^,
Z = - £ and ^2 =. -a
= -q(i)l
= al.
Suppose bi = 0. Then f{bi, ai) = 0 for all a G A, so / ( 6 , a)q(i) = 0 for all a, by (26). Since ^(£) = —a / 0, we get f(b^a)
= 0 for all a and, since
/ is nondegenerate, 6 = 0. Thus 6 i-^ 6^ is a one-to-one surjective linear transformation from B to 5 £ , so Bi and B are isomorphic as vector spaces
4. COMPOSITION ALGEBRAS
over F. Recalling equation (31), t h a t £ G B^
33
and t h a t B is closed under
the involution, we have f{xi,y) for any x^y £ B.
= f{xy,i)
= 0
Thus Bi C B^ so t h a t the sum B + Bi is direct. We
proceed to show t h a t multiplication m B ® Bi is given by (35)
(a + bi){c + di) = (ac + adb) + {da + bc)i
so t h a t B®Bi
is the algebra ( 5 , a) obtained by applying the Cayley-Dickson
process to B. (See Section 3.) Towards this end, we begin by remembering t h a t xlc = q{x)\ for any x, so t h a t {x + y){x -\- y) — q(x + y)l and hence / ( x , y)l = {q{x + y)-
q{x) - q(y))l
= xy + yx
for all X and y. In particular, with x = b ^ B and y — i ^ B^ ^ we see t h a t bl+ib^G,
Therefore
(36)
ib = -bl = bi.
Since, by(34), x{'xy) = {x'x)y — q{x)y for any x , y G ^4, linearization (x \-^ X + z) gives x{'zy) + z{xy) = for any x^y^z
f{x,z)y
^ A. With x^y ^ B and z = £, we get
(37)
-a:(£y) + £(xy) - 0
and so x(yi)
— x(iy)
= i{'xy)
= i{'yx) = {yx)L
Applying the involution to
(37), we obtain {yi)x - {yx)t — 0, so {yi)x = {y'x)(. for all x,y ^ B. Finally, for any x^y ^ B^ {xi)(yi)
= {tx)[ytj
flexible) identities, so [xi)[yi)
— i{'xy)i
by the middle Moufang (and
= 'xyt^ — ayx.
Thus (35) has been verified.
Note t h a t the involution on B ® Bi takes the form a-\-bi Since Bi
= a + ib = a-ib
=
a-bi.
C B^ ^ for any a^b^c^d G 5 , we have f{a^di)
— f{bi^c)
= 0.
Therefore / ( a + bi,c+
di) = f{a, c) + f{bi,
di)
= / ( a , c) + / ( 6 , d)q{i) = / ( a , c) - a / ( 6 , d)
34
I. ALTERNATIVE RINGS
for any a + bi^c + di G 5 © Bi,
Because a 7^ 0, nondegeneracy of / on
B implies nondegeneracy of / on B Q Bi.
IncidentaUy, this shows t h a t if
B Q Bi IS an alternative algebra and 5 is a composition algebra, then so is B ® Bi^ justifying the final statement of our theorem. Now apply the above process to the algebra B = Fl.
If A ^ B^ then A
contains Bi = F ® Ft which, as we have just shown, is the algebra li A ^ B\^ then A contains B2 — B\ ® B\i\
— ( F , a,/?) for some nonzero
(3 e F. li A j^ B2, then A contains 5 3 = ^ 2 ® ^2^2 = (F,a,(3,j) nonzero 7 G F .
{F^a).
for some
Since £ • £^£2 = h^ ' ^2 = ^h • ^2 = -^^1 • ^2 /
^h ' ^2,
the algebra B3 is not associative, so the next step in this process yields an algebra which, by Proposition 3.1, is not alternative, contradicting the fact t h a t A is alternative. It follows t h a t A is the Cayley-Dickson algebra (F,a,/?,7).
•
4 . 1 2 C o r o l l a r y . Over a field of characteristic quadratic alternative PROOF.
algebra is a composition
different from 2, any
simple
algebra.
Let A be an alternative algebra with linear trace t and qua-
dratic norm n.
(See Theorem 4.10.)
Since n is multiplicative, we have
only to prove t h a t the bilinear form / associated with n is nondegenerate; t h a t is, we must prove A-^ = 0.
For this, and because
/(l,l)=:2/0
shows t h a t / is not identically 0, it suffices to show t h a t A-^ is an ideal of A, Proceeding exactly as in the proof of Theorem 4.11, we reach equation (31)—f{ay^w)
= f(aw^y)—which
equation (32)—f{xa^z)
shows t h a t A-^ is a left ideal of A, and
— f(za^x)—which
shows t h a t A-^ is a right ideal of
A.
n In 1958, R. Bott and J. Milnor proved t h a t a finite dimensional real
division algebra must have dimension 1, 2, 4 or 8. The only such algebras which are alternative are listed in the next corollary. 4 . 1 3 C o r o l l a r y . The only finite dimensional over the real numbers PROOF.
are R, C, H(R)
alternative
division
algebras
andC.
Let /I be a finite dimensional division algebra over R. Then
every a ^ A satisfies a polynomial with coefficients in R. Such a polynomial of lowest degree is necessarily irreducible and hence of degree 1 or 2.
4. COMPOSITION ALGEBRAS
35
It follows t h a t A is quadratic. Since A is simple, it is a composition algeb r a by Corollary 4.12 and hence one of the four algebras which arise from the Cayley-Dickson process starting with R. Reviewing the proof of Theorem 4.11, it is evident t h a t the scalar a which defines the algebra B ® Bi must be negative, for otherwise {\-\-{y/a)~^t){\
— {y/a)~^t)
= 0, which can-
not occur in a division algebra. But then q{i) = —a > 0, so, replacing £ by {^yq(J))~^i•,
we may assume t h a t P = —1. Thus the algebras Bi obtained
at the end of the proof are, respectively, R, C, H(R) and C.
D For what integers n is there a formula of the type (38)
{al + al + --- + a^ibl
+ b] + • • • + bl) = c] + 4 + • • • + cl
where the Ci have the form Q = Y^ijk^jk^j^^
^^^ ^he coefficients aijk are
real numbers independent of the ai and 6^? This so-called n-squares
problem
was first considered by A. Hurwitz in 1898. Trivially there is a 1-squares formula, and we have previously noted t h a t there are n-squares formulae for n = 2 , 4 , 8 which can be obtained using the fact t h a t the norm functions on C, H{R) and C are multiplicative. At this point, it is not hard for us to show that there are n-squares formulas only for n— 1 , 2 , 4 , 8 . 4 . 1 4 C o r o l l a r y ( H u r w i t z ) . An n-squares
formula
exists
if and only if
n = 1, 2, 4 or 8. P R O O F . AS
noted, we need prove only necessity. So suppose there exists
an n-squares formula. Let A be the vector space R^ and, for a = ( a i , . . . , a^) and b = {b\^.. .,&n) in A-, define ab = c, where c = ( c | , . . . , Cn) is the n-tuple giving the solution to (38). This product is well defined because the aijk are independent of a and b. For a = ( a i , . . . ^an) G A^ define q[a) = a^ + ' - • + a^. Note t h a t q{ab) = q(ci)q{b) for any a, 6 G ^ and t h a t q{a) = 0 if and only if a = 0. Without loss of generality, we may assume t h a t A has a unity. To see this, choose f ^ A with q{f) = 1. Then q{fa) = q{(if) = q(ci) for all a £ A. Thus the linear transformations L{f) (39)
aL{f)
= fa,
and i 2 ( / ) , defined by aR{f)
= af
for a £ A, are one-to-one and hence bijections of the finite dimensional vector space A. Equations (39) say t h a t q(aL(f))
= q{a) = q{aR{f))
for
36
I. ALTERNATIVE RINGS
all a G ^ , and hence also q{aL(f)~^)
= q(a) = q(aR{f)~^)
for all a. Now
define a new product • in A by setting
a*b =
{aR(fr')(bL{f)-').
Then, for any a G ^ ,
f*a
= [{f)Rifr'][aL{fr']
and, similarly, aicp
= f[aL(fr']
=a
= a. Thus f^ is a unity for the algebra (A, + , ^ ) , which
is therefore a composition algebra since q(ai^b) = q{a)q{b) for all a, 6. Thus, we may now assume t h a t (A, + , •) is a composition algebra with respect to q and hence alternative. If ab — 0, then q{ab) — q{a)q(b) = 0, so q(a) = 0 or q{b) — 0 and hence a = 0 or 6 = 0. Thus A has no nonzero zero divisors, so A is an alternative division ring. The result now follows from Corollary 4.13.
D
For further discussion of the n-squares problem, we refer the reader to "The Four and Eight Square Problem and Division Algebras", by Charles W. Curtis [Cur63]. In Section 3, we defined the notion of split Cayley-Dickson algebra. More generally, a split composition
algebra
is a composition algebra which
is not a division algebra. Split composition algebras can be characterized in several ways. 4 . 1 5 P r o p o s i t i o n . Let A be a composition
algebra with respect to the qua-
dratic form q over a field of characteristic
different from 2. Then the fol-
lowing conditions
are
(i) A has (nonzero)
equivalent, zero
divisors.
(ii) A is split. (iii) There exists a nonzero a ^ A with q{a) — 0. (iv) There exists an idempotent Furthermore,
e 6 ^, e / 0,1.
if A '^ F ® F, then each of the above conditions
is
equivalent
to (v) A has nonzero nilpotent PROOF.
elements.
Certainly (i) implies (ii). Next note t h a t if q{a) ^ 0 for some
a G ^ , then a is invertible, with inverse a ( g ' ( a ) ) ~ \ since q{a) — aa by (33). Thus (ii) implies (iii). Since e^ = e implies e(l — e) = 0, (iv) implies (i). We
4. COMPOSITION ALGEBRAS
37
now show t h a t (iii) implies (iv). To establish (iv), it is sufficient to show t h a t there exists x ^ A with xx = 0, x + x == a l / 0, since then, a~^x
is
the desired idempotent. If no such x exists, then whenever q(x) — 0, so also X + x" = 0. Now assume (iii). Then there exists a ^ A with q{a) = 0 but a / 0. Then q{ax) = 0 for all x, so ax + ox = 0 for all x; thus, ax +^a
= 0
for all X. Replacing x by x, we obtain ax + xa = 0 for all x, but / ( a , x) = q{a + x) - q{a) -
q{x)
= (a + x)(a + x) — aa
xx = ax + xa
so t h a t / ( a , x) = 0 for all x. By nondegeneracy of / , a == 0, a contradiction. Thus the equivalence of the first four conditions has been established. Since (v) always implies (i), to complete the proof, it suffices to show t h a t (iv) implies (v) unless A '^ F® F, Recall that any composition algebra except F ® F \s simple (see Proposition 3.2 and preceding remarks). e /
0, 1 is an idempotent in A and eA{l
— e) = (1 — e)Ae
If
= 0, then
(1 — e)ae — ea[\ — e) — ^ and hence ea — ae for any a ^ A.
(In these
calculations, we use Artin's Theorem freely, thus regularly assuming that any subalgebra generated by two elements is associative.) Thus e is a central idempotent and Ae is a proper ideal, contradicting simplicity. It follows that either eA{l — e) / 0 or (1 ~e)Ae
^ 0. In the former case, there exists a such
t h a t /i = ea( 1 — e) 7^ 0. Notice that // is nilpotent. In the case (1 — e)Ae ^ 0, there exists a ^ A such t h a t (1 — e)ae is a nonzero nilpotent element, so (v)
follows.
n
We now establish a very important and rather striking result. 4 . 1 6 T h e o r e m . Any two split composition over a field of characteristic PROOF.
algebras of the same
different from 2 are
dimension
isomorphic.
Let ^ be a split composition algebra with respect to the qua-
dratic form q over a field F of characteristic different from 2. We again freely use the fact t h a t any subalgebra of A generated by two elements is associative (since A is alternative). By Proposition 4.15, A contains an idempotent e 7^ 0, 1. Note that q(e) = ee = 0 because e is not invertible and t h a t the scalar e + 6 is 1 because (e + e)e = e. It follows easily t h a t the restriction of q to the subalgebra of A generated by 1 and e is nondegenerate. Now let B be any subalgebra of A which contains 1 and e and on which the restriction
38
I. ALTERNATIVE RINGS
of q is nondegenerate. As in the proof of Theorem 4.11, there exists t ^ A such t h a t Bi C B^ and (? = - g ( ^ ) l = a l 7^ 0. Since et = ie by (36), using the middle Moufang identity we get {ety
= {ie){ei)
setting ii = i + a-^{l-
C B-^ and
ij = f + a-\l
a)ei, we have Bii - a)(iei
= f + a-^{l-a)(e
+ + e)f
= (i • €e)i = 0 and,
ef) = a l + a - ^ ( l - a ) a l = 1.
With reference to the final paragraph of the proof of Theorem 4.11, but starting with the subalgebra B generated by 1 and e/\i
B — A, then A =
F e F = ( F , 1). li B ^ A, then A contains B ® Bii
with e\ = 1. This
algebra is ( F , 1,1). If yl is not ( F , 1,1), then yl = ( F , 1,1,1). The theorem follows.
D
4 . 1 7 C o r o l l a r y . Any split generalized quaternion characteristic
different from 2 is isomorphic
Dickson algebra over a field F of characteristic
algebra over a field F of
to M2{F),
Any split
Cayley-
different from 2 is
isomor-
phic to Zorn's vector matrix algebra 3 ( ^ ) PROOF.
In Section 3, we saw t h a t M2{F) and 3 ( ^ ) are split composition
algebras, so the result follows from Theorem 4.16.
D
5. T e n s o r p r o d u c t s In this section, we study an important construction t h a t will be a useful tool throughout this book, the tensor product. This notion can be defined in the very general setting of modules over arbitrary rings; however, since we are interested only in algebras over commutative associative rings with unity (often fields) we shall work within the setting of unital modules over commutative associative rings. 5.1 D e f i n i t i o n . Let $ be a commutative associative ring with unity and let M , A^ and T be ^-modules. A m a p u)\ M x N ^ T \s bilinear if 1. cj(mi + m 2 , n ) = u;(7ni,n) + Ct;(m2,n), 2. u ; ( m , n i + 712) = a ; ( m , n i ) + c j ( m , n 2 ) , 3. a ; ( a m , n ) = Q;a;(m, n) = u{ni^an) for all m , m i , m 2 G M , all n, 711,712 G N and all a G $ .
5. TENSOR PRODUCTS
39
5.2 D e f i n i t i o n . A tensor product of ^-modules M and TV is a ^ - m o d u l e M ®^ N together with a bilinear m a p u: M X N -^ M ®^ N with the property t h a t any other bilinear map from M x A^ to a ^-module T factors uniquely through cj; t h a t is, given any bilinear m a p f:MxN-^T^
there
exists a unique homomorphism of ^-modules (f: M ®^ N ^^ T such t h a t the following diagram commutes.
M XN
^ M ®^ N I
I
I T After such a definition, it is natural to ask whether tensor products actually exist. As we shall see, they do, and they are essentially unique. We begin by giving a construction of one particular tensor product. Given two ^-modules M and A^, we first consider the set
[m,n)^My.N
where all sums here are understood to be
finite.
We can turn S into a
module over $ by defining addition and scalar multiplication by
(m,n)eMxN
(m,n)eMxN Yl
(^(m,n) + y(m,n)){m,
u).
(m,n)6Mx7V and
(m,n)eMxN
(m,n)eMxN
Let H be the submodule of S generated by all elements of the form (mi + 1712,n) - i'rni.n)
- (m2,n),
( m , n i + 712) - ( m , n i ) - ( m , n 2 ) (aTn, n) — a ( m , n )
and
(m^an)
— a{m^n)
40
I. ALTERNATIVE RINGS
where m , m i , m 2 G M , n , n i , n 2 € N and a G # . quotient module S/H.
Let M ®^ N be the
Clearly there exists a natural embedding M X N -^
S. If we denote by m ® n the image of (m^n)
in the quotient S/H^
then it
is clear from the definition of H t h a t (mi + m2) ®n = ( m i ® n) + ( ^ 2 ® n) m ® (ni + ^2) = ( m ® n i ) + ( m ® 712) and ( a m ) ® n = a{m ® n) =^ m®
(an)
for any m , m i , m 2 € M , any 7i,ni,n2 € A^ and any a G ^ . It is also clear t h a t the m a p u: M X N -^ M ®$ N defined by ( m , n) H-» m ® n is bilinear. 5.3 P r o p o s i t i o n . With the notation tensor product
of M and N.
established
Furthermore,
above^ (M 0^ N^uj) is a
if (T^f)
is any other
product of M and N^ then there exists an ^-isomorphism
tensor
(f: M ®4> N -^ T
such that f = (f 0 (jj. P R O O F . TO
prove t h a t ( M ®^ N.uj) is a tensor product of M and A^,
let U be another ^-module and let ^ : M x iV ^ f/ be a bilinear m a p . Since the elements of the form ( m , n ) ^ M X N form a basis of 5 , we can define a module homomorphism ip^: S —^ U hy ^'i
Yl
^im,n){m, n))=
(rn,n)eMxN
^
X(^^^^)g{m, n).
(7n,n)eMxN
Since g is bilinear, it is easy to see t h a t (p^ maps the generators of H to 0, so (f^ induces a module homomorphism (f: M ®^ N ^ ip{m ® n) = g{m^n)
U such t h a t
for all m G M and n E N; thus g = (f o uj.
The
uniqueness of (f follows easily. To prove uniqueness of the tensor product up to isomorphisms, assume that ( T , / ) is another tensor product of M and N,
Since
f:MxN^T
is bilinear, the first part of the proof shows t h a t there exists a module homomorphism (p: M ® N -^ T such t h a t f = (p o uj. On the other hand, since u: M x N ^^ M ®<j) A^ is also bilinear and (T^f)
is a tensor product,
there exists a module homomorphism ip: T -^ M ® ^ N such t h a t uj = tpo f. Notice t h a t u = ip o ip o uj.
Since we also have u = I 0 u {I denoting
the identity homomorphism on M ® N to denote the tensor product of ^-modules M and N.
Sometimes, if there is no obvious reason to
emphasize t h a t the tensor product is taken over t h e ring $ , we drop the subscript and write simply M ® N. T h e universal property of tensor products turns out to be the most useful tool in establishing facts about them. The next examples and propositions will illustrate this very clearly. 5.4 P r o p o s i t i o n . Let M be a module over a (commutative,
associative)
ring $ . Then ^ 0^ M = M. P R O O F . Let g: ^ x M ^
am.
M he the bilinear m a p defined by ( a , m ) H-^
Since g is bilinear, there exists a homomorphism ip: $®
M ^ n ( ^ ) defined hy A ^
f{A,Im)
and the mapping M ^ ( $ ) -> M ^ n ( ^ )
defined by 5 i-^ /(^n? ^ ) are algebra homomorphisms. Since / is bilinear, by Remark 5.8, there exists an algebra homomorphism (^: Mm{^)®^Mn{^) Mmn{^)
-^
such t h a t g — (^ o (jj. To see t h a t (/? is an isomorphism, we shall
show t h a t it maps a $-basis oi Mm{^)®^
Mn{^)
onto a ^-basis oi
Mmni^)-
Denote by {eij} the set of elementary matrices of Mmi^)-) which is a basis for this free module, and, by {e^^}, the set of elementary matrices of
Mn{^).
From the definition of (^^ we see t h a t (f{eij ®^'}^^) is an elementary matrix in ^mn{^)
and that Lp is actually a one-to-one mapping of the set {cij ® e^^}
onto the set of all elementary matrices of
Mrani^)-
We wish to provide more examples which will be useful in later chapters. For this, we need the following lemma. 5.10 L e m m a . Let F C E be a field extension over F.
and let V be a vector space
Then E ®p V is a vector space over E, where the product
scalar A G E and an element (41)
of a
^ ^ Xi ® yi ^ E ®p V is given by
^J2^^^ ® Vz = I^^{>^X^)®y^.
Moreover,
if {vi | i G X} is a basis of V over F, then {\ ® v^ | i G X} is a
basis of E ®p' V over PROOF.
by fxi^^y)
E.
For each element A G £^, we define a m a p f\:
E xV
-^ E®p V
= Ax 0 y, for (a:,y) £ E X V. Since f\ is bilinear, there exists
a homomorphism (f\: E ®f? V —^ E ®/? V such t h a t fx = (fx o cj. a £ E ®F y and X £ E^ define Xa = f\{a).
For
With this scalar product, it is
routine to check t h a t E (g)/? 1/ is a vector space over E. Given an arbitrary element t — ^
Xj®yj
G E®pV^
we can write yj in the form y^ = Y^ikijVi^
5. TENSOR PRODUCTS
45
kij G F. Then
= E j Et(^ij-^j) ® ^z = E t ( E j ^zj2^^)(i ® ^z)Since ^
^2i2:j G £" for every index j , it follows t h a t {1 ® t;^ | i € X} is a
set of generators. The fact t h a t this set is also linearly independent follows from Corollary 5.6.
D
Let F be a field and let a be an element in F , the algebraic closure of F. We denote by F(a) a.
The field F{a)
the smallest subfield of F containing both F and
is an algebraic extension of F; t h a t is, every element
of F{a) is the root of a polynomial with coefficients in F.
Let /o be the
monic polynomial in F[X] of least degree which has a as a root. It is well known that F{a) is isomorphic to the quotient field F[X]/(fo).
If F = C,
the field of complex numbers, then /o = n i - i ( ^ ~ ^i) ^^ ^^e product of r polynomials X — a^ in C[A^], where r = d e g / o and the a^, 1 < i < r^ are distinct elements of C. We refer the reader to Chapter 5 of [ C o h 7 7 ] for more details. 5.11 E x a m p l e . Let F be a field, let /o G F[X] and let F be a field containing F . Then
as algebras. In particular, if F is a subfield of C and F{a) is an algebraic extension of F , then C ® F F{a) is isomorphic to the direct sum of r copies of C, where r is the degree of a polynomial of least degree which has a as a root. To prove the first part of this statement, consider the map
'•"''
(/o)
(/o)
given by (A, / + (/o)) i-^ A / + (/o)- This map is clearly bilinear, so there exists a hnear m a p ^ F[X] E[X] cp: F ® ——- -> ——-
46
I. ALTERNATIVE RINGS
such t h a t g — (^ o uj. Denoting the degree of /o by r, the set {X^ + (/o) | 0 < i < r — l } i s a basis of F [ X ] / ( / o ) over F , so, according to L e m m a 5.10, {1 ® {X' + (/o)) I 0 < i < r - 1} is a basis of E t h a t if [I® {X' + (/o)) = X' + (/o) G map of a basis of E
®F F [ X ] / ( / O ) . NOW
JE;[X]/(/O);
thus (/? is a one-to-one
onto a basis of E[X]l{fo)
®F F [ X ] / ( / O )
notice
and hence a
bijection. Since it is easy to see t h a t (p is also an algebra homomorphism, the result follows. For the second p a r t , suppose F C C, t h a t a is algebraic over F and that /o is a polynomial of least degree over F which has a as a root. Then /o = n i = i ( ^ ~ ^i) ^^^ ^^^ ^i-) 1 ^ ^ ^ ^? ^re distinct elements of C. Hence, by the first part and the Chinese Remainder Theorem (see [ C R 8 8 ] ) , we have
/;[x]^c[x]^ ® (/o)
"
(/o) ~
^
C[X]
qx] {{X-a,){X-a2)'"{X-ar)) C[X]
- (^-ai)
C[X]
(^-c.2)
"^
{X-ar)
= c®c® •••ec. ^
V
'
r summands
5.12 E x a m p l e . For any commutative ring /Z, we find it convenient to denote by H(/?) the algebra R+ Ri + Rj + Rk^ where P — p — k^ — ijk
=
— 1. Let F be a field and let A' and E be field extensions of F. Then (42)
E®FWO
=
H(F®FA')
as algebras. In particular, \i E — C and K = E{a) is an algebraic extension of F of degree r, then C®FH(A')^H(C)ffi..-®H(C). ^
V
'
r summands
To prove (42), we define g: E X H(A') -> H{E ®F K) by g{\,a) ( A ® a i ) + (A®a2)i + (A®a3)j-|-(A(8)a4)fc, for A G £" and a = a\ +a2i-\-a^j
= +
a^k G H(A'). This map is bilinear, so there exists an algebra homomorphism (y9: E ®F H(A') -^ H(E ®F A ) such t h a t g = (f o LJ. If {vi \ i ^ 1} is a basis of A' over F and if A' = { l , 2 , j . A:}, then the set B = {viX | i G I , x G A'} is a basis of H(A^) over A"; thus I ® B = {1 ® ViX | i G I , x G A'} is a basis of E ®F H ( A ) over E. In a similar way, it is easy to see t h a t the set
5. TENSOR PRODUCTS
B' = {{l®Vi)x
I i € I , a: G X} is a basis of
47
over E. Finally, since
H{E®FK)
(/? is a one-to-one m a p oi I® B onto B\ it follows t h a t (f is an isomorphism. The last statement follows from (42) and Example 5.11 because C ®F H{K)
^ H{C ® A^) ^ H(C e . • • ® C) ^ H(C) 0 • • • 0 H(C). sy
^
'
^
V
r summands
'
r summands
As we have seen in Proposition 5.7, the tensor product of two algebras is again an algebra. It is easy to check t h a t if the given algebras are associative, then their tensor product is also associative. In general, the tensor product of alternative algebras is not alternative. (This can be shown as a consequence of Corollary IV. 1.4 and Lemma VII.0.1.) We do, however, note the following. 5.13 P r o p o s i t i o n . Let A be an alternative pose B is a commutative A ®7? B is
associative
algebra over a field F and sup-
algebra over F. Then the tensor
product
alternative.
P R O O F . Let x = ^ a^ ® bj and y = J2^i
® ^j ^ ^ elements of A ® F B.
To verify the left alternative identity, we must show t h a t x'^y — x{xy) Now x^y — x{xy)
= 0.
is a linear combination of terms of the form
{(a, ® bj){ak ® 6 ^ ) } « ® &!) + {{a^ ® 6^)(a, ® 6 ^ ) } « ® 6^) - (a^• ® bj){{ak ® 6^)(a; ® t^)} - (a/, ® 6^){(a, ® bj){a^ ® t ' J } which can be rewritten as {a^ak)a'^ ® {bjbi)^^ -f {akai)a'^ ® ( M j ) ' ^ ! - a,{aka[) ® bj{b(^b'^) - ak{a^a[) ® b^{bjb'^). Since B is associative and since, for fixed b ^ B^ the map a \-^ a®b\s
linear,
this expression can in turn be rewritten in terms of associators, [a,-, a/,, a^] ® bjb^b^^ + [a/,, a,-, a^] ® b^bjb'^ and then, since B is commutative, also as {[a,, a/,, a^] + [ak, a^-, a'^]] ® bjbf>b'^, which is 0 because the associator is an alternating function on A. establishes the left alternative identity in A ®F B.
This
Similarly, the right
alternative identity holds as well, so A ®F B is alternative, as claimed.
D
Chapter II
A n Introduction to Loop Theory and to Moufang Loops
1. What is a loop? The theory of loops has its origins in geometry, combinatorics and nonassociative algebra. In geometry, the coordinatization of a projective plane leads to various loop structures on the set of labels from which coordinates are chosen. Any Latin square with first row and column in standard position is the multiplication table of a loop. In rings with 1 for which there is a well defined notion of inverse, it is often the case that subsets closed under products and inverses are loops. R. H. Bruck's "Survey of Binary Systems" [Bru58] and G. Pickert's "Projektive Ebenen" [Pic75] were the first comprehensive treatises on loops and, for many years, these books were the standard references on the subject. More recently, H. 0 . Pflugfelder wrote an introduction to loop theory intended to be useful as a textbook for a graduate level course [Pfl90]. This book was soon accompanied by what was first intended to be a sequel, "Quasigroups and Loops, Theory and Applications" [CPe90], but which became, in fact, an extensive collection of papers amply illustrating the scope and diversity of the field today. We recommend all these books to the reader. The brief introduction to loop theory and, in particular, to the theory of Moufang loops which we present in this chapter, is based primarily upon the books by Bruck and Pflugfelder cited above. 1.1 Definition. A quasigroup is a pair (L,-) where L is a nonempty set and (a, 6) — I > a • 6 is a closed binary operation on L with the property that 49
50
II. LOOP THEORY AND MOUFANG LOOPS
the equation a -b = c determines a unique element b £ L for given a^c ^ L^ and a unique element a ^ L for given b^c £ L. We call a • b the product of a and 6. A /oop is a quasigroup with a two-sided identity element 1. Both cancellation laws hold in a quasigroup (L^-). For example, the equation a\'b — a2'b implies ai = a2 by uniqueness of the element x in the equation x • b = c. In any quasigroup (Z/, •), a critical role is played by the right and left translation maps R(x) and L{x)^ x G L, which are defined by (1)
R{x):a\-^a'X
and
L(x):a\-^X'a
for a ^ L, For example, the solution to a-x = 6, guaranteed by the definition of quasigroup, is x = bL{a)~^ ^ while the solution to y-a = b\s y = bR[a)~^, (In loop theory, it is common to write maps to the right of their arguments and we shall adhere to this practice in this work whenever translation maps are involved.) The definition of quasigroup is equivalent to the assertion that R(x) and L{x) are bijections of L, for any x ^ L. The right translation R{x)^ for instance, is one-to-one since a\R{x) = a2R{x) implies a\ - x = a2 • x so that ai = a2, by cancellation. The map R{x) is onto because, if 6 G //, the definition of quasigroup assures the existence of a G // such that a - x = b. Thus, the set of left and right translation maps generates a group M(L), called the multiplication group of L, which is a subgroup of the symmetric group on the set L; thus, MiL)
= {L(x),Rix)\xe
L).
If (L, •) is a loop, the equation a - x — \ has a unique solution a^ called the right inverse of a, and the equation x - a = I has a unique solution a^ called the left inverse of a. In general, the left and right inverses of an element are different. When they are equal, however, we use the notation a~^ to denote the unique (two-sided) inverse of a. This is the case, for example, in any associative loop, which is just a group. As with rings, so with quasigroups and loops, we shall often omit reference to the binary operation, referring simply to "the quasigroup L" or to "the loop L". Similarly, we routinely write ab rather than a • b to denote the product of elements a and b.
1. WHAT IS A LOOP?
51
A subloop of a loop (i/, •) is a subset of L which, under the inherited binary operation, is also a loop. Since, for h ^ H^ the equation hx = h has a unique solution in L, it follows t h a t the identity elements of H and of L are necessarily the same. 1.2 P r o p o s i t i o n . Let L be a loop and let H be a nonempty The following
are
subset of L.
equivalent,
1. H is a subloop of L, 2. If x^y G HJ then xy^ xR(y)~^ 3. If x^y^z
£ L, xy = z and two of x^y^z
these elements PROOF.
and xL(y)~^
are all in H.
are in H, then the third of
is in H too.
If Zf is a subloop of L and x,y
£ H^ then xy e H because H
is closed under the binary operation of L. T h e element z = xR(y)~^ satisfies zR{y)
G L
= x. Thus zy = x. Since ^ is a loop and ay = x has a unique
solution a both in H and in L, necessarily z ^ H. Similarly, xL(y)~^
G H.
Thus (1) implies (2). Now assume (2) and consider the equation xy = z. Because of (2), if x^y ^ H ^ then so is z; if a;,z G H ^ then so is y = zL{x)~^; so is X = zR(y)~^.
if y, z G H ^ then
This gives (3).
Finally assume (3) and let x^y e H. There exists a unique z = xy £ L, By (3), z ^ H. Similarly, there exists a unique z £ L such t h a t xz — y and, by (3), z £ H. Also, there exists a unique z £ L such t h a t zx = y and, by (3), z £ H.
All this shows t h a t ^ is a quasigroup. For x G H^ we have
x l = X, so 1 G ^ by (3); hence, /^ is a loop.
1.3
D
C o s e t d e c o m p o s i t i o n s . In general, Lagrange's Theorem does not
hold for loops. (For example, it is easy to construct a loop of order five in which every element has order two.) On the other hand, we can often still be assured t h a t the order of a finite loop is divisible by the order of certain subloops. Let ^ be a subloop of a finite loop L. Since the right and left cancellation laws hold in L, it is certainly the case t h a t \xH\ = \IIx\
= \H\ for every
X G I/. T h u s , if L is the disjoint union of cosets of IT, it is clear t h a t the order of L will be divisible by | ^ | . A necessary and sufficient condition for
52
II. LOOP THEORY AND M O U F A N G LOOPS
L to be the disjoint union of right cosets is (2)
Hx n Hy 7^ 0, a:, ^ G i/ if and only if Hx = Hy,
and this condition is equivalent to (3)
H{hx)
= Hx
for all x G L and all he
H.
To see the equivalence of (2) and (3), first assume (3) and suppose t h a t hix = h2y e Hx n Hy.
Then Hx = H{hix)
= H(h2y)
= Hy.
hand, assuming (2), for any x E L and any h e H ^we have hx 6 so H{hx)
=
On other H(hx)r\Hx,
Hx.
Summarizing, when condition (3) holds, there exists a subset T of L, called a transversal
of ^ in L, such t h a t
L = [j
Hx
with HxDHy
= (/} for x,y
eT
and, when this is the case, \L\ is divisible by \H\. A loop L is diassociative
if, for any x, t/ G L^ the subloop (x, y) generated
by X and y is associative (and hence a group). If L is diassociative, a e L and H = (a) is the cyclic group generated by a, then condition (3) clearly holds, so, if L is finite, \L\ is divisible by \H\. Thus the order of any element in a finite diassociative loop L divides the order of L. T h e concepts of commutator, associator, nucleus and centre, previously defined for rings, have natural analogues in loop theory. 1.4 D e f i n i t i o n s . Let a, b and c be three elements of a loop L. T h e (loop) commutator
of a and b is the unique element (a, 6) of L which satisfies ab = (6a)(a,6)
and the (loop) associator
of a, b and c is the unique element (a, 6,c) of L
which satisfies (ab)c = {a(6c)}(a,6, c). If X , y and Z are subsets of a loop L, we denote by (X^Y)
and ( X , y , Z ) ,
respectively, the set of all commutators of the form (x, y) and all associators of the form (x^ y^ z)^ where x e X, y EY,
z e Z.
1. WHAT IS A LOOP?
The commutator-associator
53
subloop of a loop L is the subloop V gen-
erated by the set of all commutators and associators. Thus
L' =
{{L,L,LUL,L)).
The left nucleus of L is the set Mx{L) = {a e L\ {a,x,y)
= 1, for all a:,y G L},
the right nucleus is the set Afp(L) = {a ^ L \ (x^y^a)
= 1^ for all x, y 6 L}^
the middle nucleus is Af^{L) = {a e L \ (x^a^y)
= 1, for all a:,y G L}
and the nucleus of L is A/'(L) =
Mx{L)r\Mp{L)r\N^{L).
The centre of L is Z ( L ) = {x G A/'(L) I (a, x) = 1 for all a G I } . As with rings, in expressing the nonassociative product of more t h a n two elements in a loop, we often employ dots instead of, or in addition to, parentheses to indicate the order of multiplication, juxtaposition always taking precedence over dot. 1.5 P r o p o s i t i o n . Each nucleus hence a group. PROOF.
of a loop is an associative
The centre of a loop is an abelian
subloop
and
group.
Let L be a loop and consider first the middle nucleus A^^ of L.
Clearly 1 G N^ and, if x, y G N^ and a^b ^ L, then (a • xy)b = (ax • y)b = (ax){yb)
since y e N^
= a{x ' yb)
since x G N^
= a{xy ' b)
since y G N^
which shows t h a t xy G N,j,. x^l
since x G N^
Let x G N^.
— Ix^ implies t h a t x^x = 1 = x^x.
unique two-sided inverse, x~^.
Then
{X^X)XP
= x^{xx^)
=
Thus x^ = x^^ so t h a t x has a
For a £ L^ {ax)x~^
= a(xx~^)
= a implies
54
II. LOOP THEORY AND M O U F A N G LOOPS
R{x)R{x~^) = / , the identity map on L, and so R{x)~^ = R{x~^), Similarly, x~^{xa) — a implies that L{x)~^ — L{x~^). Now let x ^ N^ and a,6 G L. By the definition of loop, we can write a = tx for some t £ L, Then (ax"^)6 = (tx ' x-^)b =: tR{x)R{x)~^
- b = tb
and a{x~^b) = {tx){x~'^b) = t{x • x"^6) == t • 6L(x~^)L(x) = tb; thus {ax~^)b = a(x~^6), so that x~^ £ N^. Since we have already shown that N^ is closed under multiplication, we now have that, for any x^y £ 7V^, the elements xR{y)~^ = xR{y~^) = xy~^ and xL{y)~^ = xL{y~^) = y~^x are both in L, so, by Proposition 1.2, N^ is a subloop of L, It is obvious that N^ is associative. Turning our attention to the left nucleus, we note first that 1 G A^A sind then, that if x, y G Nx^ also xy G A^A because {xy • a)b — {x • ya)b
since x G A^A
= x{ya ' b)
since x £ N\
= x(y • ab)
since y G A^A
= {xy){ab)
since x G A'A-
Let y G A^A- Then {yy^)y — y{y^y) = yl = ly shows that yy^ = 1 and hence that y^ = y^. As before, we see that y has a unique two-sided inverse y~^. Since y G A'A, for any a e L^ we have y(y~^a) = a, so L{y)~^ — L{y~^), Let a^b £ L. Then y{y~^ ' ab) = {yy~^){ab) = ab and t/(t/"^a . 6) = (y • y"^a)6 = aL{y~^)L{y) 'b = ab. Thus y{y~^ • a6) = y{y~^a • 6) and, after cancelling y, we obtain ^/"^ • ctb = y~^a ' 6, so that y"^ G A'^A- Since A^'A is closed under multiplication, it now follows that, if x,y G A^A, then xL{y)~^ - xL{y~^) = y~^x G A^A- The computation xR{y)~^ • y = xR{y)~^R(y) — x = x(y~^y) — {xy~^)y shows, after cancelling y^ that xR{y)~^ = xy~^ G A^A ^is well. Now Proposition 1.2 says that A^A is a subloop of L. Clearly this subloop is associative.
1. WHAT IS A LOOP?
55
W i t h any loop (i/,-)^ there is an associated opposite loop fined by i/^PP = L and x ir y = y - x for x^y Since the left nucleus of
J\/A(Z/°PP).
L^PP
G L,
(L^PP,*)
de-
Evidently, Afp{L)
=
is associative, as we have just seen,
the right nucleus of L is also associative. As the intersection of associative subloops of L, the nucleus of L is also an associative subloop of L, subset of L, To see t h a t Z{L)
Clearly Z{L)
is a commutative, associative
is a subloop, let x,t/ G Z{L),
Af(L)^ we have already established t h a t xy^ xR(y)~^
Since x^y E
and xL(y)~^
are also in
Af{L) and, moreover, t h a t y has a two-sided inverse y~^ such t h a t R(y)~^ R(y~^)
and L(y)~^
= L{y~^).
=
The usual group theory argument now works
in the present context to show t h a t ay~^ = y~^a for all a G i>, hence t h a t y~^ G 2(L).
Similarly, it is routine to verify t h a t xy G 2{L),
follows immediately t h a t xR(y)~^
= xy~^ and xL(y)~^
from which it
= y~^x are also in
a
z{L),
While as noted in §1.3, Lagrange's Theorem fails for loops, as we have already observed in §1.3, it is interesting to observe t h a t the order of any nucleus of L will divide |L|, since condition (3) clearly holds for H = JVA, AT^ or A^, and a condition corresponding to (3) for left cosets holds for A/"p. More generally, the order of any subloop of any nucleus of L divides |L|. The definition of "normality" in loop theory is a generalization of the definition for groups. 1.6 D e f i n i t i o n . A subloop H oi di loop L is normal^
and we write H 0(6), then there exists Some 61 G G such that G = (a) x (61). This is a consequence of the theory of finite abelian groups. 4.4 Lemma. Let L be a commutative hamiltonian Moufang loop. Then L is associative and hence an abelian group. We show first that x^ is in the nucleus of L for every x ^ L. Since L is commutative, for any x £ L^ the inner map T(x) is the identity. PROOF.
4. HAMILTONIAN LOOPS
73
By Theorem 3.3, T{x) is a pseudo-automorphism with companion x~^.
It
follows t h a t a:~^ is in the nucleus of L; hence, so is x^. Now suppose a^b^c £ L and the associator ( a , 6 , c ) ^ 1. We claim a, b and c have finite order. For this, write {ab)c = (a • bc)s with s = (a, 6, c) = a^ = 6^ = c^ for some nonzero integers i^j^ky by Lemma 4.2. Let m = ijA;. Then (23)
(ab-c)"^
=
{a'bc)'^s'^.
By commutativity and diassociativity {ab • c ) ^ = (ab)'^c'^ = {a'^b'^)c'^
= s^^s'^s'^
= ^ij+jk-\-ik
and, similarly, (a-bc)'^ = ^«i+jA:+i/: From (23), we obtain s'^ = I and hence ^tm _ f^jm _ ^/:m _ j Xhus a, 6 and c have finite order, as claimed. Write o(a) = 3^m, where 3 J^ m, and choose integers k and ^ such t h a t fc3^ + £m = 1. Then a = a^^"^ a^"^ with the first factor in the nucleus of L (and hence in the centre). It follows t h a t {a^b^c) — (a^'^^b^c)
and so,
replacing a by a^^, if necessary, we may assume t h a t the order of a is a power of 3. Similarly, we assume that the order of 6 is a power of 3. In view of the remark preceding this lemma, we can write (a,6) = (a) X (6i) for some bi. In particular, for any integers i and j , we have (a^) 0 (6^) — {1} so, by Lemma 4.2, {a\b'[yc)
— 1. Now b — d^b\ for some integers u and v.
By diassociativity, ab — a^+^ft^, so {ab)c = (a^+^6y)c = a''^\b\c)
= a^a"" • 6^c) = a{a''b\ • c) = a(6c),
which contradicts ( a , 6 , c ) 7^ 1. The result follows. 4.5 L e m m a . Let L be a hamiltonian elements
of L which do not commute.
the subloop which they generate contains PROOF.
Moufang
D loop and let a and b be
Then a and b have finite order and Qg-
By Lemma 4.2, there exist integers u and v such t h a t c =
(a, 6) = a~^b''^ab
= a^ = b^. In particular, c commutes with a and with b.
By Lemma 4.3, 1 = (c,6) = (d'^-fb) — (a^b)'^. Thus c, and hence also a and 6, have finite order. Consider the set of all pairs of elements x^y ^ (o^,&) which do not commute and choose a pair for which o(x) + o{y) is minimal.
74
11. LOOP THEORY AND MOUFANG LOOPS
As above, the commutator (x, y) commutes with x and with y. Let n = o{x) and m = o{y). Suppose p is a prime dividing n and p ^ n. Then x^ and y commute by minimality, so 1 = {x^',y) = (^^vY showing that (x^y) has order p. It follows that p is the only prime divisor of n and, similarly, that p is the only prime divisor of m. Let n = p'^^ and m — p^^ and assume, without loss of generality, that n\ >_ m\. As before, (x,2/) = x^ for some i. Since p = o{x^) = p'^^/gcd(p'^^^i)^ we have £ = jp'^^~^ for some j ^ 0 (mod p). So {x,y) = x^^""^ and, similarly,
for some k ^ 0 (mod p). Let t = —jp"'i-^i and yi = x^y^. X and yi do not commute since gcd(k,p'^) T/[
Note t h a t
= 1 implies (y) = (t/i). Thus
7^ 1, by minimality. By Lemma 4.3,
= x^^"'^~\^^^^~\y^,x^y^^~^^^'^'~^~^^/'^ = x~^^^' ~' y^^"^~^
(t/, x)^^^^'~'
(^"^"^"^)/^
= (x,y)-'(a:,y)(x,y)'=^^"'-(^'"'-"-^)/^ and thus (x,y)^'^^"^ (^"^^ that mi > 1 and, since ni j are both odd, and ni = and t/ have order 4, (a;, y)
-i)/2 ^ i whereas (x,y) has order p. It follows > m j , that ni > 1 too. Therefore p = 2, so A; and 2. Since ni > mi > 1, also mi = 2. Therefore x = x'^ = y'^ has order 2, and (x, t/) = QgD
4.6 Corollary. Let L be a hamiltonian Moufang loop which is not commutative. Then every element of L has finite order. By Lemma 4.5, there exist a,b e L such that a"^ = 1, a'^ = b'^ = (a, 6) and (a, 6) = Qs- Let PROOF.
C{L) = {g £ L \ gx = xg for all x ^ L}. If c ^ C(Z/), then c has finite order by Lemma 4.5. Suppose c G C{L) and write (24)
(ab)c = (a • bc)s
4. HAMILTONIAN LOOPS
75
for some s £ {a)n {b) fl (c), by Lemma 4.2; so s = I or s = a^. In the latter case, c has finite order because 5 G (c), so assume s = 1. Squaring each side of (24) gives (25)
( a 6 ) V = (a • bcf.
If a and 6c do not commute, then be has finite order, hence so does c (because 6 and c commute).
If a and be do commute, then (ab^e'^
= a^{be)'^ =
oP'ilP'e^) and so, since o? — {ah)^ in Q^^ b^i? = (? and b^ — 1, contradicting the fact t h a t b has order 4.
D
4 . 7 L e m m a . Le^ L be a hamiltonian tative.
Moufang
loop which is not
eommu-
Then L = LQ x A, where LQ is a 2-loop and A is an abelian group
all of whose elements PROOF.
The set
have odd order in the centre of L. LQ
of aU elements of 2-power order in L is a subloop
by Lemma 4.3. Let A be the set of elements of odd order in L. If a i , a 2 G A and ( a i , a 2 ) ^ 1, then the group G — ( a i , a 2 ) contains an element of order 2, by Lemma 4.5. This cannot be the case, however, since the fact t h a t ( a i , a2) is a power of a\ makes it is easy to see t h a t the elements of G have the form d^a^-, which has odd order by Lemma 4.3. Thus the elements of A commute, hence A is commutative and an abelian group, by Lemma 4.4. Since every element of L has finite order (Corollary 4.6), we have L = LQA.
Any element of LQ commutes with any element of A, by Lemma 4.2,
and, for the same reason, if x,y^z {x^y^z}
are three elements of L with one of
in LQ and another of { x , y , z } in A, then also (x^y^z)
= 1. Let
ci,C2 G Z/o, cti, a2 G i4 and observe t h a t (ciai)(c2a2) = {ciai • C2)a2 = (ci • aiC2)a2 = (ci • C2ai)a2 = (ciC2 • ai)a2 =
(c\C2)(aia2).
It follows t h a t L = LQ X A and A is central.
D
4.8 T h e o r e m ( N o r t o n [Nor52]). Let L be a Moufang hamiltonian
loop.
Then L is
if and only if
1. L is an abelian group, or 2. L — Qs X E X A, where Qs is the quaternion is a (possibly trivial) elementary
group of order 8, E
abelian 2'group and A is a (possibly
76
II. LOOP THEORY AND MOUFANG LOOPS
trivial) abelian group all of whose elements
consist of finite odd order,
or 3. L = MieiQs)
X E X A, where Mie(Qs)
is the Cayley loop and E and
A are as in 2. PROOF.
First we show t h a t the classes of loops identified in the state-
ment are hamiltonian.
This is obvious for an abelian group.
L = LQ X E X A where LQ = Mie{Qs)^
Suppose
E is an elementary abelian 2-group
and A is an abelian group all of whose elements have odd order. Note t h a t LQ has a unique square (different from the identity). Call this element s. If ^ is a subloop of L and H is central, then H is normal. On the other hand, let h ^ H be an element which is not central. Write h = qea with q e Lo^ e G E^ a G A and note t h a t q must have order 4 (otherwise, it is central). Thus q^ = s^ h"^ = sa'^ and, with n = o ( a ) , /i^^ = s'^a'^'^ = s since n is odd and ^^ = 1. Thus s e H, Now let h e H and x G L. Then xh = hx or xh = (hx)s
since s is the only c o m m u t a t o r in L other than the identity.
Since s G H and s is central, {hx)s = {hs)x G Hx. It follows t h a t xH = Let h £ H and a:,t/ G L. Then {hx)y
= h(xy)
or {hx)y
— [h • xy^s
Hx. since
s is also the only associator in L other than the identity. It follows t h a t {Hx)y
— H{xy)
and, similarly, t h a t x{yH)
= {xy)H^ so H is normal. Thus
L is hamiltonian. A similar argument shows that any group as described in statement 2 is also hamiltonian. The main import of this theorem is necessity, to which we now turn our attention. If L is commutative, then L is an abelian group by Lemma 4.4 so, for the rest of the proof, we assume t h a t L is not commutative. Moreover, in view of Lemma 4.7, we also assume t h a t L is a 2-loop. Suppose first t h a t L is a group. By Lemma 4.5, there exist a, 6 G Z/ such t h a t a"^ = 1, a-^ = 6"^ = (a, 6), so t h a t Q = {a,b) is a group isomorphic to the quaternion group of order 8. Let C(Q) = {c e L \ gc = eg for all g G Q}. U g G L does not commute with a, then g'^ag
— a ~ \ since g~^ag G (a) has
order 4. Since b~^ab — a ~ \ we see t h a t gh~^ commutes with a. Similarly, \i g G L does not commute with 6, then ga~^ commutes with 6. We claim
4. HAMILTONIAN LOOPS
77
that L = C{Q)Q and consider four possibilities, li g ^ L commutes with both a and 6, then g G C{Q) C C{Q)Q. If g commutes with b but not with a, then gb~^ commutes with both a and 6, so ^^6"^ G C{Q), hence g = (gb~^)b G C{Q)Q. Similarly, if p commutes with a but not with 6, ^ G C(Q)Q. Finally, if ^f commutes with neither a nor 6, then ^6~^ commutes with a but not with 6, so gb~^a~^ commutes with both a and 6. Therefore, g{ab)-^ G C(g), so ^ G C{Q){ab) C C ( g ) g . Thus, indeed, L = C(Q)Q. Let c G C((5) have order 4. Then be and a do not commute and (bc)"^ = 1. If be has order 4, then a"^(6c)a is an element of order 4 in (6c), which is cyclic of order 4. Thus a~^(6c)a = (6c)~^, an equation which also holds if be has order 2. In all cases, then, a~^(be)a = {be)~^ = e~^b~^ = b~^e~^. On the other hand, a~^(be)a = 6~^c, so c = e~^ has order 2, a contradiction. Thus C{Q)^ containing no elements of order 4, is an elementary abelian 2group. Clearly C{Q) is central. Since {a'^) is the only subgroup of Q which does not contain an element of order 4, we also haveC(Q)nQ = {l^a^}. Let £ be a maximal subgroup of C{Q) not containing a^. Thus E H Q = {1}. If X G C{Q) has order 2 and x ^ E^ then a^ G {E^x) implies a'^ = 6X, for some e G ^ , so x = a^e G Q ^ . Thus L = Q X E. Assume L is not associative. Let a and b be any two elements of L which do not commute and let G be the group generated by a and 6. As just shown, we must have G = Q8>^ E for some elementary abelian 2-group E. Thus G/G' is an elementary abelian 2-group which can be generated by two elements, so \G/G^\ < 4; therefore, G = Qs- We have learned that any two elements a and 6 of L which do not commute must generate a group isomorphic to Qg] in particular they have order 4 and a? — b'^ — (a^b). Now fix a, 6 G Z/ such that Q = {a,b) is isomorphic to Qg- Let C{L) = {e ^ L \ ex = xe for all x G L}. We claim that (26)
C{L) = {eeL\e^
= l},
Suppose c G L, c^ = 1 and c / 1. Then, for any x G L, we have xe — e\x with ci G (c) = {l,c} by Lemma 4.2, so ci = c. Thus e G C(//). Conversely,
78
II. LOOP THEORY AND MOUFANG LOOPS
suppose c G C{L). Write {ab)c = (a • bc)s^ where 5 G (a) fl (6) D (c). Hence s = 1 OT s = a^. Since a^ = (aby G Q and L is diassociative, a^c^ = {abfc^
= {ab • cf
= (a • 60)^52 = (a • 6 c ) ^
If a and 6c commute, then a^c^ = 0^(60)-^, so c^^ = i^^)^ = ^^^^ implying b^ = 1^ which is not true. Thus a and be do not commute, so they generate a group isomorphic to Qg- So a^ = (bc)'^ = b^c^ = o?c^ and c^ = 1. This verifies (26). Let C{Q) = {c e L \ ex - xc for all x e Q} and note that the preceding argument in fact showed t h a t , if c G C(Q)^ then e^ = I. Thus
C{Q) = CiL). Also, we see t h a t a^ = b^ is the only nonidentity square in L, for if c G C((5), then c^ — \^ while if c does not commute with some q £ Q^ then c^ = q^ = c?, From this fact, and because L is a 2-loop, it follows, in particular, t h a t any x G //, x 7^ 1, must have order 2 or 4. We claim t h a t C(L) = 2 ( L ) , the centre of L, For this, we have only to show C(Z/) C M{L)^ the nucleus of L, because 2 ( L ) = M{L) n C(L). Let c G C(L), c / x , y G //, (x^)c = x(yei)^
ci G (c) = { l , c } since c'^ = 1. Since c /
must have Cj = c, so (x^y^e) L is Moufang, e G M{L).
1. Then, for any 1, we
= 1. So c is in the right nucleus and, because
(See Proposition 2.4 and Theorem 3.1.)
Suppose x^y £ L and xi/ = yx.
We claim (x^y^z)
= 1 for all z £ L.
Since C(//) = Z{L)^ this is clear if either a:-^ = 1 or y^ = 1, by (26). Suppose x^ y^ I and i/-^ /
1. Then x ^ C{L)^ y ^ C{L)^ so x and y have order 4,
x^ = ?/^ and y = yx"* = (yx^)^ with (yx^)^ = y^x^ = 1. Thus y = ex with c = yx'^ G 2{L). easily t h a t (x^y^z)
By diassociativity and the definition of centre, it follows = 1 for all z.
Since a^ is the only nonidentity square in L and since x~^y~^xy
=
x~^(a:y~^)'^?/^, it follows t h a t a^ is the only nonidentity commutator in L. Suppose x , y , z G // do not associate.
As we have just shown, x and y
4. HAMILTONIAN LOOPS
79
cannot commute, so x and y each have order 4 and x^ — y^ — a^. Moreover [x^y^z)
G {x)n(t/) and this intersection must be {l,a:^}. Thus (x^y^z)
= a?.
So a? is also the only nonidentity associator in L. Replacing a and 6, if necessary, there exist a^b^c e L such t h a t (a, 6, c) / 1. Since a and b do not commute,
g = (a,6) is a group isomorphic to QsM — {a,b,c)
Our observations to this point show t h a t
is the Cayley loop. ( T h e elements a, 6, a t , c, ac, 6c, a6 • c
correspond, respectively, to the elements i, j , A:, £, i£, j £ , H which define MieiQs)-)
Let (J(7lf) = {c 6 i/ I cm = mc for all m e L}.
Clearly C{M) C C(Q) = Z{L),
so C{M) = 2 ( / / ) is an elementary abelian
2-group. We claim t h a t L = Z{L)M. 4. If ^f-^ = 1, then g G Z{L),
Let g £ L^ g ^ I, Thus ^ has order 2 or Suppose g has order 4. If g commutes with
an element m G M of order 4, then (^m"^)-^ = g'^m~'^ — 1 since g^ = Tn-^. Thus ^m~^ G >2(L) and ^ = {gm'^)!!!
G Z{L)M.
On the other hand,
suppose ^ commutes with no elements in M of order 4. If ^ associates with two noncommuting elements m i , m 2 G M of order 4, then ^, rrii and m2 generate a group G (Moufang's Theorem) and, as in the proof of the already completed hamiltonian group case, G = QQE where QQ = (^1,7712) — Qs and E is an elementary abelian 2-group. Thus g — me with m e M and e G £" C Z{L)^ so ^ G Z{L)M.
So we now further assume t h a t g associates
with no two of a, 6, c. If ga^ b and c associate, then ga G Z{L)M^ so ^ G Z{L)M.
as above,
On the other hand, if (ga^b^c) ^ 1, remembering t h a t a-^
is the only nonidentity associator in L and t h a t this element is central, we obtain {ga){bc) = {{ga • b)c}a'^ = {{g • ab)c}a'^ = {g{ab • c)]a^ — {g{a • bc)}a^ — g(a • be) which shows t h a t g, a and be associate. Since a and be are noncommuting elements of order 4 in M , this implies ^ G Z{L)M. t h a t L = Z{L)M.
So we have established
Since Z{L) n M = { l , a ^ } , letting £^ be a maximal
80
II. LOOP THEORY AND M O U F A N G LOOPS
subgroup of Z(L)
not containing a^^, it follows t h a t L = M X £ as in the
group case. This completes the proof.
D
5, E x a m p l e s o f M o u f a n g l o o p s In this section, we introduce some important examples of Moufang loops which are not groups. 5.1
M{G^ 2) l o o p s . Let G be a nonabelian group and let u be an inde-
terminate. Let L = G 0 Gu be the disjoint union of G and Gu and extend the binary operation on G to a binary operation on L by the rules
(27)
g{hu)
=
{hg)u
{gu)h
=
{gh-')u
(gu){hu)
=
h-^g
for g^h E G. Then L is a Moufang loop, denoted M ( G ' , 2 ) , which is not a group. Moufang loops of this type were discovered by Orin Chein [ C h e 7 4 ] . The case in which G is the symmetric group 5*3 on three letters gives rise to the Moufang loop ^ ^ ( 5 3 , 2 ) , of order 12, which is significant because it is the smallest Moufang loop which is not a group [ C P 7 1 ] . 5.2
M ( G , * , gfo) l o o p s . We generalize the class of loops defined in the
previous section. Suppose G is a nonabelian group, t h a t go ^ G is central and t h a t ^ »-> ^* is an involution
of G (that is, an antiautomorphism of G
of order 2) such t h a t ^Q = go and gg"^ is in the centre of G for every g £ G. (Note t h a t this implies t h a t g and g* commute.)
Let L — G (j Gu and
extend the binary operation from G to L by the rules
g{hu) {gu)h {gu){hu)
— {hg)u — {gh^)u =
goh^g
for g^h £ G. A straightforward case by case argument shows t h a t L is a Moufang loop. If it were a group, then, for any g^h E G, we would have (gh)u
== g{hu)
= {hg)u^ giving gh = hg.
Thus G would be abelian, a
contradiction. The requirement t h a t G be nonabelian assures us, therefore, t h a t the loop L is not associative. This loop is denoted M ( G , *,^o)-
5. EXAMPLES O F MOUFANG LOOPS
81
T h e notation M{G^ —^^9o) is useful when the involution is the inverse m a p g \-^ g~^.
T h u s , M ( G , —1,1) is the loop M ( G , 2 ) introduced in §5.1.
W i t h G — Qsi the quaternion group of order 8, and go the generator of the centre of Qg? it is not difficult to see t h a t M{Qs^ —l,^o) is the Cayley loop described in §4.1. (The element u is the element i of §4.1 and go is —1.) 5.3
L o o p s of u n i t s . Suppose R is an alternative ring with unity.
element x ^ R is called invertible
An
or a unit if there exist elements y and z
such t h a t xy = zx = 1. Let ZY(i?) denote the set of invertible elements of R, We wish to show t h a t U(R) is a Moufang loop. First, if xy = zx = 1^ then using Proposition L l . l l , the alternating n a t u r e of the associator and equation (1.7), we have [x, y, z] = {xy)[x, y, z] = y{x[x, y, z]) = y(x[z, x, y]) = y[zx, x,y] = 0 and so z = z{xy)
= {zx)y = y. Thus, if an element x of an alternative ring
has a right inverse and a left inverse, these elements are equal: we denote this element
x~^,
If X is invertible and a is arbitrary, appealing again to (1.7) and to Proposition L l . l l , we find (28)
[ x ~ \ x , a ] = ( x x ~ ^ ) [ x ~ \ x , a ] = x~^ {x[x~^ ^ x ^ a]) = x~^[x~^XjX^a]
— 0.
By the Generalized Theorem of Artin, we see t h a t x, x~^ and a generate an associative subalgebra of R, In particular, this shows t h a t if x is an invertible element in an alternative ring, then either of the equations ax = 6x, xa = xb implies t h a t a — b. We also see t h a t for any a, 6 G i?, the equations ax = b and xa = b have unique solutions. Thus U{R) is a loop provided it is closed under multiplication, and a Moufang loop since the Moufang identities hold in R, Continuing our assumption t h a t x is invertible, but with a and 6 arbitrary, and denoting by / the Kleinfeld function (see Definition 1.1.4), we see t h a t / ( a , 6, x^x~^) 0 =
— 0 for any a, 6 G R. Thus, if [x, a, 6] = 0, then
[x~^x^a^b] = x[x~^^ a, 6] + [x^a^b]x~^ + f(x~^^x^a^b)
= x[x~^^ ct,6];
II. LOOP THEORY AND MOUFANG LOOPS
82
hence
[x"^a,6] = 0.
(29)
Finally, suppose that x and y are invertible. Since [x^y^xy] — 0, we have [x~^^y^xy] — 0 and hence [x~^^y~^^xy] = 0. Thus {xy){y~^x~^) — 1. This shows that if x and y are in U{R)j then so is xy^ with inverse y~^x~^. Thus, as claimed, the set U{R) of all invertible elements of /? is a Moufang loop called the (Moufang) loop of units or the unit loop of R. Note that any subloop oiU{R) is also a Moufang loop. We conclude this section by establishing Moufang's Theorem for such subloops. 5.4 Theorem (Moufang's Theorem for subloops of tl(R)). Ifa^b^c are invertible elements of an alternative ring R which associate, then the subloop ofU{R) which they generate is a group. Let A - {a,a~^^, B = {b,b~^} and C = {c,c~^}. It is clear from (28) that [A,A,R] = [B,B,R] = [C,C,R] = 0 and, from (29), that [A^ B^C] = 0, so the result follows from Theorem 1.1.8. D PROOF.
5.5 The general linear loop. Let i2 be a commutative associative ring with unity. The general linear group, denoted GL(2,/2), is the group of 2 x 2 invertible matrices over R. The special linear group, SL(2, R) is the subgroup of GL(2, R) consisting of those matrices of determinant 1. Each of these groups has a Moufang loop analogue. Recall that Zorn's vector matrix algebra over R is the set 3(-K) of 2 x 2 matrices of the form
a,6 G i2, x,y G i2^, with entrywise addition and multiplication defined by
al
Xl
(12
X2
[yi
bi
_y2
&2
(See §1.3.5.)
Ci\CL2 + Xl • 72 (i2yi + &iy2 + Xl X X2
aiX2 + 62X1 - Vl X y2 6162 + yi • X2
5. EXAMPLES OF MOUFANG LOOPS
83
In §1.3.5, we noted that the function det: 3 ( ^ ) -^ R defined by det
a
X
y
b
ab — X' y
is multiplicative. Thus, just as with ordinary matrices, an element A
a
X
y b
of 3 ( ^ ) is invertible if and only if its determinant is an invertible element of i2, in which case A-^ =
(detA)-^
-X
a
The loop of invertible elements in 3 ( ^ ) is a Moufang loop analogous to the general linear group GL(2, R) of degree 2 over R. We call this loop the general linear loop and denote it GLL(2, i2); that is, (30)
GLL(2, R)= {Ae 3{R) \ det ^ is a unit in R}.
The special linear loop is the subloop SLL(2, R) of GLL(2, R) consisting of those matrices of determinant 1: (31)
SLL(2, R)= {Ae
GLL(2, R)
\detA=\}.
Chapter III
Nonassociative Loop Rings
1. L o o p r i n g s In this chapter, we first recount some of the early results about loop rings in general. We then turn our attention to loop rings which are alternative over Moufang loops of the form A/(6',*,^o) ^^^
to a description of the
corresponding groups. This work is based primarily upon [ B r u 4 4 , B r u 4 6 ] and [ G P 8 7 , G o o 9 1 ] Let L be a loop and let i? be a commutative and associative ring. 1.1 D e f i n i t i o n . The loop ring of L with coefficients in iZ, denoted RL^ is the free /2-module with basis L and multiplication given by extending the multiplication in L via the distributive laws. Except in Chapter VI, we always assume t h a t R has a unity and identify the element li of RL with the loop element i. Thus we think of L as a subset of RL and the identity element of L as the unity for this ring. An element of RL is a formal sum ^i^iC^e^-,
where the a£ are elements of /2, all but
finitely many of which are 0. The definitions of addition and multiplication take the form
ieL
eel
£eL
and
leL
leL
£eL 85
hk=i
86
III. NONASSOCIATIVE LOOP RINGS
T h e loop elements which actually appear (that is, with nonzero coefficients) in a formal sum have a name. 1,2 D e f i n i t i o n . If a = ^ a.il is an element of a loop ring, then the support of a , denoted s u p p ( a ) , is the set oi i E L for which a£ / 0: supp(E«^^) = { ^ e I | a ^ 7 ^ 0 } . As a free i2-module with basis L, the coefficients of each element of RL are unique: (1)
J2oi£^=Y^(id
implies
ae = Pi for all i e L.
Group rings—the case where L is a group—have been studied in depth for a long time. T h e subject is full of interest and intriguing problems. We cite, for instance, the two classical textbooks of D. S. Passman [Pas77] and S. K. Sehgal [Seh78] and a more recent book by S. K. Sehgal [Seh93]. By contrast, with only a few exceptions, the general nonassociative loop ring did not receive much attention until the early 1980s. R. H. Bruck did establish a semi-simplicity result about arbitrary loop rings in 1944 [Bru44] and two years later a theorem about the centre of a loop ring [Bru46] which we wish to describe here. Recall from Chapter II t h a t the inner mapping group of a loop L is the group Inn(Z/) generated by all maps of the form T{x) = R{x,y)=
R{x)L{x)-^ R{x)R{y)R{xy)-'
L{x,y)=L{x)L{y)L{yx)-' for x^y G L. We say t h a t elements a and b of L are conjugate
if b = a6 for
some 0 G Inn(L) and note t h a t conjugacy defines an equivalence relation on L. The equivalence classes associated with this relation are called classes.
Since, in a group, R{x^y)
and L{x^y)
conjugacy
each reduce to the identity
m a p while T{x) maps a to x~^ax, the definition of conjugacy in a loop is a generalization of its definition in group theory. In a loop ring, a (finite) class sum is the sum of all the elements in a finite conjugacy class of L.
With
this background, and recalling the definition of the centre of a nonassociative ring in (1.1.1), we can state Bruck's 1946 result [Bru46].
1. LOOP RINGS
87
1.3 T h e o r e m . The (finite) class sums of a loop ring RL form an for the centre of
R-basis
RL.
Since the elements of L form an i2-basis for i?L, an element a
PROOF.
is in the centre of RL if and only if ax ' y = a ' xy, for all x.,y £ L.
xa - y — x - ay.,
xy - a = x - ya
and
ax = xa
It is easy to see t h a t in the presence of the remaining
conditions, the second one here can be omitted. In other words, a is central if and only if (2)
ax ' y = a ' xy,
xy - a = x - ya
and
ax = xa
for all a;, y G i>. Extending linearly the inner maps of L to RL, the conditions in (2) are equivalent to the single condition aO = a for aU 0 G I n n ( L ) . It follows immediately t h a t any finite class sum is central. On the other hand, if ^ = ^£eL ^^^' then, for 6 G I n n ( L ) ,
£eL
Thus a G Z{RL)
£eL
iieL
implies t h a t a^Q-\ — ai for all d G Inn(L); t h a t is, the
coefficients of conjugate elements in the support of a are equal. Since the support of a is finite, the conjugacy class of each element in the support of a must be finite and so a is a linear combination of finite class sums. Thus the finite class sums span Z{RL)
as an 7Z-module. Since these sums have
disjoint supports and the elements of L are linearly independent, the finite class sums are also linearly independent and hence form a basis of as asserted.
Z{RL)., D
1.4 R e m a r k . The term "unique nonidentity c o m m u t a t o r " will occur frequently in this book. It refers to an element 5 /
1 in a group or loop L
with the property t h a t every commutator in L is 1 or 5. More generally, we shall call an element 5 7^ 1 of a loop L a "unique nonidentity commutatorassociator" if every commutator in L is 1 or 5 and every associator is 1 or s. It is convenient to record here two results which will be important later. 1.5 C o r o l l a r y . Let L be a (possibly associative) a unique nonidentity
commutator-associator
s.
Moufang
loop which has
Then s is central of order
88
2.
III. NONASSOCIATIVE LOOP RINGS
The centre of a loop ring RL is spanned
elements
those
of RL which are of the form i + si, £ E L. Since s"^ ^ 1 and since s~^ is also a c o m m u t a t o r , we have
PROOF.
s~^ = s.
by the centre of L and
Thus 5^ = 1. If is :^ si for some i e L^ then is = {si)s^
so
i — si and 5 = 1, a contradiction. Thus s commutes with every element of L, Also, if {ki^s ^ k(is)
for some k^i E L, then {ki)s
= (k • is)s.
Again,
after suitable cancellation, we reach the contradiction 5 = 1 . Thus s is also in the nucleus of L and hence in the centre of L, For i^x^y
G L^ we have iR{x^y)
either i - xy or (i - xy)s
= [ix • y){xy)~^
and, since ix - y \s
with s central, it follows t h a t iR{x,y)
G
{i,si}.
Similarly, iO G {^, si} for any inner m a p 0, so the class sums in RL are of the form i, for i G Z{L),
and i + si, for i ^ Z{L).
A nonassociative ring is power associative
D
if each of its elements gener-
ates an associative subring. In 1955, Lowell Paige showed t h a t most commutative power associative loop algebras are group algebras, thereby suggesting the probable complexity of a loop algebra which is not associative. Recall t h a t the characteristic
of a ring R with unity 1, denoted c h a r i J , is
the smallest positive integer n such t h a t n\ — 0, if such n exists; otherwise, by definition, the characteristic is 0. 1.6 T h e o r e m ( P a i g e [Pai55]). Let L be a loop and let R be a tative and associative relatively prime associative
ring of characteristic
zero or positive
characteristic
to 30. Then the loop ring RL is commutative
if and only if L is an abelian
commu-
and
power
group.
P R O O F . One direction is obvious. For the other, we present the essence of Paige's original proof, though Paige overlooked some problems with characteristic t h a t were later noticed by J. M. Osborn [Osb84]. Assume t h a t RL is commutative and power associative and linearize the identity x^x^ = (x^x)x.
First, replacing x by x + y, we obtain
4x'^{xy) + 2x^y^ + \{xy){xy)
+
\{xy)y^
- {xP'x)y 4- {x^y)x + {x^y)y + "l^xy • x)x + 1{xy • x)y + 2{xy ' y)x + 2{xy • y)y + {xy'^)x -f {xy'^)y +
{y'^y)x.
1. LOOP RINGS
89
Now replace y hy y + z. We obtain 4x^{yz) + 8{xy){xz) + 8{xy){yz) + A{xy)z'^ + A{xz)y'^ +
8{xz){yz)
= {x'^y)z + {x'^z)y + 2(xy • x)z + 2{xz • x)y + 2{xy • z)x +2(xz ' y)x + 2{xy • y)z + 2(a;2/ • z)y + 2(a;t/ • z)z +2{xz • y)y + 2{xz • y)z + 2(a:z • z)y + 2{x • yz)x +{xy'^)z + 2{x • yz)y + 2(x • i/.2r)2: + {xz'^)y +{y'^z)x + 2{yz • y)x + 2{yz • z)x + {z'^y)x. Replacing x by 2x leads to 16(x2/)(x2:) + 8x^(yz) = 2{x^y)z + 2{x^z)y + A{xy • a;)z + A{xy • >2r)x + A{xz • x)y + 4(a:z • y)x + 4(yz • x)x. Now let X, 7/ and z be elements of L. Thus each side of the above equation is a linear combination of loop elements. In characteristic 0 or positive characteristic prime to 2, we may divide by 2, obtaining (3) 8{xy){xz) + Ax'^iyz) = {x'^y)z + {x'^z)y + 2{xy • x)z + 2{xy • z)x + 2{xz ' x)y + 2{xz • y)x + 2{yz • x)x and then, with y = x, we obtain 12x^{xz) = 3(x^x)z + 3(a:'^z)a: + 6(xz • x)x^ which holds for all x^y^z G L. As before, our assumptions on the characteristic permit us to divide by 3, giving (4)
4x'^{xz) = {x'^x)z + {x'^z)x + 2{xz • x)x
for any x and z in L, Since Ax'^{xz) ^ 0, each side of (4) is a nonzero element of RL with just one element of L in its support. By the restrictions on the characteristic, {xz • x)x and {x'^z)x are in the support of the right side, so they must be equal. After cancelling x, we have (5)
xz ' x = x^z
for all a:,z G L, With x,y,z e L and using the identity (5) to conclude that x'^y = xy • x^ x^z — xz - x and x^{yz) = (x • yz)x = (yz • x)x, equation (3)
90
III. NONASSOCIATIVE LOOP RINGS
becomes (6) 8{xy){xz) +
2x\yz) = 3{xy ' x)z + 3(xz • x)y + 2(xy • z)x + 2{xz • y)x.
Now suppose K;, X and z are three elements of L and we want to prove (wx)z = w(xz). Since L is a loop, there exists y £ L such that w = xy. Rewriting (6), we have (7)
8w{xz) + 2x^{yz) = 3{wx)z + 3{xz • x)y + 2{wz)x + 2{xz • y)x.
Because of the restrictions on characteristic, either {wx)z = w(xz)^ or (wx)z — x^(yz)^ or chari? = 7 and {wx)z = (wz)x = (xz • y)x. Suppose {wx)z = x'^(yz). Then 8w{xz) = (wx)z + 3{xz ' x)y + 2{wz)x + 2{xz • y)x. Now w{xz) is the only element in the support of the left side, so it is the only element in the support of the right side. Since [wx)z is in the support of the right side, we must have (wx)z = w(xz). It follows that either {wx)z = w{xz) or char R = 7 and {wx)z = {wz)x = (xz • y)x^ with w = xy. Suppose the latter is the case, {wx)z — {wz)x = {xz ' y)x and apply our observation to the elements xz^ y, x. Either {xz • y)x = {xz){yx) or char/Z = 7 and {xz • y)x = {xz • x)y. The first possibility gives {wx)z = {xz ' y)x = {xz){yx) = w{xz). On the other hand, the second possibility shows that all loop elements on the right hand side of (7) are equal and the coefficient is not zero. Thus all terms in the equation are equal and again {wx)z = w{xz). D 2. A l t e r n a t i v e l o o p rings As a consequence of Paige's Theorem, one is not going to find loop rings of characteristic relatively prime to 30 (which are not associative) within the variety of Jordan rings (since these are power associative and commutative) nor do there exist Lie loop rings, since Lie rings, which satisfy the identity x'^ = 0, cannot contain invertible elements. There is a third important
2. ALTERNATIVE LOOP RINGS
91
variety of nonassociative rings, however, the variety of alternative rings, and here, as first shown in [ G o o 8 3 ] , the situation is more promising. 2.1 T h e o r e m . Let R be a commutative
and associative
let L = M(G^ *,5'o) be a Moufang loop constructed as described in ^11,5,2. Then the following 1. RL satisfies
the right alternative
2. RL satisfies the left alternative 3- 9 -\- 9* is i'^ Z{RG)j
ring with unity
from a nonabelian
statements
are
and group
equivalent.
identity. identity.
the centre of RG, for each g £ G.
4. If chd^r R ^ 2, then, for each g E G, h'^gh
G {g^g""} for all h G G;
z / c h a r i ? = 2, then, for each g £ G, either g = g* or else h~^gh G {9^9"^} for all h PROOF.
eG.
First note t h a t any element in RL can be written in the form
X + yu^ with x , y 6 RG.
With elements of RL represented in this way,
multiplication in RL is reminiscent of multiplication in a Cayley-Dickson algebra (c.f. §1.3) since, for x^y^z^w
G RG^
{x + yu){z + wu) = {xz + gow'^y) + {wx +
yz*)u.
Let a = X + yu and b = z -\- wu and compute {ab)b - {xz^ + gow*yz + goiw'^wx + u^*y^*)} + {wxz + goww'^y + wxz* +
y(z*)^}u
and ab^ = {xz^ + g^xw^w + goiz'^w^y + + {wzx + wz^'x + y{z*)^ +
zw*y)} goyw*w}u.
Remembering t h a t [s^t] denotes the (ring) commutator of 5 and ^, and t h a t [r, 5,^] denotes the (ring) associator of r, s and /, we obtain [a, 6, b] = go {[w'^y, z + z''] + [w^'w, x]} + {w[x, z + 2:*] + go{ww*y - yw^'w)} u = 9o {['^*y, 2: + z*] + [w^'w, x]) + {w[x, 2: + 2r*] + golww"^, y] + gQy[w, it;*]} u because ww*y — yw*w = [ww^^^y] + yww'^ — yw*w = [ww*^y] + y[w^w*]. Suppose RL satisfies the right alternative identity, so t h a t [a, 6,6] = 0 for
92
III. NONASSOCIATIVE LOOP RINGS
all a, 6 € RL. If, for some w G RG^ w'^w were not central in the group ring RG^ then, for some x G RG^ we would have [w'^w, x] ^ 0. Setting y = z = 0^ we would have [a, 6,6] = ^ o [ ^ * ^ ? ^ ] / 0, a contradiction. Such arguments show t h a t RL satisfies the right alternative identity if and only if z + z* e Z{RG), for all 2r, ti; G RG.
ww"" e Z{RG)
and ww"" = w^'w
Since centrality oi w -\- w* implies, in particular, t h a t w
and tt;* commute, RL satisfies the right alternative identity if and only if z + z* and zz^ G for all z G RG.
Z{RG)
Thus it is clear t h a t if RL satisfies the right alternative
identity, then ^^ + ^* is central for all ^ G G. Conversely, if ^ + ^* is central for all ^ G G, then z + z* is central in RG for all z G RG and hence zz* is central for all z G RG as well, since zz* is a linear combination of terms of the form ^^*, g E G (which is central by definition of M{G^ *?5'o)) ^ind of the form gh"" + hg'^ = gh*-\-{gh*Y ^ g^h ^ G. Thus statements (1) and (3) are equivalent. Again setting a = x + yu and b = z + wu with x, y^z^w E RG^ we have a^6 = {x'^z + goy'^yz + goi'w^yx +
w'^yx*)}
+ {wx^ + gowy*y + yxz* +
yx*z*}u
and a(a6) = {x^z + goxw'^y + goix^w'^y + + {wx'^ + yz^'x + yz*x* +
zy*y)} goyy''w}u
so t h a t [a, a, 6] = go {[y*y, z] + [w^'y, x + x*]} + {y[a: + x*, z*] + go^wy^y - yy^w)}
u
^ 50 {[2/*2/, z] + [tt;*t/, X + x*]}
from which it follows as before t h a t statements (2) and (3) are equivalent. For the equivalence of statements (3) and (4), we recaD t h a t the centre of a group ring RG is spanned by the finite class sums, a class sum being
2. ALTERNATIVE LOOP RINGS
93
the sum of all the elements in a conjugacy class of G (see Theorem 1.3). It is then immediate t h a t (4) implies (3). On the other hand, since h~^{9 + 9")h = h-^gh
+
h'^g^'h
for any g^h e G/ii g + g* is central for all g £ G^ then (8)
g + g^'^h-'gh
+ h-'g^'h
for all g and h in G. Suppose the characteristic of R is not 2. Then the support of the left hand side of (8) is {^r,^*}.
So, for any ^ G G, h~^gh
is g or g* for all
h. If the characteristic of i? is 2 and g y^ g*^ then there are two different elements in the support of each side of equation (8).
So again, for any
g ^ G^ h~^gh G {g^g*} for all h. These remarks establish statement (4) and complete the proof.
D
We investigate condition (4) of Theorem 2.1 in the special case t h a t * is the inverse map on a nonabelian group and the characteristic of the coefficient ring is different from 2. Suppose then t h a t G is a nonabelian group and, for any ^ G G, h~^gh G {g^g~^}
for all h £ G. This condition implies t h a t all subgroups of G are
normal; thus G is hamiltonian and hence the direct product Qg X E x A of the quaternion group Qg of order 8, an abelian group E of exponent 2 and an abelian group A all of whose elements have odd order.
(See
Norton's Theorem, Theorem II.4.8.) Let qi and q be elements of Qg such that q^^qqi
= q~^ ^ q and let a be any element of A.
Then, in C =
Qs X E X A, (^, l , a ) is not fixed under conjugation by ( ^ i , l , a ) , so this conjugate is (^, l , a ) ~ ^ .
Thus a is both fixed and mapped to a~^ under
conjugation by a, so a = a~^. Since the nonidentity elements of ^ have odd order, we conclude t h a t a = 1. Thus ^ = {1} and G = Qg X E. Conversely, \i G — QgX E for some abelian group E of exponent 2, then it is readily verified t h a t h~^gh G {g'>g~^} for all g and h. So, in view of Theorem 2 . 1 , we obtain 2.2 T h e o r e m . Let R be a ring of characteristic be a nonabelian
different
from 2, let G
group and let L = M{G^ —l^ffo) for some central go G G.
Then the loop ring RL is alternative
if and only if G = QgX E is the direct
94
III. NONASSOCIATIVE LOOP RINGS
product of the quaternion
group of order 8 and an abelian group of
exponent
2. In particular, this theorem shows t h a t the loop ring of the Cayley loop is alternative since, as noted in §11.5.2, the Cayley loop is M{Qs-, — l , ^ o ) , where go is the generator of Z{Qs)'
Similarly, the loop M((58?2) = M^Qs^, —1,1)
(see §11.5.1) also has an alternative loop ring.
3. T h e L C p r o p e r t y In this section, we characterize those nonabelian groups which have involutions satisfying statement (4) of Theorem 2.1 in the case of characteristic different from 2; t h a t is, those nonabelian groups G which have an involution such t h a t h~^gh G {^,^*} for all ^ , / i G G. In the process, we identify a certain property of a loop which plays a dominant role in this book. In any diassociative loop, elements x and y will commute if any one of x^ y^ xy is central. If these obvious situations are the only ones in which elements commute, it is reasonable to assert t h a t the loop has a certain lack of commutativity. 3.1 D e f i n i t i o n . A (possibly associative) Moufang loop L has the lack of commutativity
property
("LC", for short)
if it is not commutative and if,
for any pair of elements x , y G L^ it is the case t h a t xy = yx if and only if x 6 Z{L)
OT y e Z{L)
or xy G
Z[L).
It is obvious, but useful to note, t h a t squares are central in a Moufang loop with LC. Hence commutators are also central because {x^y)
=
x~'^{xy-^)'^y'^. T h e quaternion group Qs of order 8, the dihedral group D4 of order 8 and the Cayley loop are all Moufang loops with LC. Each of these loops also possesses a unique commutator-associator 5 / 1. T h a t these conditions are independent is illustrated by the following example. 3.2 E x a m p l e . Let G be the group presented as follows: G = (a:i,X2,X3 I Xi"^ = (xi^^Xj) Then G/Z{G)
= {(xi^Xj)^Xk)
= 1 for distinct
i^j^k).
has exponent 2 and two noncentral elements of G commute
if and only if they lie in the same coset of the centre of G. It follows t h a t G
3. THE LC PROPERTY
95
has LC. On the other hand, G has three nonidentity commutators (xi,X2), {xi,x^)
and (x2,X3).
Our reason for drawing attention to the LC property is the following theorem which, in the light of Theorem 2.1, shows how to construct an abundance of loops with alternative loop rings. 3 . 3 T h e o r e m . Let G he a nonahelian g ^-^ g* with the property if G has property
group.
Then G has an
that h~^gh £ j^^,^*} for all g^h £ G if and only
LC and a unique nonidentity
group, the involution
involution
commutator.
For such a
is given by g
(9)
if g is central
9* = { sg
where s is the nonidentity PROOF.
commutator
otherwise of G.
Assume first that G has a unique c o m m u t a t o r s ^
\.
By
Corollary 1.5, s is central and of order 2. Define g* by (9). Then (^*)* = g for any g £ G because 5^ = 1. Now assume that G also has property LC. We show t h a t ^ H^ ^* is an antiautomorphism (and hence an involution) by considering two cases. Case I: gh ^ hg. In this case, hg = sgh. /i*^* = {sh){sg)
Also, neither g nor h nor gh is central.
= hg = sgh =
So
{ghy.
Case 2: gh = hg. Here there are four possibilities corresponding to whether or not each of g^h is central.
If both these elements are central, then so is gh and
/i*^* — hg — gh — {ghy.
If neither g nor h is central, then gh must be
central because of LC and we have /i*^* = {sh){sg)
= hg = gh = {ghy.
If
g is central, but h is not, then gh is not central, so /i*^* = {^h)g = sgh = {ghy.
The last case—h central, g not central—is similar. Thus ^ i-> ^r* is
an involution and certainly h~^gh G {^,^*} for all g^h £ G. Conversely, assume t h a t G is a nonabelian group with an involution g ^-^ g"" which is such t h a t h~^gh G {5^,5'*} for all ^ , / i G G. li g = ^* for some g £ Gy then g is central. In particular, gg* is central for any g. From this, we also learn t h a t g* commutes with g and with g~^, for any g £ G.
96
III. NONASSOCIATIVE LOOP RINGS
Now suppose g^h ^ G and gh ^ hg. Then g~^hg = /i*, so hg = gh^. Also, h'^g'^h
= g (otherwise, h would commute with ^* and with ^^*, hence with
^ ) , so hg = g*h = gh*. Remembering t h a t h~^ commutes with /i*, we have We claim t h a t the element s — g~^g* is independent of noncentral g. For this, Rx g^h e G with gh 7^ hg and let x be any noncentral element of G, If xg 7^ ^ x , then, as we have shown, x~^a:* = g'^g'^-
Similarly, if
xh 7^ /ix, then x~^x* = h~^h* — g'^g"". In the case x^ = gx and x/i = /ix, then h~^(gx)h
= gx
while also, h~^{gx)h
or
h~^{gx)h
= {h~'^gh)(h~^xh)
= (gx)* — x*g^ = y*a:* = g*x.
Thus g = g* or x = x*,
either of which yields a contradiction because neither g nor x is central. This establishes the claim. Noting t h a t if gh / hg^ then g^^h^^gh
= g~^g*^ we see t h a t s = g'^g""
is a unique nonidentity commutator and also t h a t ^* = h~^gh = sg. If ^ is central and h is not, then gh is not central and we have (5/1)^* = h'^g* = (^/i)* =z s(gh) = shg^ so ^* = g. It follows t h a t g* is defined by (9). To see why G has LC, recall t h a t a unique nonidentity commutator in a group is central and of order 2. If gh = hg^ but g ^ 2(G) then ^* = sg and /i* = 5/1, so {ghy
= h^g"" = {sh){sg)
and /i ^ 2 ' ( G ) ,
= hg = gh. Thus gh
is central.
D
If G is not abelian, then the loop L = M{G^ *?5'o) is not associative (see §11.5.2) and thus RL is not associative. In view of Theorems 2.1 and 3.3, the following corollary is immediate. 3 . 4 C o r o l l a r y . Let R be a commutative, of characteristic
ring with unity
different from 2. Let L be the loop M ( G , *,^o) for
nonabelian group G, some involution go in G. Then RL is an alternative if G has LC and a unique involution
associative
g ^-^ g'^ of G, and some central ring which is not associative
nonidentity
commutator
s.
and some
element
if and only
In this case,
the
on G is given by (9).
With the next corollary, we commence gathering information loops whose loop rings are alternative.
about
3. THE LC P R O P E R T Y
3.5 C o r o l l a r y . Let R be a commutative and of characteristic alternative^
97
and associative
ring with
different from 2 and let L — M{G^^^go).
unity
If RL
is
then
(i) V = G' has order 2; (ii) M{L)
= Z{L)
(iii) L has property
=
Z{G); LC;
(iv) the centre of RL is spanned by Z{L)
and those elements
of RL of the
form £ + si, i e L. Recall t h a t L = G U Gu is the disjoint union of G and a coset
PROOF.
Gu and t h a t multiplication in L is given by g(hu)
=
{hg)u
{gu)h
=
{gh'')u
{gu){hu)
=
goh^'g
for g^h e G, By Corollary 3.5, G has a unique nonidentity commutator, 5, and G' = { 1 , 5 } . It is routine to verify t h a t (10)
{g,hu)~
g~^g*
and
{gu.hu)
=
{gh'")~\gh*y
for any g^h E G. By Corollary 3.4, k~^k* G {l,'^} for any fc G C , so the only commutators in L are 1 and s. Direct calculation shows t h a t any associator in // is a c o m m u t a t o r or the product of two commutators. Specifically, for ^, /i. A; G G, we have (g,h,ku) {gu,h,k)
= {g,h),
{g,ku,h)
= {k~^,h~^),
= {h~\g*~^),
{gu,h,ku)
{g,hu,ku)
= {k^g.h''),
=
{gu,ku,h)
{k^h.g), =
[k^g.h)
and {gu,hu,ku)
=
{g*-\h-^){{hg*)-\k*-').
Thus L', which is the subloop of L generated by the commutators and associators, is { l , ^ } = G'. This gives (i). Let g^h^k
G G. The equation (gu^h^k)
= (k~^^h~^)
not in the nucleus of L, while the equation (g^h^ku)
shows t h a t gu is
— (g^h)* shows t h a t
^ G G is in the nucleus of L if and only if g is in the centre of G. Af{L) = Z(G).
Thus
The equation (g^hu) = g'^g"" shows t h a t if ^ is in the centre
of G, then it commutes with all elements of L, so (ii) follows.
98
III. NONASSOCIATIVE LOOP RINGS
Since the central elements of G are precisely those k for which k'^ = k^ the relations in (10) also show t h a t an element of the form hu commutes with g e G if and only if g is central, and t h a t elements gu and hu commute if and only if gh* is central; t h a t is, if and only if the product (hu){gu)
is
central. Thus L has property LC, which is (iii). Part (iv) follows immediately from part (i) and Corollary 1.5.
D
Theorem 3.3 has focused attention on groups with LC and a unique nonidentity commutator. Such groups are easily characterized. 3.6 P r o p o s i t i o n . A group G with centre Z{G) has the lack of ity property and a unique nonidentity
commutator
commutativ-
if and only if G/Z{G)
=
C2 X C2. PROOF.
We have previously observed t h a t squares are central in a group
with LC, so GIZ[G)
is an elementary abelian 2-group. Suppose this group
contains the direct product {aZ{G))
X {hZ{G))
x
{cZ{G))
of three copies of the cyclic group C2 of order 2. Then no two of a^b^c can commute, because of LC. Since, in any group, there is the identity (xy, z) — (x, z){{x^ z), y)(y, z)^ and since commutators are central in a group with LC, we have (xy, z) — (x, z){y^ z) for all x^y^z
£ G. Now suppose t h a t
G also has a unique nonidentity commutator s.
By Corollary 1.5, 5 is a
central element of order 2, so [ab^c) = (a,c)(6,c) = s^ = I and, because of LC, either ab or c or abc is central, none of which is true. It follows t h a t G/Z{G)
contains the direct product of at most two copies of C2 and, since
G is not abelian, exactly two; t h a t is, G/Z(G)
= C2 X C2.
Conversely, if G is a group such t h a t G/Z{G)
= {aZ{G))
X
{bZ{G))
with each factor isomorphic to C2, then a and b cannot commute (because G is not abelian). Therefore two noncentral elements of G commute if and only if they both lie in the same one of the cosets aZ[G)^
bZ{G),
abZ(G).
From this, lack of commutativity in G is evident. Moreover, denoting by s the commutator of a and 6, it is easy to see from the relation (xy^z) (x^z)(y^z)^
=
which holds because commutators in G are central, t h a t s is the
only commutator in G other than 1.
D
3. THE LC PROPERTY
99
3.7 E x a m p l e s . Using the notation of M. Hall Jr. and J. K. Senior [ H S 6 4 ] , we enumerate the groups of order less than 32 which have a unique commut a t o r and property LC. • Qs = {a^b\ a"^ = l^b'^ = a^^ba = a~^b) • D4 = {a.bla"^ = b'^ = 1, ba =
a'^b)
• 16r2C2 = (a, 6 I a^ = 6^ = 1, (a, b) = a?) . 16r2&= {a,b,c\ • UV2d = {a,b\a^ • QsX
a^ = l , ( a , c ) = ( 6 , c ) = \,o? = 6^ ^ c^ = (a,6)) = b^ = 1, (a, b) = a^)
C2, Qs X C3, /^4 X C2 and Z:)4 x C3.
In §V.2, we shall see why this list is complete. If G is any of these groups and s is the unique nonidentity commutator of G, the m a p g ^-^ g^ given by (9) is an involution.
Moreover, if R
is any commutative and associative ring with unity and of characteristic not 2 (this restriction is, in fact, not necessary—see Theorem IV.3.1 and Corollary IV. 1.2), and if go is any element in the centre of G, the loop ring RL of the loop L = M ( G , * , ^ o ) is an alternative but not associative ring, by Corollary 3.4. 3.8 R e m a r k . Both Qg and D4 have the LC property and, in both these groups, the element a'^ is a unique nonidentity central commutator. Thus the loop Mi6((58,2) = M ( g 8 , - l , l ) and the Cayley loop, M{Q8^—li(i^)',
are loops with alternative loop rings.
=
MIG{QS)
They are not iso-
morphic because, for example, a^ is the only element of order 2 in the Cayley loop whereas there are nine elements of order 2 in loop
MIQ{QS',2)
MIG{QS^2).
The
is, however, isomorphic to A/(Z?4, —1, a^). Conditions un-
der which M ( G , * , ^ o ) is independent of ^0 are known [ C G 8 5 , Theorem 3]. In [ C G 8 5 ] , it is also shown how to determine whether or not, given two nonabelian groups Gi and G2 (each with LC, a unique nonidentity commutator and containing central elements ^1,^2 respectively), RA loops of the form M ( G ' i , * , ^ i ) and M(G2?*^^2) are isomorphic. 3.9 E x a m p l e s . Let N = 2'^ with n > 1 and let G be the group generated by two elements a and b with a^^ = b"^ = I, ba = a^'^^b. (a^) is cyclic and G/Z{G)
Then Z(G)
=
= C2 X C2. Thus G has property LC and a
unique nonidentity commutator so, by Corollary 3.4, any loop L of the form M{G^ *?^o) = GuGu
with * given by (9) has an alternative loop ring.
100
III. NONASSOCIATIVE LOOP RINGS
Note t h a t |L| = 21^1 = 8iV = 2^+^. Now it is not difficult to show t h a t the group G described here is generated by any pair of elements which do not commute. This observation allows us to establish a significant property of L: all proper subloops of L are associative. The argument, briefly, is as follows: if x i , X2 and x^ are any elements of L and they all belong to G, then obviously the subloop (a^i, 0:2, 2:3) generated by the Xi is associative. If xi and X2 are in G, if X3 is in Gu and if x\ and X2 do not c o m m u t e , then Xi and X2 generate G, so quite clearly {x\^X2^x^)
= L. The possibility t h a t
just xi is in G while both X2 and x^ lie in Gu reduces immediately to the case of two generators in G since ( x i , 0:2, a^a) = (a^i, 0:23:3, 0:3); so does the final case, all Xi in Gu^ since (a:i,X2,X3) =
{x\X2->X2Xz^x^).
The point is t h a t we now have, for each n > 4, at least one loop of order 2'^ which has an alternative loop ring t h a t is not associative, but which contains properly no other such loop. This gives some indication of the richness of the class of loops around which much of this book is based.
4. T h e n u c l e u s a n d c e n t r e Let /Z be a commutative and associative ring with unity and of characteristic different from 2. Let G be a nonabelian group with the LC property and a unique nonidentity (necessarily central) commutator, s. Let ^ H-^ ^* be the involution on G defined by
(U)
/ J ' '"'^^"^^ \sg
(See Theorem 3.3.) Let g^ G Z{G)
\fgtZ{G). and let L be the loop M ( G , * , ^ o ) de-
scribed in §n.5.2. The involution on G extends linearly to a (ring) involution of the group ring RG which we also denote *: (12)
( E , e G « . 5 r = E,6G«p5*-
By Corollary 1.5, the centre of RG is spanned by the centre of G and elements of the form ^ + 5^, so it follows t h a t both g -\- g* and g + sg are in Z{RG)
for any g eG.
(13) for any x G RG.
Thus
x + x"" e Z{RG)
and (1 + s)x 6
Z{RG)
4. T H E NUCLEUS AND CENTRE
101
Now, if X G RG^ \{ sx = X and if g is any element in the support of x, then the coefficients of g and sg must be the same, so x is central: (14)
x G RG and sx — x
implies
x G
Z{RG).
Suppose x* = X. Writing x = XQ + x\^ where a:o has support in
Z{G)
and supp(a:i) fl 2 ( G ) = 0, we have x"" — XQ + sxi^ so xi = sxi^ Xi is central and hence x also is central. On the other hand, if x is central and written as a linear combination of class sums of the form g^ g ^ Z{G),
and g + sg^
^ G G, it is clear t h a t x* = x. Thus (15)
X G 2:(i2G) if and only if x* = x.
As noted in the proof of Theorem 2.1, any element of RL can be written in the form x + yu^ where x , ^ G RG^ and, representing elements this way, multiplication in RL takes the form (x + yu){z + wu) - {xz + gow*y) + (wx + where x^y^z^w
yz*)u,
G RG, In what follows, it is convenient to refer to x and y
as the RG- and i^Gi^-components of the loop ring element x + yu. 4.1 P r o p o s i t i o n . Let L be a loop of the form M{G^ '^'>9o) o,^^d suppose is alternative.
Let J\f{RL)
denote the nucleus and Z{RL)
RL
the centre of RL.
Then M{RL)
= Z{RL)
^ {x + yu\x,y
^ Z{RG),
= {x + yt/ I X G Z{RG), PROOF.
sy = y] sy =
y).
Observe t h a t
{x + yi/ I X, y G Z{RG),
sy = y] = {x-^ yu\ x e Z{RG),
sy = y]
because of (14). We first show that this set is the nucleus of RL. Let a — X -\- yu^ f] — z -\- wu and j — p -\- qu he elements of RL^ where x.,y^z^w.,p^q
are in RG. Direct calculation shows t h a t the associator
[a,/?,7] can be written in terms of commutators in RG: (16)
[a,/3,7] = ^ o { K 2 / , p ] + [q'^.x]
+ [^*y,^*]}
+ {(l[x, z] + ti;[x, ;?*] + y[2:*, p*] + g^lq, w'']y + ^ o k * g , y]]u.
102
III. NONASSOCIATIVE LOOP RINGS
Suppose t h a t sy = y (and hence t h a t y is central). For any t = Y^g^o^gS RG^ we have t-f"
= {I-
^
s) YlgdziG) ^gd ^^^ ^^^^ 2/(^ - T ) = 0. Using this
and the fact t h a t i/* = y, by (15), we have
{tyy = 2/*r = ye = y{e -t + t) = yt = ty; thus ty is central for all ^ G RG. Moreover, since gh — hg = 0 OT gh — hg = (1 — s)gh for any g^h ^ G^ sy = y implies also t h a t (gh — hg)y = 0 for any g,h and hence [t^v]y = 0 for any t^v £ RG.
If then b o t h x and y are
central in RG and 5^/ = t/, it follows easily from (16) t h a t [a,/3,7] = 0; t h a t is, a e
Af{RL).
Conversely, suppose t h a t a G M{RL)
so t h a t
[Q^,/3,7]
7 in iJL. For any t and i; in i?G, the centrality oit-\-t*
= 0 for aU /? and
and t; + t;* implies
t h a t [ r , t ; ] = [^,^*] = -[^,'^] and [r,t;*] = [t^v]. Referring to (16), it follows that [a, /?, 7] = goiiw^'y, p] + [gr*it;, x] + [y*^, z]}
+ {^[a^, ^] - '^[a^, j^] + y[z,v] - 9o[q, ^]y + ^o[^*g, y]}u = o. Suppose X ^ Z{RG).
Then choosing it; G i2G such t h a t [w^x] ^ 0 and
setting q — I and p = 2^ = 0, the iZG-component of [a,/3,7] is ^ o [ ^ , ^ ] 7^ 0. Therefore x must be central and
[«,/?, 7]
If 2/ ^ Z{RG)^
choose p so t h a t [y^p] 7^ 0 and set it; = 1 and z = 0. Then
the iJG-component of [a,/J, 7] is ^'oiy^p] 7^ 0. Thus both x and y must be central elements of RG and [a,/3,7] = ^ o { k * y , p ] + [y*^,^]} + {2/[^,p] - 9o[q,w]y}u Taking ^ = 0, it follows t h a t y[z^p] = 0 for any z^p £ RG,
= 0. With z and
p noncommuting elements of G, we have zp — pz = (1 — 5)2:^ as before. Then y{l — s)zp — 0 implies y(\ — s) — ^ because zp is invertible. Thus the nucleus of RL is the required set. Since 2(6*) = Z{L)^ from Corollary 3.5 we see t h a t the centre of RL is spanned by
Z{G)u{i
+ si\ie
L},
5. THE NORM AND T R A C E
Thus Z{RL) = Af{RL).
103
D
4.2 Corollary. Let L be a loop of the form M(G,*,^o) ^^^ suppose RL is alternative. Then Z{RL) = {re RL\re
= ir for all i G L],
Suppose r G RL commutes with all ^ G Z/. Write r — x -\- yu^ x^y e RG. Let g £ G, Since rg = xg + {yg*)u and gr = gx + {yg)u^ it is clear that xg = gx and yg = yg* for all ^ G C, so x is in Z(RG). Let g be a noncentral element of G. Then yg = yg* — syg and hence y = sy. By Proposition 4.1, r = x + yu e Z{RL). D PROOF.
For r — X + yu e RL., define (17)
r* = X* + syu.
Then * is an involution on RL which extends the involutions on G and on RG and which induces an involution on L defined mutatis mutandis like that on G: (18)
r = < l^si
^ ^ i{i^Z{L).
Proposition 4.1 and observation (15) give immediately the following result. 4.3 Corollary, r G Z{RL) if and only if r"^ = r. In particular, for any r G RL, r + r"^ and rr* (~ r^'r) are central elements of RL. 5. T h e n o r m and t r a c e Remember that an algebra A over a commutative and associative ring $ with unity is quadratic if every x E A satisfies an equation of the form x^ — t{x)x + n{x)l = 0 with t{x),n{x) G $. (See §1.4.) Let L be a loop of the form M{G^ *55'o) for which RL is alternative and, for r G RL, define respectively the trace t{r) and norm n{r) of r by (19)
t(r) = r + r*
and
n{r) = rr'^.
104
III. NONASSOCIATIVE LOOP RINGS
The trace and norm are functions RL -^ Z(RL)
whose basic properties are
summarized in the next proposition. 5.1 P r o p o s i t i o n .
(a) Every element
of RL satisfies
t{r)r + n ( r ) = 0. Thus RL is a quadratic
the equation r^ —
algebra
over its
centre
Z(RL), (b) The trace is a linear function
and the norm a quadratic form on RL,
(c) The trace is commutative in the sense that t(qr) = t{rq) for all q,r ^ RL, p,q,r (d) IfR
and associative in the sense that t(pq • r ) = t{p • qr) for all e RL, has no additive 2'torsion,
then the norm is
nondegenerate.
P R O O F . T h e proofs of (a) and (b) are direct verifications. For part (c), we use the facts t h a t L has the LC property and a unique nonidentity commutator, 5, which is also a unique nonidentity associator (see Corollary 3.5). Commutativity of the trace is equivalent to qr + r*^* = rq + q*r*;
t h a t is,
[q^ r] = [^*, r*]
for all q^r ^ RL, Now [^*,r*] = q*r'' — r'^q* = (rq — qr)* = [r, ^]*, so we must show t h a t [r, ^]* = —[r, ^] for any q^r £ RL.
Since the c o m m u t a t o r
is bilinear, it is sufficient to prove t h a t [r, g]* = —[^, ^] for r^q £ L. This is obvious if either r or g' is central. If neither r nor q is central, then r* = sr and (7* = sq and so [r,qY = [^*,r*] = q*r%- r^'q* = {sq){sr)
- {sr){sq)
=
-[r,q]
because s is central and s^ = I. This establishes commutativity of the trace. Associativity is equivalent to pq ' r + r'^ ' q'^p'^ = p - qr + r*q'^ • p*;
t h a t is,
[p, ^, r]* = — [j9, q^ r]
for all p^q^r e RL and, as above, it is sufficient to establish this property in the case t h a t p^q^r e L. \i p or q or r is central, or if p , q and r associate, then clearly [p^q^r] = 0. So assume these elements do not associate and that none is central. Then p* = sp,, q* = sq^ r* — sr and
b, q-, rY = {pq • ^)* - {v • q^T = ^* • ^ V - ^*^* • p* = (^r) • {sq){sp) - {sr){sq) • [sp) = s{r'qp-rq'p)
=
-s[r,q,p].
5. T H E NORM AND T R A C E
105
Since the associator is an alternating function in RL^ we get [p, g,r]* = s[p^q^r\.
Now |j9, g, r] = pq-r — p-qr = (p'qr)s
— p'qr
= (s — l)(p'qr)^
since
5 is a unique nonidentity associator. Therefore s[p^ q^ r] = (s'^ — s)(p • qr) = (1 — s){p • qr) = —[p,^,^] as desired. This establishes associativity of the trace. For (d), we must prove nondegeneracy of the bilinear function / associated with q. Note t h a t f(r^p)
= n{r + p) — n(r) — n{p) = rp* + pr* =
Suppose r G RL and f(r^p)
— 0 for all p G RL.
t{rp^).
W i t h p — 1, we obtain
r* = —r. Writing
leZ(L)
^tz(L)
£ez(L)
^iZ{L)
we have
and, since R has no 2-torsion, a^ = 0 for all ^ G >2(//). Now / ( r , p) = 0 for all p implies / ( r , p\p) — 0 for all p, pi G -RL. Since / ( p , r ) = /(pr*) and the trace is associative, / ( r , p i p * ) = ^{Tpp\) — f{rp^pi)
= 0 for all p , p i G RL.
By
what we have already learned, this means t h a t (rp)* = — (^p) for all p G ^ L and hence, for any p G KL, rp has no central element in its support. Note, however, t h a t if i is in the support of r, then ri has P in its support and i'^ is central because L has LC, by Corollary 3.5. It follows t h a t r = 0.
D
Chapter IV
R A Loops
In this chapter, which is based primarily on [ G o o 8 3 , C G 8 6 ] , we investigate and describe those loops whose loop rings, in characteristic different from 2, are alternative, but not associative. It is convenient to have a name for such loops. 0.1 D e f i n i t i o n . An RA {ring alternative)
loop is a loop whose loop ring
RL over some commutative, associative ring R with unity and of characteristic different from 2 is alternative, but not associative. Since an alternative ring satisfies the Moufang identities and since a loop ring RL contains L, it is clear t h a t an RA loop is a Moufang loop.
1. B a s i c p r o p e r t i e s of R A l o o p s We begin with one of several possible characterizations of RA loops. 1.1 T h e o r e m . A loop L is an RA loop if and only if L is not and has the following (i) if three elements
associative
properties: of L associate in some order, then they associate
in
all orders; (ii) if g^h^k £ L do not associate, PROOF.
then g - kh — gh - k — h - gk.
First, suppose t h a t L is an RA loop and R is some ring of
characteristic different from 2. Since L C RL and RL is an alternative ring, statement (i) holds by the Generalized Theorem of Artin, Corollary 1.1.9. To obtain (ii), let ^f, /i and k be three elements of L which do not associate and put X =^ h-\- k^ y — g m the right alternative identity, yx - x — yx^. We 107
108
IV. RA LOOPS
obtain gh ' h + gh ' k + gk ' h + gk ' k = gh? + g - hk + g - kh + gk^ and, since gh - h — gK^ and gk - k = gk'^^ gh • k + gk ' h = g • hk + g ' kh. Because chari? / 2, ^/i • fc is in the support of the left side and thus in the support of the right. So, \i gh - k ^ g • hk^ then gh- k = g - kh. Similarly, by considering the left alternative identity, we obtain gh - k = h - gk. Conversely, suppose t h a t L is a loop which satisfies statements (i) and (ii) of the theorem. Let iZ be a commutative and associative ring with unity. For X = "^ agg and t/ = ^ I3gg in RL., yx-x
— yx^ is a linear combination of
terms of the form gh-k — g - hk. li h = k^ then gh-h = g -hhhy
associativity
or (ii), so yx ' X — yx^ reduces to a sum of terms of the form [gh ' k — g • hk) + (gk - h — g - kh) with h ^ k and this is 0 by (i) and (ii). This shows t h a t the right alternative identity holds in RL. The left alternative identity follows in a similar way.
n Since statements (i) and (ii) of Theorem 1.1 involve only the loop L and not the ring R and because the second half of the proof of Theorem 1.1 required no assumptions about the characteristic of the coefficient ring, we note the following. 1.2 C o r o l l a r y . If L is an RA loop, then RL is an alternative coefficient
ring for any
ring R.
Since an alternative ring is power associative, Paige's Theorem (Theorem III. 1.6) shows t h a t most commutative alternative rings are group rings. This also follows from Theorem 1.1. 1.3 C o r o l l a r y . If L is an RA loop and if h and k are elements commute, alternative
then gh - k — g - hk for all g ^ L. A commutative loop ring in characteristic
of L which loop with an
different from 2 is a group.
P R O O F . By Theorem 1.1, either gh - k = g - hk or gh - k = g - kh = g - hk. Thus gh ' k — g ' hk m every case. The second statement is clear.
D
1. BASIC PROPERTIES OF RA LOOPS
109
1.4 C o r o l l a r y . The direct product L x K of loops is an RA loop if and only if precisely one of L and K is an RA loop while the other is an abelian group. PROOF.
It is straightforward to check t h a t the direct product of an
abelian group and a loop satisfying statements (i) and (ii) of Theorem 1.1 is a loop which also satisfies (i) and (ii). On the other hand, if L x K is an RA loop satisfying (i) and (ii), then both L and K satisfy (i) and (ii) and at least one of these loops is not associative. Without loss of generality, assume that L is not associative (and hence R A ) . We complete the proof by showing t h a t K is an abelian group. It is sufficient to show t h a t K is commutative since associativity will then follow by Corollary 1.3. To show t h a t K is commutative, let a, 6 G K and let g, h and k be three elements of L which do not associate. Since (^, 1), (/i,a) and (A;, 6) cannot associate in L X Ii,, we must have
[ig,l)ih,a)]ik,b)
=
ig,l)[ik,b){h,a)],
by Theorem 1.1. Thus ab — ha. F. Fenyves has called a loop extra
D if it satisfies the identity {xy • z)x —
x{y • zx) and, furthermore, shown t h a t an extra loop is Moufang [ F e n 6 8 ] . 0 . Chein and D. A. Robinson later characterized extra loops as precisely those Moufang loops in which all squares are in the nucleus [ C R 7 2 ] . 1.5 C o r o l l a r y . An RA loop is an extra loop. PROOF.
Let L be an RA loop. If x, y and z are three elements of L
which associate, then, by Moufang's Theorem, these elements generate a group, so (xy • z)x = x(y • zx) is clear. If x, y and z do not associate, then neither do xy^ z^ x (again by Moufang's Theorem), so (xy • z)x — by Theorem 1.1 and, since x, y, xz do not associate, {xy){xz)
{xy)(xz)
= x{xz • y).
Repeated use of Theorem 1.1 gives xz ' y = z ' xy — zy ' x — y ' zx and so, again, [xy • z)x = x(y • zx).
D
We wish next to describe some fundamental characteristics of an RA loop, but first several points should be noted.
no
IV. RA LOOPS
1.6 R e m a r k . A Moufang loop of exponent 2 is an abelian group. To see why, let L be such a loop and note first t h a t because L is diassociative, xyxy
= 1 implies xy = yx] thus L is commutative. Then {x ' yz)x = (yz • x)x = yz
by commutativity, diassociativity and the fact t h a t x^ — \. Also {xy • z)x = {yx - z)x = y(x • zx) = y{zx ' x) = yz Thus (x ' yz)x
by the right Moufang identity using diassociativity and x^ = 1.
= (xy • z)x and, after cancelling x, we get x - yz — xy - z.
Thus L is associative. 1.7 R e m a r k . If ^ is a proper subloop of a Moufang loop L, then L is generated by the complement of H. This follows by fixing a £ L\
H and
noting t h a t if h ^ H^ then h is the product of ha and a~^, both of which are in the complement of H. 1.8 T h e o r e m . Let L be an RA loop.
Then
(i) g'^ e J^{L) for each g G L; (ii) Af{L) = Z{L)
= {a ^ L \ ax — xa for all x G L];
(iii) (^, h) — \ for g^h ^ L if and only if (^, h^k) = 1 for all k G L; (iv) if g^h^k ^ L and (^, h^k) ^ 1, then (^, /i, A:) = (^, /i) = (^, k) = (/i, /c) Z5 a central element
of order 2;
(v) L', the commutator-associator
subloop of L, is a central group of
order 2. PROOF.
Throughout this proof, we assume t h a t L is contained in an
alternative loop ring RL and t h a t char R ^ 2. As previously noted, statement (i) holds because L is an extra loop, but we present here a proof which does not make use of this fact. Linearizing the middle Moufang identity, xy ' zx — {x ' yz)x^ we see that the alternative ring RL satisfies xy ' zw + wy - zx — [x - yz)w + [w • yz)x for all x^y^z.w
G RL.
Let ^, h and k be three elements of L and set
x = y = g^z = h and w = k. We get g'^ ' hk + kg ' hg = {g ' gh)k + {k • gh)g = g'^h'k
+ {k'
gh)g,
1. BASIC PROPERTIES O F RA LOOPS
111
using the left alternative identity to rewrite g - gh d.s g^h. Suppose g^j h and k do not associate. Then g^h - k ^ g^ - hk^ so g^h - k = kg - hg. Also, neither triple g^ h^ k nor triple g^ h^ kg can associate, by Moufang's Theorem. Repeated use of Theorem 1.1 now gives kh-g'^
- (kh • g)g = {h • kg)g = kg - hg = g^h • k = g^ ' kh = kg^ - h — k - hg^;
t h a t is. A;, h and g^ associate. This contradiction shows t h a t g^ is indeed in the nucleus of RL^ for any g ^ L. (ii) Denote by C{L) the set {a £ L \ ax = xa for all x ^ L} and remember t h a t Z{L)
= C{L) nAf{L),
If a G C(L), then (a, x,y)=
by Corollary 1.3, so a G AfiL). remains to show t h a t M{L)
1 for all x,y e L
Thus C(L) C A/'(L), so Z{L)
C Z{L).
= C{L).
It
Since the nucleus of a Moufang loop
is normal (Corollary II.3.4), we can form the quotient loop L/Af{L)^
which
is a Moufang loop of exponent 2, by (i), and hence an abelian group by Remark 1.6. Clearly, M{L)
is contained in the nucleus of RL.
Let n G M{L)
and let ^, h and k be any three elements of L which do not associate. Then gh ' k — n\{g • hk) for some n^ G J^{L) gn — (ng)n2 for some 712 G M{L)
(since L/Af(L)
(since LjM^V)
is a group) and
is commutative) and so
the (ring) associator \g^h^k\ and (ring) commutator [^,n] are \g,h,k\
^ (ni - \){g'hk)
and
[g,n\ = [ng){n2 - 1).
By Proposition 1.1.7, [g-inWg^h^k] — 0, so {{ng){n2-\)}{(n,-l){g-hk))
= d.
Since 712 — 1 is in the nucleus of RL^ we have {ng){{n2-\){{n,-\){g-hk))]
=Q
and, since ng is invertible, (712 — l ) { ( ^ i — ^)[9' hk)} = 0. Since ni — 1 is in the nucleus of iZL, {n2 - l ) { ( n i - 1)(^ . hk)} = {(712 - l ) ( n i - 1)}(^ • hk), so (712 —
1){TII
— 1) = 0 because g - hk \s invertible. So we have 7ii + 712 =
1 + ^2^1 3.nd, since 7^l /
1 and we are in characteristic different from 2,
necessarily 712 = 1, so ^7z = TI^. This argument shows t h a t n commutes with all elements not in the nucleus. Since L is not associative, N{L)
is a
112
IV. RA LOOPS
proper subloop, so its complement generates L, by Remark 1.7. It follows t h a t n commutes with every element of L and is therefore central. (iii) By Corollary 1.3, (g^h)
= 1 implies (g^h^k)
Conversely, let g^h e L and suppose t h a t (g^h^k)
= 1 for all fc G L.
= 1 for aU /: G L.
have to prove t h a t gh = hg. By (ii), we can assume t h a t h ^ J^iL).
Clearly,
[g^h^x] = 0 for all x G RL so, by Proposition 1.1.7, we have [g^gh] G Also gh ' g = n{g • gh) for some n G J^{L) so gh ' g = ng^h.
(since L/Af{L)
Hence [g^gh] = (1 — n){g^h)
We
Af(RL).
is c o m m u t a t i v e ) ,
G M{RL).
Hence, for all
a,6 G i/, {(1 - n){g^h)]a Now g'^ G M{L) n)h}(ab)
= Z{L)
• 6 = {(1 - n ) ( / / . ) } ( a 6 ) .
and g^ is invertible, so {(1 - n)h}a
for all a^b e L. Since h ^ Af(L) and because L/Af{L)
there exist a and b such t h a t ha-b — ni(h'ab) Therefore (1 — n)ni{h
- b = {{I
-
is associative,
for some ni G Af(L)^ rii ^ 1.
• a6) =:= (1 — n){h • a6). So (1 — n)ni
= 1 — n, giving
m + n = 1 + n n i . Since ni / 1, necessarily n — I. Therefore gh - g = g^h and, cancelling g, we get /i^ = gh, as required. (iv) Let n = (g^h^k),
note t h a t n G N{L)
= Z{L)
(because L/Af{L)
is
a group) and suppose n 7^ 1. By Theorem 1.1, kg ' h = g • kh = gh • k = n{g • hk) = n{gk - h) = (n - gk)h, so kg = ngk. Therefore n = (k^g) — {g,k)~^.
Since g^ G M{L)
= Z{L)
and
n is central, we have g^k — kg
— kg - g — ngk - g — ng - kg — ng - ngk = {ng)^k = n^g^k,
so n'^ = 1. Thus {g,k)~^
= (5^,^)5 so (g^h^k)
— n — (g^k).
Interchanging
the roles of h and A:, we have also (g, k, h) = (^, h). Since g, h and k do not associate. Theorem 1.1 gives gk-h
— g- hk and g-kh
— gh-k.
Since gh-k
=
(g ' hk){g, h, k) (by definition of (^, /i. A:)), we have g - kh — [gk • /i)(p, /i, A:), so gk ' h = (g ' kh){g,h,k)~^
— {g • kh)(g,h,k)
order 2. This shows t h a t {g,k,h)
= (g^h^k)
because (g^h^k) = {g^h) - {g,k).
= n has Similarly,
hg ' k = g ' hk and h - gk — gh - k implies (^, /i, fc) = (/i, ^, fc) = (/i. A;). (v) First note t h a t parts (iii) and (iv) together show t h a t every commutator in L is an associator, t h a t every associator in // is a c o m m u t a t o r , and that the commutator-associator subloop V is a central group of exponent
1. BASIC PROPERTIES OF RA LOOPS
113
2. We show IL'I = 2 by establishing that any two nonidentity associators are equal. We already know this to be the case if the associators in question have two elements in common; for example, if (^,/i, a) 7^ 1 7^ (^,/i,6), then (g^h^a) = (g^h) = (g^h^b) by (iv). Next, suppose that two nonidentity associators have just one element in common. Specifically, suppose that (g^h^k) 7^ 1 7^ (g^a^b). If (a^g^h) ^ 1, then, since this associator has two elements in common with both (g^a^b) and (g^h^k)^ we have (g,a,b) = (a,g,h) = {g,h,k). Similarly, if {b,g,h) 7^ 1, we have {g,a,b) = {b,g,h) =: {g,h,k). If {ab,g,h) 7^ 1, then (ab.g) = {ab,g,h) = {g,h,k), while ab-g = a- gb = ga-b = {g ' ab){g, a, b) implies that {ab^g) = (g^a^b). Thus (g^h^k) = (g^a^b). All this shows that (^, /i, k) — (^, a, 6) if any of the associators (a, g^ /i), (6, g^ h), (ab^ g^ h) is not 1. We now consider the case that each of these associators is 1. Denoting the Kleinfeld function by / , as usual, we have / ( a , b,g,h)=
[ab, g,h]-
6[a, g, h] - [6, g, h]a.
Since [a^g.h] = [b^g^h] — \ab^g^h\ = 0, we have f{a^b^g^h) = 0 and hence also (1) 0 = f{a,h,g,b)
= [ah,g,b]-
h[a,g,b]-
[h,g,b]a = [(^h,g,b]-
h[a,g,b],
because / is alternating. Since (g^a^b) 7^ l-^ g-, a and b do not associate, so [g^a^b] 7^ 0 and therefore [a,5,6] 7^ 0. Thus (1) shows that [a/i,p,6] 7^ 0; that is, (ah^g^b) 7^ 1. Similarly, we may assume that f(h,k^g^a) = 0 and conclude that (ah^g^k) 7^ 1. Thus {g,h,k) = {ah,g,k)
= {ah,g,b) = {g,a,b).
We conclude that any two associators which are not the identity and which have at least one element in common are equal. Finally, suppose that (g^h^k) 7^ 1 7^ (a^b^c) and that the sets {g^h^k} and {a^b^c} are disjoint. Considering f{g, h, k, a) = [gh,fc,a] - h[g, k, a] - [h, k, a]g,
114
IV. RA LOOPS
we see that each associator on the right has an element in common with both (g^h^k) and (a^b^c)^ so, if any of these associators is not zero, we obtain (g^h^k) = (a^b^c) as above. Thus we may assume that each of the associators [^/i, fc,a], [g^k^a] and [h^k^a] is zero and hence that f{g^h^k^a) = 0 too. Since / is an alternating function, 0=
f(g,a,h,k) = [ga, h, k] - a[g, h, k] - [a, h, k]g = [ga, h, k] - a[g, /i, k].
Since [g^h^k] ^ 0, we have [ga^h^k] ^ 0, so (ga^h^k) ^ 1. Similarly we can show that (ya, b^c) ^ 1 and so (a, 6, c) = (ga, 6, c) = (ga, h, k) = (g, h, k), giving the desired result.
D
In §11.1.3, we observed that a finite loop L is the disjoint union of right cosets of a subloop H if and only if H{hx) = Hx for all x G /> and all /i G H. Whether or not Lagrange's Theorem is true for Moufang loops is an open question, but the issue is clear for RA loops. 1.9 Corollary. If H is a subloop of an RA loop L, then L is the disjoint union of cosets of H and hence, if L is finite, the order of L is divisible by the order of H; thus, RA loops satisfy Lagrange's Theorem. Let h and hi be elements of H and let x ^ L. Then either hi{hx) = {hih)x^ or else h\{hx) = {hh\)x^ by Theorem 1.1. In either case, hi(hx) C Hx. Thus H(hx) C Hx. Conversely, let h^hi ^ H and X ^ L. If (/ii/i~\/i,x) = 1, then, by diassociativity, hix — {h\h~^ • h)x = {hih~'^){hx) e H{hx). On the other hand, suppose (/ii/i~\/i, x) 7^ 1. Then {hih~^){hx) = {h ' hih~^)x by Theorem 1.1. Also (h^hi) ^ 1; otherwise, (/i,/ii,x) = 1, by Theorem 1.8, so, by Moufang's Theorem, h, hi, x would generate a group, a contradiction. By Theorem 1.1, L' = {l,'^} for some central element s. Since (/i,/ii) 7^ 1, we have (/i,/ii) = s e H. Using the centrality of s, we obtain hix = s(hhih~^)x = s{hih~^){hx) € H{hx). D PROOF.
1. BASIC PROPERTIES OF RA LOOPS
115
A loop identity of the form {xy)(zx^) = (x • yz)x^^ where A; is a positive integer, is called an Mk-law, Any Moufang loop satisfies the Mi-law, which is just the middle Moufang identity. 1.10 Corollary. An RA loop satisfies the M^-law: {xy){zx^) = {x • yz)x^. PROOF.
Let x, y and z be three elements of an RA loop L. Then
(x ' yz)x^ = {(x • yz)x}x^
by diassociativity
= (xy ' zx)x^
by the middle Moufang identity
= (xy)(zx ' x^)
since x^ is in the nucleus of L
= {xy)(zx^)
by diassociativity.
D
0 . Chein has shown that Lagrange's Theorem holds in any finite loop which satisfies the Ma-law [Che73]. Thus we have further confirmation of the validity of Lagrange's Theorem for finite RA loops. Recall that in loop theory, a subloop ^ of a loop L is normal if, for any x^y ^ L^ Hx — xH and any two of the relations {Hx)y = H{xy),
{xH)y = x{Hy),
x{yH) = {xy)H
are valid. (See Definition II. 1.6 and ensuing remark.) The following corollary is therefore of interest. 1.11 Corollary. A subloop H of an RA loop L is normal if and only if xH = Hx for all x G L; equivalently, if and only if H is central or L' C H. P R O O F . We show that for RA loops, the relation (Hx)y = H{xy) for all x^y follows from Hx = xH for all x, and leave it to the reader to modify our argument in order to verify that x{yH) = {xy)H also holds for all x^y. So assume Hx = xH for all x and let x^y ^ L and h £ H, By Theorem 1.1, either {hx)y = h{xy) or {hx)y — x{hy) = {xy)h — h'{xy) for some h' G H, In either case we obtain {Hx)y C H(xy), Similarly, if h{xy) ^ {hx)y^ then h{xy) — {xh)y = {h'x)y for some h' G H^ so H(xy) C (Hx)y. Let ^ be a subloop of L. If H is central, clearly H is normal. Suppose V C H. By Theorem 1.1, V = {l,^} for some central element s. For h £ H and X G //, write hx = {xh)c^ where c = (h^x) G H. Since c is central, by
116
IV. RA LOOPS
Theorem 1.8, {xh)c = x{hc)^ we obtain Hx C xH. To obtain the other inclusion, write xh = (hx)c = c(hx) = {ch)x^ by centrality of c. Conversely, if H is normal in L but not central, let x ^ L and h £ H he such that s = (/i, x) 7^ 1. Then hx = (xh)s = x(hs)^ since s is central. By normality, hx G xH^ so hs and hence also s are elements of ^ . Since L^ = { l , ^ } , the result follows. D Loops (Z/, •) and (M, •) are said to be isotopic and each is a loop isotope of the other, if there exist bijections a^/3^j: L ^^ M such that {xai^y/3) = (x • y)j for each x^y £ L, The notion of isotopy does not arise in group theory because if two groups are isotopic, then they are isomorphic. (See, for example, [Pfl90, Lemma IL3.1, p. 48].) There are, however, loops which are not groups but which are also isomorphic to all their loop isotopes. In the hterature, such loops have often been called "G-loops". Thus, a G-loop is a loop which is isomorphic to all its loop isotopes. Since a Moufang loop which satisfies an M^^-law with k ^ \ (mod 3) is a G-loop [Pfi90, Theorem IV.4.11, p. 105], we obtain the following additional property of RA loops. 1.12 Corollary. An RA loop is a G-loop. 1.13 Theorem. Let L be a loop which is not associative. Then L is an RA loop if and only if it contains a central element s of order 2 such that, for all x^y^z ^ L, (i) if x.,y.,z associate in some order, then they associate in all orders; and (ii) if (x, y,z)^l, then (x, t/, z) = (x, y) = (x, z) = {y, z) = s. Suppose L is an RA loop. Statement (i) is an immediate consequence of Moufang's Theorem. For statement (ii), note that if x, y and z do not associate, then, by Theorem 1.8(iv), (x,y) ^ 1, (x^z) ^ 1 and (y^z) 7^ 1. By Theorem 1.8(v), there exists a central element s of order 2 such that L' = {1,^}, so PROOF.
(x, y,z)i^\
implies (x, y, z) = (x, y) = (x, z) = (y, z) = s.
Conversely, suppose L is a loop which is not associative and which contains a central element s of order 2. Suppose that statements (i) and (ii) hold for all x^y^z e L. In order to show that L is an RA loop, it is enough, by
1. BASIC PROPERTIES OF RA LOOPS
117
Theorem 1.1, to show that \i {x^y^z) ^ 1, then y -xz = xy - z = x - zy. But xy'Z
= (x' yz)(x, y, z) = (x - yz)s = (x • zy)s^ = x- zy,
the third equality holding since (y, z) = s is central. Also, xy ' z = (yx ' z)s = (y • xz)s^ = y • xz, completing the proof.
D
There are two other useful identities which hold in any RA loop. 1.14 Theorem. Let L be an RA loop. Then for any w, x, y, z in L, (i) (xy,z) = (x,z){y,z){x,y,z)
and
(ii) (xy, z, w) = (x, z, w){y, z, w). For (i), we proceed as follows, using liberally the fact that all commutators and associators in L are central: PROOF.
xy'Z={x
'yz){x,y,z)
= (xz ' y){x, z, y){y, z){x, y, z) := (zx • t/)(x, z){x, z, y){y, z)(x, t/, z) = (z • xy){z, X, y){x, z){x, z, y){y, z){x, y, z). Thus (xy,z) = (z,x,y){x,z)(x,z,y)(y,z){x,y,z) giving (i) because the associators {z,x,y), {x,z,y) and {z,x,y) are equal elements with square 1. (Using the notation of Theorem 1.13, each is 1 or s). For (ii), we recall that, for any z,w ^ L, the inner map R(z,w), which is defined by R{z,w) =
R{z)R{w)R{zw)-\
is a pseudo-automorphism of L with companion c = (2:, it;), because L is Moufang. (See Theorem II.3.3.) Thus, for any x,y £ L, [xR{z, w)][yR{z, w)'c] = [{xy)R{z, w)]c. Since L is RA, the commutator c is central; thus, upon cancelling c, we see that R(z,w) is an automorphism of L,
118
IV. RA LOOPS
Let x^y^ZyW G L, Since (x^z^w) is central, xR{z^w)
= {xz • w){zw)~^
= (x • zw)(zw)~^{x^z^w)
=
x{x^z^w)
and, similarly, yR{z,w)
= y(y,z,w)
and {xy)R{z,w)
=
{xy){xy,z,w).
Thus {xy){xy,z,w)
= {xy)R{z,w)
=
= x{x,z,w)y{y,z,w)
[xR{z,w)][yR{z,w)] =
{xy)(x,z,w)(y,z,w)
and, after cancelling xy^ we obtain (ii).
D
Recall that a Moufang loop is an inverse property loop (Theorem II.3.1) so, by Corollary II.2.3, the subloop test for Moufang loops is the same as it is for groups. 1.15 Corollary. If L is an RA loop and z^w £ L, then C{z) = {xe
L \{x,z)=
1}
and Af{z,w) = {x e L \ {x,z,w)
= 1}
are subloops of L. If x commutes with ^, so does x~^, If x and y commute with z^ then {x^y^z) r= 1 by Theorem 1.8(iii), so xy commutes with z by Theorem 1.14. Thus C{z) is a subloop. li x^y € M{z^w)^ then {x^z^w) = (y^z.w) = 1, so, by Theorem 1.14, {xy^z^w) = {x^z^w){y^z^w) = 1. Observe also that x, z and w associate if and only if x~^, z and w associate (by Moufang's Theorem) so, because there are only two associators in L, x € Af{z^w) implies x~^ G Af{z^w). Since M{z^w) is closed under products and inverses, it is a subloop too. D PROOF.
2. RA LOOPS HAVE LC
119
2. R A l o o p s have L C In §111.3, we introduced the "lack of commutativity" (LC) property. A Moufang loop L has property LC if, whenever x and y are elements of L which commute, then at least one of x, y, xy is central. If L is an RA loop, y'^ is central, so y~^ € Z{L)y^ thus the condition xy 6 Z{L) is equivalent to the condition x E Z{L)y. The following theorem is the key to a fundamental description of an RA loop [CG86]. 2.1 Theorem. An RA loop has property LC. Let L be an RA loop. As is often the case, we assume throughout this proof that L is contained in an alternative ring RL with char R ^ 2, First, we record the obvious relationship between the ring and loop associators of three elements in L (and use the fact that loop associators are central): PROOF.
(2)
[a,6,c] = ( l - ( a , 6 , c ) ) ( a 6 - c ) .
Let X and y be noncentral commuting elements of L. We prove that xy is in the nucleus of L and hence central. By Theorem 1.14(ii), this can be accomplished by proving that (a:,z,tx;) = (y^z^w) for all z,t(; G //, since there is only one nonidentity associator in an RA loop and this element has order 2. Suppose that z and w are any elements in L with (z,it;,x) ^ 1. Since x and y commute and hence associate with any third element (Theorem 1.8(iii)), it is apparent that f(z, w,x,y)=
[zw, X, y] - w[z, x, y] - [w, x, y]z = 0,
where / denotes the Kleinfeld function (see Definition 1.1.4). Since / is an alternating function and [y, w^ x] = 0, we have 0 = /{z^ y, ti;, x) = [zt/, w^ x] — y[z^w,x]. Using (2) to rewrite this in terms of loop associators and noting that (zy^w^x) = {z^w^x){y^w^x) = (z^w^x) (see Theorem 1.14(ii)), we obtain {zy • w)x + (z. It;, x)y{zw • x) = y{zw • x) + (z^ t/;, x){zy • w)x. Since {zy'w)x is in the support of the left side here, it is also in the support of the right side. Since (z^w^x) ^ 1, we must have (zy • w)x = y{zw • x)
120
IV. RA LOOPS
and, since x and y associate with every third element, y{zw - x) = {y - zw)x. Thus (zy • w)x = {y • zw)x and, after cancelling x, we obtain zy -w = y - zw. If 2r, w and y associate, then zy = yz; that is, {z,w,y)
= 1 implies {y,z) = 1.
Since w and z are interchangeable in this argument, we have also (z^ w^y) = 1 implies (t/, w) = 1. Summarizing the situation, we have shown that if x and y are noncentral commuting elements of L and (^, it;, x) / 1 for some z ^ L^ then (t/, 2r) = 1 if and only if {z^ w^y) = 1 if and only if (?/, it;) = 1. We assert that y and z cannot commute and thus (z^w^y) ^ 1. Suppose, to the contrary, that y and z (hence also y and w) do commute. Since y is not central, there exists u £ L such that yu ^ uy^ by Theorem 1.8(ii). Now if (w^u^z) / 1, applying to y and z what we have just learned about commuting elements, we discover that (y. It;) = 1 if and only if (it;, u,y) = 1 if and only if (y, i^) = 1. But (y^u) / 1, while (w^u,y) = 1 by Theorem 1.8(iv), since y and it; commute. The obvious conflict can be resolved only if (it;,^^,^) = 1. This shows that w and z associate with every element u which does not commute with y. Now the set C{y) of elements which commute with y is a subloop of L, by Corollary 1.15, and proper, because y is not central. Since L is generated by the complement of a proper subloop (recall Remark 1.7) and since (a6,it;,z) = (a,it;,>2:)(6,it;,2r), we see that (w^z^u) = 1 for all u £ L^ contradicting (z^w^x) ^ 1. We have indeed proven that y and z do not commute and, as asserted, that (2:,it;,y) 7^ 1. Thus {^z^w^x) 7^ 1 implies (z^w^y) ^ 1. The other implication follows by symmetry of x and y. Since there are only two distinct associators in an RA loop, it follows that (x^z^w) = (y^z^w) for all z and it;, and this is precisely what we wanted to show. D 2.2 Corollary. A Moufang loop which is not associative is an RA loop if and only if it has property LC and a unique nonidentity commutator.
2. RA LOOPS HAVE LC
121
P R O O F . In Theorems 1.8 and 2.1, we have shown that an RA loop has the desired properties. For the converse, let 2/ be a Moufang loop with LC and a unique nonidentity commutator, s. Since s~^ is also a commutator, s has order 2 and s is central because all commutators in a Moufang loop with LC are central. We now prove that L is RA by verifying statement (ii) of Theorem 1.1. (Condition (i) holds in any Moufang loop.) Suppose then that a:, y and z are three elements of L which do not associate. Then no two of these can commute: for example, if xy = yx, then either x G >2(iy) or y G 2^{L) or a; G Z{L)y and, in any of these cases, x, y and z must associate (in the case x G Z{L)y^ by diassociativity), a contradiction. Thus (a:,t/) = {x^z) = (y-,z) = s and no product of any two of x, y, z can be central. Furthermore, no one of x, i/, z can commute with the product of the other two: for example, if xy commutes with z, then either xy or z is central (both possibilities already eliminated) or 2: G Z{L)xy^ again implying that x, y and z associate, because of diassociativity. In particular, (xy^z) = (xz^y) = 5. The left Moufang identity—z{x'zy) = (zx'z)y—yields
z{x • zy) — (x, z){xz • z)t/ = s{xz^^y — sz^{xy) = sz{z ' xy) = s^z(xy • z) = z(xy • z). Cancelling z, we obtain x-zy = xy-z. Similarly, the right Moufang identity— {xy ' z)y = x{y • zy)—gives {xy ' z)y = sx{y • yz) = sy'^xz = sy{y • xz) = y{xz - y) = {y - xz)y, and hence xy - z = y - xz,
D
2.3 Corollary, An RA loop L is an extension of its centre Z by C2 X C2 X C2, where C2 denotes the cyclic group of order 2; that is, LjZ = C2 X C2 X C2. For any elements a^b in L which do not commute and any u E L such that a, b and u do not associate, the subloop of L generated by Z, a and b is a group, G, and L is the disjoint union G U Gu. Since Z contains all commutators and associators of L and since squares in L are central, L/Z is an elementary abelian 2-group. Suppose that X, y, Wy u are four noncentral elements of L and that L/Z contains the direct product (xZ) X (yZ) X (wZ) X (uZ), Because of LC, no two elements of {x^y^w^u} can commute. Furthermore, no three elements of PROOF.
122
IV. RA LOOPS
this set can associate. For example, if x, y and w associate, then (xy^w) = (x^w){y^w) = s'^ = 1 (Theorem 1.14) so, by LC, either xt/, w ox xy - w is central, each possibility contrary to fact. Now repeated use of Theorem 1.14 gives (xy^wu) =
(x^wu){y^wu){x^y^wu)
= {x,w){x,u){x,w,u){y,w){y,u)(y,w,u){x,y,w){x,y,u)
= ^^ = 1.
Thus xy and wu commute, which is not true. We conclude that L/Z is the direct product of at most three copies of C2. If it were the direct product of fewer than three, then L would be generated by its centre and at most two other elements. Then diassociativity (and the definition of centre) would force L to be a group. Thus L/Z = C2 X C2 x C2 as claimed. By diassociativity and the definition of centre, the subloop G generated by Z and any two elements a and 6 is a group. If t/ is a third element such that (a^b^u) ^ 1, each cyclic subgroup (aZ)^ i^^)-) (^^) ^f ^1^ has order 2 and the product {aZ) x {bZ) x {uZ) is direct because of the LC property. So this direct product must be L/Z and it follows that L = G U Gu. D 2.4 Corollary. Let L be an RA loop and let B be a commutative subloop of L. Then, for any x £ L, the subloop ( 5 , x ) generated by B and x is a group. In particular, B is an abelian group. P R O O F . Let 61,62 G B, Since 6162 = 6261, either 61 is central or 62 is central or 6162 is central. In the first two cases, (61,62,2:) = 1 is clear while, if 6162 = >2^ is central, then (61,62,0:) = (z6^\62,x) = 1 because of diassociativity, the definition of centre and the identity
(3)
{xy, z, w) = (x, z, w){y, z, w)
which holds by Theorem 1.14. Now a straightforward induction argument making further use of (3) gives the result. D 3. A d e s c r i p t i o n of an R A l o o p We now show that loops of the form M(G, *,5fo)? which we introduced in §11.5.2 and discussed at length in Chapter III as examples of RA loops, are not as special as it might there have seemed.
3. A DESCRIPTION OF AN RA LOOP
123
3.1 Theorem, / / L is an RA loop with commutator-associator subloop V = {1,5}, then L = M(G,*,5o) where G is any group containing 2(L) and two noncommuting elements of L, go is a central element of G and the involution g ^-^ g* is defined by (4)
9* = {
g
ifgeZiG)
sg
ifgiZ{G).
Converselyf for any nonabelian group G with LC, a unique nonidentity commutator, s, and involution *, the loop M(G^^^gQ) is an RA loop for any go G Z{G)j where * is given by (4). Suppose a and b are two elements of an RA loop L which do not commute and G is a group containing 2{L)^ a and 6. By Theorem 1.8, there exists u £ L such that a, 6 and u do not associate. In particular u ^ G. By Corollary 2.3, L = Gi U G\u^ where Gi is the group {Z(L)^a^b). Since Gi C G, it follows that Gi = G and L = G U Gu. Note that s e Z{L) C G. Let g^h ^ G. We claim that PROOF.
(5)
(p, h) = I \f and only if (g^h^u) = 1.
That {g contradiction. Next, for g £ G, define ^* = ugu~^. If y 6 G is central, clearly, ^* = g. Suppose g is not central. Then, since gu is not central {gu ^ G), gu ^ ug (by LC), so (^, u) = s and g* = sg. All this shows that y* is the element of G given by (4). Moreover, it follows from the LC property and Theorem III.3.3 that g ^-^ g* defines an involution of G. Now we determine how the elements of L = G U Gu multiply. First, for g^h ^ G^ we have g 'hu = {gh- u){g, h, u) = hg - u,
124
IV. RA LOOPS
since (g^h^u) = {g^h) by (5). Secondly, gu'h
= (g ' uh){g,u,h)
= (g - h*u){g,u,h) = {gh^ • u){g, /i*, u){g, u, h) = gh* - u
because h* = h or sh implies that (g^h*^u) = {g^h^u). Finally, {gu){hu) = g{u • hu){g^ n, hu) = g{uh • u){g^ n, hu) =
by the flexible identity
g{h*u^){g,u,hu)
= g{h*u^)(g^u^h)
by Theorem 1.14(ii)
= {gh*)u^(g^u^h)
since u^ is central
= /i*5 • '^^
since {g, K") = (g, h) = {g, u, h).
Setting go = u^ £ ^{G), we have shown that multiplication in L is given by g(hu) {gu)h {gu){hu)
= {hg)u = {gh*)u = goh^'g.
Thus L is precisely the loop M(G, *,^o)The last statement of the theorem is part of Corollary III.3.4.
D
In §V.4, we shall enumerate all the RA loops of order not exceeding 64.
Chapter V
The Classification of Finite R A Loops
In this chapter we classify all finite RA loops which are not direct products of proper subloops and give an explicit presentation for each such loop. As we shall see, this will permit a classification of all finite RA loops, up to isomorphism. We finish the chapter by listing the RA loops of order less than or equal to 64. The results in this chapter are based primarily on work of E. Jespers and G. Leal [JLM95, LM93].
1. Reduction to indecomposables An element of finite order in a (possibly associative) Moufang loop is said to be a torsion element. A Moufang loop is a torsion loop if all its elements have finite order and a 2-loop if all its elements have order a power of 2. Recall that a Moufang loop is diassociative and hence, if it is finite, has order divisible by the order of each of its elements. (See Corollary II.3.7 the discussion in §11.1.3.) It follows that a finite Moufang loop whose order is a power of 2 is a 2-loop. Conversely, if L is a 2-loop, then L is centrally nilpotent by a result of G. Glauberman and C. R. B. Wright [GW68]. Since the central factors of a centrally nilpotent loop are groups, the central factors of L must have 2-power order and hence L also has 2-power order. Thus, as with groups, finite Moufang 2-loops are precisely those Moufang loops whose order is a power of 2. 1.1 Proposition. Let L be a (possibly associative) Moufang loop with central squares and an order 2 commutator-associator subloop. For any prime p, the set Lp of elements of L whose order is a power of p is a normal subloop of L. The set L2' of elements of odd order in L is an abelian group 125
126
V. T H E CLASSIFICATION O F FINITE RA LOOPS
which is central as a subloop of L, If T is any torsion subloop of L, then T ^ T2 X Tr, where T^ = T n L2'. Suppose that x is an element of odd order 2n + 1 in L. Then x"^ = (x^)^ G >2(i/), so the sets i/2' and Lp, for p odd, are central subloops of L and hence abelian groups. Denote by s the unique commutator-associator of L and recall, by Corollary III.1.5, that s is central of order 2. If x and y are any two elements in i/, then, for any n > 0, PROOF.
{xyT
^UyTl
if n = 0 or 1 (mod 4)
sx'^y'^
if n = 2 or 3 (mod 4)
Thus, for t >2, (xy^^ = x'^^y'^\ It follows from Corollary II.2.3 that L2 is a subloop of L, To show that L2 is normal, it is sufficient to show that XL2 = L2X,
(L2x)y - L2{xy)
and
2/(^^2) = (2/^)^2
for any x^y ^ L. (See Definition II.1.6 and ensuing remark.) Each of these equations is an easy consequence of the definition of L2 and properties of s. To prove that {L2x)y — L2{xy)^ for example, let a £ L2 and note that {ax)y = a(xy) or s(a'Xy) = {sa){xy) according as the associator {a^x,y) is 1 or s. Let T be a torsion subloop of L. As above, T2 and T2' are normal subloops of L. Clearly T2 D T2/ = {1} and T = T2T2', so T ^ T2 X T2', U 1.2 Corollary. Let L be a torsion RA loop. Then the set L2 of elements of L whose order is a power of 2 is an RA 2-loop and a normal subloop of L. The set L2' of elements of L of odd order is a central subloop of L. Furthermore L — L2 X L2/. Because of Theorem IV. 1.8, an RA loop satisfies the hypotheses of Proposition 1.1, so virtually everything foUows from this proposition. We note only that since L2/ is central, it is a group, whereas L = L2 X L2' is not. Thus L2 is not associative, so it is an RA loop. D PROOF.
A loop L is indecomposable if it is not the direct product of two proper subloops. 1.3 Corollary. An indecomposable torsion RA loop is an RA 24oop.
1. REDUCTION TO INDECOMPOSABLES
127
1.4 Corollary. An indecomposable torsion group with central squares and a unique nonidentity commutator is a 2-group, 1.5 Remark. It is known that if L is a Moufang loop with central squares and a unique nonidentity commutator 5, and if L is not associative, then s is also a nonidentity associator in L, so V has order 2 [CG90b]. Thus Proposition 1.1, which as stated is sufficient for present purposes, actually holds under somewhat more general conditions. The principle result of Chapter IV, embodied in Theorem IV.3.1, showed that RA loops are loops of the form M(G, *,^o), where G is a nonabelian group, * is a particular involution on G and ^o is an element in the centre of G. 1.6 Proposition. Let L = M(G, *,^o) ^^ ^^ ^ ^ loop and let A be an abelian group. Then {g^aY = (^*,tt) defines an involution on G X A and M{G X A^ *, (^0,1)) i^ o,n RA loop which is isomorphic to M{G^ *?5^o) X A. Conversely, if A is an abelian group such that M(G X A^ *, {go^ 1)) is an RA loop for some nonabelian group G and some go G 2 ( G ) , then the restriction of ^ to G is an involution on G and M{G X A, *, (po, 1)) — ^ ( G , *,yo) X A. P R O O F . For the first part, let s denote the unique nonidentity commutator of G. Clearly G x A is a nonabelian group with LC and a unique nonidentity commutator, (5,1), so M{G X ^4, *,yo) is an RA loop, by Corollary IV.2.2. If V is an indeterminate such that M{G X yl,*,(^o?l)) = (G X yl) U (G X A)v^ then it is a simple matter to verify that the map M{G X >l,*,(^o, 1)) -^ M(G, *,po) X A defined by
(1) {g,a)v ^ (gu.a) is an isomorphism. Conversely, given an abelian group A such that M{G X ^5*5(^051)) is an RA loop, it is clear that G has LC and that the unique nonidentity commutator of G X A is of the form (5,1), where 5 is a unique nonidentity commutator in G. By Theorem IV.3.1, the involution on G X A is defined by I (^f, a) (^?^)
=
if (y, a) is central
\
I (5, l)(y,a)
otherwise.
128
V. THE CLASSIFICATION OF FINITE RA LOOPS
SO the restriction of * to G is an involution on G. Again, it is straightforward to check that the map (1) defines an isomorphism M{G x i4,*,(po, 1)) -^ M{G,^,go)x A. n We now show that, if L = M(G,*,^o) is an indecomposable RA loop, then the group G is "almost" indecomposable. By o(x), we mean the order of an element x in a Moufang loop. 1.7 Theorem. Let L = M(G', *,^o) ^^ o, finite indecomposable RA loop. Then G = D X C where D is an indecomposable 2'group and C is a cyclic group which J if nontrivial, is a 2-group. If C is nontrivial, then go = dc with d e Z{D) and 1 7^ c G C. First write L = G UGu where u ^ G and u^ = go e Z{G). By Corollary 1.3, Z is a 2-loop, so G is a 2-group. Since G is finite nonabelian, we can certainly write G = D X A for groups D and A with D indecomposable and nonabelian. Since G has a unique nonidentity commutator, A must be abelian. If A is trivial, there is nothing to prove, so assume A ^ {1}. Writing ^0 = da^ we note that a 7^ 1; otherwise, by Proposition 1.6, L = M{D X ^,*,(^o? 1)) = ^ ( - D , *,5ro) X A is not indecomposable. Therefore, writing PROOF.
A = {^1) X •••X {tk), for some ti of order 2^' > l , i = l , . . . , / : , we can write go in the form go =
dt{''''tl\
for some 5^ > 0. We claim that ^0 can be chosen in such a way that each exponent Si is 0 or 1. To see this, we consider two possibilities. If s\ is even, let U\ = ut^ ^ and g\ = u\. Noting that the t^ are central in G and hence central in L (see Corollary III.3.5), we have 51 = u ^ i f ' -^> = dt\' • • • tinp If 61 = 2g + 1 is odd, set ui = ut^ gy = uHT"-^'
-^' = dt^.tf,' •••ti".
^ and gi = u{. Then
= dt1'+'f,^ • ••tlHr-''
= dt\tl^ •. -tl'.
1. REDUCTION TO INDECOMPOSABLES
129
In each case, we have found an element Ui ^ G such that gi = uf = dt{H'^^'-fj^ with 6i e {0,1}. Note that L = M(G,*,^i), by Corollary IV.2.3 and Theorem IV.3.1. Repeating for general i the above argument for the case i = 1, we realize our claim. Remember that when ^o is written ^o = da with a £ A^ then necessarily a 7^ 1. Thus there is no loss of generality in assuming that ^o = dti- - -t^^ where I < r < k and o(/i) > ••• > 0(^7.). Let C = {h'-'tr) and B = {/2> X • • • X {tk). Then A = CxB
a^nd G = DxCxB.
Since go e D x C
and B is abelian. Proposition 1.6 shows that L = M(D X C, *,^o) X B. Indecomposability of L implies that B = {l}^ so G = D x C. D The next result plays a key role in the classification theorems of the next section. It also allows us to show that certain groups which arise in the proofs of these theorems are indecomposable. 1.8 Theorem. A finite group G has property LC and a unique nonidentity commutator, s, if and only if G = D x A where A is an abelian group and D — {x^y^Z{D)) is an indecomposable nonabelian 2-group such that . Z{D) =
{ti)x{t2)x{ts),
• o(/,) = 2^» fori=
1,2,3, mi > 1, m2,m3 > 0,
• s = {x,y) = tp~'
e {ti),
• x'^ e (ti) X {^2} 0,'rid •
y^e{t^)x{t2)x{t3).
Suppose that a group G is of the form G = D x A with A abelian and D as described. Then Z{G) = 2{D) X A and G/Z{G) = D/Z{D) = {xZ{D)^yZ(D)). Since x'^ and y'^ are central and D is nonabelian, G/Z{G) = C2 X C2, so G has LC and a unique nonidentity commutator, by Proposition III.3.6. The import of this theorem is the converse. Thus, suppose that G is a finite group with LC and a unique nonidentity commutator; hence, by Proposition III.3.6, G/Z{G) = C2 X €2- Writing G as the direct product of indecomposable groups, it is clear that some factor D is nonabelian. By Corollary 1.4, i) is a 2-group. Also, G — D X A for some group A which is necessarily abelian because G has just one nonidentity commutator. Since Z{G) = Z(D) x A, we have D/Z{D) = G/Z{G) = C2 X C2, so there exist x,y e D such that D = {x,y,Z{D)). PROOF.
130
V. THE CLASSIFICATION OF FINITE RA LOOPS
Clearly the unique nonidentity commutator of G is also a unique nonidentity commutator for D^ and this element is (x^y). We claim that Z(D) is the direct product of cyclic groups with s = (x^y) in the first factor. For this, suppose that (2)
2iD) =
{h)x{t2)X''-x{tk)
is a decomposition of Z{D) in which s is the product of (nonidentity) elements from a minimal number of factors (ti). Without loss of generality, these factors are (^i),... , {U) for some £ > I. Let o(/^) = 2'^\ Since 5^ = 1, we have S — l-^
' ' 'll
,
where, without loss of generality, mi > m2. We show that 1=1. to the contrary that i > 1. Then
Assume
and
so that
contradicting minimality. Thus 5 G (^1), as claimed. Next we assert that we can, if necessary, replace x by an element Xi G xZ{D) and the generators U of Z{D) by new generators f- such that x^ G {t[) X (12)' For this, write x^ = ai • • -a/^ with a^ G (^i). If some ai is not a generator of (^^), then a~^ is also not a generator of (^^), hence a~^ = t^^ for some a. Let xi = /^x. Then xj = ai • • -o^ • • -a^ has no component in (ti). (As is conventional, the ^ denotes an omitted factor.) Repeating this process, if necessary, we see that we may assume that x^ = ai^ - - - a^^, where each {oiij) = {Uj)'> ^ind o(a^j) > • • • > o(a^^). If none of the a^^, I < j < i^ belongs to (ti), then {ti^) X • • • X (ti,) = {ai^ ' '-ai^) X {ai^) X • • • X (a,-,), with x'^ G (o!iiO;z2 * * '^ii)- Thus x^ G (^1) X (^2), with t'^ = a^^a^^ ' ' 'Cti^- In the case that one of the a^ 's belongs to (^i), a similar argument applied to the other factors of x^ gives the same conclusion.
2. FINITE INDECOMPOSABLE GROUPS
131
In a similar way, we can also find y ^ G and a factorization Z(D)
=
{h) X (^'2) X (^3) X • • • X (tk) such t h a t y'^ E {h) X {t'^) X (^'3). Thus Z{D)
=
(^1) X (^2) X (^3) X C for some abelian group C , so i^ = { ^ , y , ^ i , ^ 2 , ^ 3 ) x C . Since D was indecomposable, C — {1} and the proof is complete.
D
2. F i n i t e i n d e c o m p o s a b l e groups In this section, we show t h a t any finite indecomposable group with a unique nonidentity commutator and property LC lies in one of five distinct classes of groups. By Theorem 1.8, if D is such a group, then D =
{x^y^Z{D))^
where • 2(i^)=(/i)x(/2)x(^3), • o{ti) = 2^» for i = 1,2,3, m i > 1, m2, ma > 0,
• x^ = i^H^\
Oii > 0, and
• y' = t^r'4'4\fi,>0. We employ this notation throughout this section and begin by showing t h a t X and y can be chosen in a very convenient way. 2.1 L e m m a . Without
loss of generality,
x2 = f^(«>)t^("^)
and
we may assume
that
j/2 = t^^'^'hl^^'hl^'"'^
where, for an integer n, we define
{ PROOF.
0
if n is even
1
if n is odd.
The proof is similar to t h a t of Theorem 1.7. Defining
x'=
(A), is the fewest number of generators oi A, Theorem 1.8 implies t h a t the rank of the centre of an indecomposable finite group with LC and a unique nonidentity commutator is at most 3. To describe such groups, our strategy will be to consider the three possibilities for central rank. To prove t h a t certain groups are not isomorphic, we will use the fact t h a t the exponent of a Moufang loop is an isomorphism invariant. 2.3 D e f i n i t i o n . The exponent of a Moufang loop L is the smallest natural number n such t h a t x'^ = I for all a: G //, if such n exists; otherwise, the exponent is said to be infinite. 2.4 T h e o r e m . A finite group D is indecomposable identity two
commutator
with LC, a unique
non-
and centre of rank 1 if and only if D belongs to one of
families, V\:
Groups with
presentation {x,y,ti
\x^ = y^ = tp
= 1)
and V2: Groups with
presentation {x,y,ti
\x'^ = y'^ = / i , tl^'
= 1)
where, in each case, ti is central and s = (x^y) = tf"^^ distinct:
no group in V\ is isomorphic
PROOF.
. The families
are
to any group in 7)2-
Let D be a finite indecomposable group with LC, a unique non-
identity commutator and Z{D)
= (^1), o{ti) = 2^^ > 1. Since D/Z{D)
C2 X C'2, if r^i = 1, then |Z)| = 8, so D is isomorphic to either D4 = {x,y I x^ = y^ = l , ( x , t / ) = ^1) or Qs = {x,y\
x^ = 2/^ = ti,{x,y)
= ti).
Clearly D4 is in the family Vi and Qs is in the family V2.
^
2. FINITE INDECOMPOSABLE GROUPS
133
Suppose mi > 1. According to Lemma 2 . 1 , we can choose x and y in such a way t h a t x^ and y^ assume the values 1 or ^i. Thus we have four possible choices: (i) x^ = y^ = 1; (ii) a;2 = i i , t/2 = 1; (iii) x^ = l , y 2 = fj. (iv) x^ = y'^ = ti. Choice (i) gives a group in V^ and choice (iv) a group in 'D2. In case (11), setting y = xy we have y
= sti = q
ti = /^
"^ and, since the
exponent here is odd, Lemma 2.1 shows t h a t we can choose a generator y^^ such t h a t y"
= ^i- So we are dealing with a group in the family V2.
By
symmetry, case (iii) also defines a group in P2Let Z)i be a group in X>i and D2 a group in X>2 with \Di\ = | ^ 2 | - If TTii = 1, it is clear t h a t Di ^ D2 while, if m i > 1, it is easily seen t h a t the exponent of Di is 2^^ while the exponent of D2 is 2^^"'"^ so the groups are not isomorphic. Finally, since Di and D2 both have indecomposable centres, it follows t h a t Di and D2 are indecomposable.
D
2.5 T h e o r e m . A finite group D is indecomposable identity two
commutator
with LC, a unique
non-
and centre of rank 2 if and only if D belongs to one of
families, P 3 : Groups with
presentation
{x,y,tx,t2\x
=ti
=t2
=l,y
=t2)
and V4: Groups with {x,y,ti,t2
presentation \x'^ = ti,y'^ = t2,t'f
= tT^
= 1)
where, in each case, t\ and ^2 are central and s — {x^y) = t\ ^ PROOF.
Let D be an indecomposable group with LC, a unique noniden-
tity commutator and centre Z{D)
= l^ti) X (^2)? where o{ti) — 2^» > 1 for
i = 1,2. Then D — /?/(^2) is a group with LC and a unique nonidentity commutator and it is indecomposable because its centre is indecomposable. Thus D is in one of the two classes V\ or V2 described in Theorem 2.4.
134
V. THE CLASSIFICATION OF FINITE RA LOOPS
Denoting by x and y the cosets xZ{D)
and yZ{D)
respectively, we have
two possibilities, up to isomorphism: (a) x2 = y2 ^ T; (b) X 2 = : y 2 ^ ? i . Because of Lemma 2.1, case (a) can be lifted to D in four different ways: (a.i) x2 = 2/2 ^ i.^ (a.ii) x^ = 1, y2 ^ ^2; (a.iii) x^ = ^2, y^ = 1; (a.iv) x^ = t/2 _ ^2 Case (a.i) gives us the group (x, y , ^ i ) x ( t 2 ) , which is not indecomposable. Cases (a.ii) and (a.iii) give a group in V^, In case (a.iv), setting x' = xy, we have x'
— si\ — t^^
i\, \i m\ > 1,
Lemma 2.1 shows t h a t we can choose a generator x'' such t h a t x "
= 1 , in
which case Z) is in P 3 . If m i = 1, Lemma 2.1 shows t h a t we can choose x^' such t h a t x'' = ^i and D is a group in P 4 . In a similar way, again by Lemma 2.1, case (b) also lifts to D in four different ways: (b.i) x^ = y^ = U; (b.ii) x'^ = ti, y'^ = Ut2\ (b.iii) x^ zz t^t2, y'^ = ti; (b.iv) x^ = y^ = ht2. In case (b.i), the group is the direct product {x,y,^i) X (^2), which is not indecomposable. In case (b.ii), we can write Z{D) and t = t2 otherwise.
t h a t D is a group in V4. x^ — st\t2
= tl ^
— {t) x (ti/2) where / = ^1 if m i < m2
If m i < m2 and t = t i , setting ^2 = hh
shows
If m i > m2 and t = ^2? we take x' = xt/, so
^^t2 and, since m i > 1 in this case. Lemma 2.1 shows
t h a t we can choose a generator x" such t h a t x " = ^2- Setting t[ = ^1/2, we see again t h a t D is in V4. Case (b.iii) is similar to (b.ii). In case (b.iv), we take x' = xy so t h a t x'
= stit2
Lemma 2.1 shows t h a t we can find x" such t h a t x " and x " 2(D)
= ^1 if m i = 1.
= {ti) X {t\t2)
Suppose
TTII
> 1.
— ^1 ^ "'^^^l*
= 1 if m i
> 1
If also m i < m2, then
and D is a group in P 3 while, if mi > m2, then
2. FINITE INDECOMPOSABLE GROUPS
135
Z{D) = {tit2) X (^2) so that D = (x,2/,^i^2) X (^2)5 which is not indecomposable. On the other hand, if mi = 1, then 1 = mi < m2, x'' = ti and Z{D) = {ti) X (^1^2), so D is in V4. To settle the distinctness of P3 and P4, let D3 be a group in V3 and D4 a group in V4, In Z?3, the element x is noncentral of order 2 while there is no such element in D4. Thus the groups D3 and D4 are not isomorphic. Finally, we prove that the groups in V3 and V4 are indecomposable. Let Z? be a group in either of these classes. By Theorem 1.9, if D is not indecomposable, then D = H X K^ where H is nonabelian, Z{H) 7^ 1, K is nontrivial abelian and D' C H, Since Z{D) = 2{H) x A' = (^1) x (^2)? 2(H) must be cyclic. Let /i = ti^t2^ be a generator of Z{H)^ say of order 2 ^ Since 5 = tf' e Z{H) is the unique element of order 2 in Z{H)^ we must have
This implies ^2^ = 1 , so either a2 is even or r — 1 > m2. Since D = Z{D)[JxZ{D)[JyZ{D)[JxyZ{D) and Zf C D is not abelian, it follows that H contains elements from at least two noncentral cosets. In any of the possible cases, it is easy to see that there exists an element yi ^ H which can be written in the form y^ — yz^ with z G Z{D). Hence 1 ^ y2 _ ^^^2 ^ fj f) Z{D) = Z{H). So this element can be written in the form y^ = {t^HTT ^ r some n > 1. Since y'^ = t2 = y\z~^ = (^i'^^2^)^-2:"'^, we see that a2 cannot be even; hence r > m2. Therefore o(h) > 2^^ ^n^^ since (h) is a direct factor of Z{D) = {h) X (^2), it must be that o{h) = 2'^K Therefore ai is odd. Since the element t2Z^ is of the form t2t^^t^'^^ while an element of Z{H) is of the form ^f''^^2'''' we see readily that no element of the form t2Z^ can be in Z{H)^ a contradiction. D 2.6 T h e o r e m . A finite group D is indecomposable with LC, a unique nonidentity commutator and centre of rank 3 if and only if D belongs to the family P5; Groups with presentation {x,y,ti,t2,t3
I x^ = ^2,2/^ = ts.tT'
= tT^ = tT^ = 1),
where ti^ ^2 ^^^ ^3 ^^^ central and s — (x^y) = t^ ^
136
V. THE CLASSIFICATION OF FINITE RA LOOPS PROOF.
Let D be an indecomposable group with LC, a unique noniden-
tity commutator and centre Z{D)
= (ti) X (^2) X (^3) where o(/^) = 2^» > 1
for i = 1,2,3. Consider D = Dl{t^)^
which is a group described by Theo-
rem 2.5. So, denoting by x and y the cosets xZ{D)
and yZ{D)^
respectively,
we have two possible choices, up to isomorphism: ( a ) X'^ rz T, t/2 = t2\
(b) x^ = t,,t
= h-
Because of Lemma 2.1, case (a) lifts to one of the following subcases: (a.i) x^ = 1, 2/2 ^ ^2; (a.ii) x^ = 1, 2/^ = ^2^3; (a.iii) ^2 = ts, y^ = ^2; (a.iv) x'^ = ts, 2/2 ^ t2t3. Case (a.i) leads to the decomposable group {a:,y,/i,/2) X {^3).
Case
(a.ii) leads to the decomposable group (x,i/,/i,^2^3) X i^)? where ^ = ^3 if ^ 2 ^ ^^3? ^i^d t = t2 otherwise. Case (a.iii) gives D in the family P 5 . In case (a.iv), setting y^ = xy^ we have y^ = ^^2^3 = ^Y^^
^2^3- Apply-
ing Lemma 2.1, we see t h a t we can choose a generator y" such t h a t y'^ — ^2 if mi > 1, and y" = ^1^2 if ^ 1 = 1- In the first case, D is in the family P 5 and, in the second case, Z{D)
— {t\) x (^1/2) X (^3), so again we see t h a t D
is in the family P 5 . Case (b) also lifts to four subcases: ( b . i ) x2 = t i , 2/^ = ^2; (b.ii) x'^ = U, y2 3, ^2/3; (b.iii) x2 = ^1/3, 2/2 rzr ^2; ( b . i v ) x^ = ^1^3, 2/2 ^ ^2^3.
Cases (b.i) and (b.ii) define groups which are not indecomposable, these being, respectively, (x, 2/, ^1,^2) X (^3) and (a:, 2/, ^1, ^2^3) X (^), where t = t^ if 7722 > ^ 3 ^ and ^ = ^2 otherwise. In case (b.iii), if mi > m3, we can write Z{D) D is not indecomposable. If m i < m3, then Z{D)
= (hts)
x (^2) x (^3) and
= {ti) x (^2) X (^1^3) and
D is in the family P 5 . T
/I
• \
•
/
1
/
2
In case (b.iv), settmg y = xy, we have y
9
— st\t2t^
2"^1—^-1-1
— ^i
The proof of Lemma 2.1 shows t h a t we can choose y" G xyZ{D) t h a t 2/"
= ^2 if ^ 1 = I9 and 2/"
9
^2^3such
= ^1^2 if ^ 1 > 1- In the first case.
2. FINITE INDECOMPOSABLE GROUPS
137
we are back to (b.iii). To discuss the second case, assume first that mi > m2. Then we can write Z{D) = (^1^2) X (^2^3) X (^3) if 1712 > ms, and 2{D) = {tit2) X (^2) X (^2^3) otherwise. Set Di = (j/,/,^1^2,^2^3). Then either D = Di X {t^) or D =^ Di x {^2)7 respectively, and in both cases D is not indecomposable. Now assume mi < r7^2• If 'n^i > m.3, we can write Z{D) = (^1/3) X (^2^3) X (^3); thus D = {x,y,tit2,,t2h) X (ts) is not indecomposable. In the remaining case, we have mi < min{m2, 7723}, so Z{D) = {h) X {tit2) X {tits) and D is in the family P5. Finally, we shall show that a group D in P5 is indecomposable. Suppose it is not. Then, by Theorem 1.8, D = H X K with H indecomposable, s E H and K nontrivial abelian. A straightforward calculation shows that if nil = 1, then a central element t^t^t^ is the square of an element in D if and only if a = /37 (mod 2) (in which case, t^t^t^ — {x^y^Y)^ while, if TTii > 1, then f^t^t^ is the square of an element in D if and only if a is even (in which case t^t^^tl = {t^^~^^'^^'~'^''^xf^y^f.) Let N be the set of elements in Z{D) which are not squares of elements in D. Then
N
Ui4^3
\f^ + f^l odd}
{^?^2^3 I ^ odd}
if mi == 1 otherwise
and, in both cases, |A^| = 2'^i"*"'^2+m3-i \Ye consider each case separately. Suppose first that mi = 1. Since D — H X K with A^ nontrivial abelian, Z{H) has rank either 1 or 2. If the rank is 1, Z{H) is cyclic and, since 5 = ^1 is an element of order 2 in Z{H) but not a square, it must be that \Z{H)\ = 2. Since \H/Z{H)\ = 4, by Proposition III.3.6, the order of H is eight, so every central element in // is a square. This contradicts the fact that ti £ N, Therefore Z{H) has rank 2 and A^ = (k) is cyclic, say of order 2^. Write Z{H) = (si) X (52) with ^1 G (^i). Since ^1 has order 2 but is not a square, it follows that Si = ti^ so si = s. Thus the unique nonidentity commutator in H is not a square and so, by Theorem 2.5, H £ V3. If 5 is a group in V3 (with ti — s oi order 2), presented as in Theorem 2.5, it is readily seen that the elements of Z{B) which are squares of elements in B are of the form ^2 or ^1^2 ^^^h fi odd. Therefore, the only elements in Z{H) that are not squares of elements in H are those in the set Ni = {tis^ I /? even}. Let N2 = {k^ \ /3 odd}, the set of nonsquares in K.
138
V. THE CLASSIFICATION OF FINITE RA LOOPS
Then N = NiK U {Z{H)N2 \
NiNi),
a set of order \N\ = lo(s2)\K\
+ \Z(H)\2'-'
- io(.2)2^-^
= i 0(^2)2^+ 2 0(52)2^-1 - i 0(52)2^-1 =
lo(s2)2'-\
Since \N\ = 2^i"''^2+m3-i jg ^ power of 2, we have a contradiction. Now suppose 7721 > 1 2ind notice that if a, 6 G N^ then neither ab nor ab~^ is in N. Again K has rank either 1 or 2. If K = (ki) x {k^) is of rank 2, then /ji, ^2 ^ind /;:i/;;^^ are all in TV, a contradiction. On the other hand, if K = (k) has rank 1, then the elements in kZ{H) are not squares in D. Hence, by our remark, for any h G Z{H)^ h = (hk)k~^ is a square. Since k is not a square, k = ^i^'^^^2^3 for some a, /3 and 7. In particular o{k) > 2^^. Write ^1 — hk'^ for some h G Z{H) and n > 0. Since ^ z z ^ p - ^ ^ / , 2 - i - i ^ n 2 - i - ^ G ^ , we obtain A;-2"^^"' G i^; thus A;-^"^^-' = 1. Since o(A;) > 2^^, this implies that 2 | n and hence that t\ is a square, a contradiction. D As an illustration of the results of this section, we conclude by enumerating the groups of order less than or equal to 32 which have LC and a unique nonidentity commutator. Our notation is that of M. Hall Jr. and J. K. Senior [HS64]. (See also [TW8O].) The indecomposable groups with LC and a unique nonidentity commutator are described in Table 1. The others are listed below. Order 16: 16r2ai = D4 X C2, 16r2a2 = Qs X C2 Order 24: D4 X C3, Qs X C3 Order 32: 32r2ai = D4 X C2 X C2, 32r2a2 = Qs X C2 X C2, 32r26 = 16r26 X C2, 32r2Ci = 16r2Ci X C2, 32r2C2 = 16r2C2 X C2, 32r2d = 16r2d X C2, 32r2ei = D^x C4, 32r2e2 = Qg x C4 3. F i n i t e i n d e c o m p o s a b l e R A l o o p s The results of the previous section and a few additional remarks will show how to construct all indecomposable RA loops.
3. FINITE INDECOMPOSABLE RA LOOPS
139
1. The indecomposable groups of order less than or equal to 32 with LC and a unique nonidentity commutator TABLE
[ G D4
1 Qs IGTib l6T2d 16r2Ci 16r2C2 32r25 32T2h 32r2/ 32r2i 32r2ii 32r2i2 32r2A:
_\G\j
Z{G)
8 8 16 16 16 16 32 32 32 32 32 32 32
~^
y'
C2
1
1
C2
h
^1
CA
1
1
CA
h
^1
C2XC2
1
Ci. On the other hand, if mi > 1, let y' — yu. Then y'
= sti is an odd power of ti
so, by Lemma 2.1, there exists ^" such t h a t ?/" = ^i. Hence, in this case, L2 = M{{y",u,Z{L2)),^,x'^)
e L5. We shall soon see t h a t L5 C £1 u £ 2 .
If L3 G L3, then L3 = M{{x,u,Z(Ls)),^,y^)
e £3.
3. FINITE INDECOMPOSABLE RA LOOPS
141
2. Every finite indecomposable RA loop is in one of the classes described below TABLE
1
ZiD)
1
ih)
\
ih) ih) ih) ih) ih) ih) ih)
L2
La L4 L5 L5-
Le
p~ 5l
G
L«^ = 90
^1
i
1
Di
h
1
Di X ( w )
w
1
I>1 X {w)
h h h h
D2
1
D2
^1
D2 X {w)
w
D2 X ( w )
i 1
ih) X {t2) ih) X (i2)
t2
h
D3
u
t2
Dz X (w)
w
Lio
ih) X ih) ih) X ih)
t2
Dz X (w)
Lii
ih)
X (f2)
1, we set u' = xu. Then u' = sti and Lemma 2.1 shows that we can find u^^ such that n'' = ^1. So L5 = M ( ( x , y , 2 ( L 5 ) ) , * , w " ' ) e £ 2 .
142
V. THE CLASSIFICATION OF FINITE RA LOOPS
T A B L E 3 . T h e seven classes of finite indecomposable RA loops
1
Z{D)
1 1
(^i) ih)
L^4
{h)x
^2~
1
G
y'
u^ = go
1
1
1
^2
D3
h 1
^1
t2
Z?4
^1
D2
{t2)
]
i
1, set
x' = x?/. By Lemma 2 . 1 , there exists x " such that x" = ^1. Hence Lg =
M{{x\y,Z{Ls)),^y)eU If L9 G L9, then L9 = M{{y,u,Z{L^)),^,x'^)
G £5.
_ ^2"^iNext, suppose L\Q G LIQ. If 0(^1) > o(it;), then 5 = ^
2 ( L i o ) = (^1^) X (^2) X ( ^ ) and Lio = {x^y^u,t\w,t2) composable. If 0(^1) < o(tt;), writing t h a t Lio = M{{y,u,Z{Lio)),^,x^)
Z{L\Q)
xu.
x (ti;) is not inde-
— {t\) X (^2) X {t\w)^
we see
G £5.
If Lii G Lii, we have {xu)'^ — st\. x'^ = 1 and Lii = M{{x\y,
G (^iti;).
Z{Lu)),^,u^)
If mi = 1, let x' = xu. G £3. If ^ i
Then u^ = sti^ so there exists 1/" such that u"
Then
> 1, let u' =
= ^i. Hence Ln =
Mi{x,y,2iLu)),*,u"^)eC4. It is clear that L n ' C £4 a n d , if L12 is a loop in the class L12, then Lu = M{{y,u,Z{Lr2)),*,x^)
e £e.
We turn our attention to Li2> € \-\2'- Here m i = 1, so o ( / i ) < o{w), Z{Li2') = (ii) X («2) X (iiu;) and Lu' = M{{y,u,Z{Lu')),*,x^)
G A-
4. FINITE RA LOOPS OF SMALL ORDER
143
It is clear t h a t L13 C £ 5 , L14 C CQ and L15 C £ 7 . Finally, suppose t h a t (h) X (^2) X (^3) X (tiw)
L\Q
G
Lie- If 0(^1) < o(tt;), we write
Z{LIQ)
and see t h a t Lie = M{{x,y,Z{Lie)),^,v?)
If o ( t i ) > o(it;), then L\e = {x^y^u^tiw^t2',t^)
=
G £7-
x {w) is not indecomposable.
AU the above shows t h a t , up to isomorphism, there are at most seven classes of indecomposable RA loops, those classes denoted £ 1 , . . . , £ 7 in Table 3. Arguments similar to those of Section 2 show t h a t the loops in this table are aU indecomposable. Finally, we show t h a t loops in different classes are not isomorphic. Elementary consideration of the ranks of centres shows t h a t we need only prove t h a t £1 n £2 = 0, £ 3 n £4 = 0 and £5 D £5 = 0. If m i > 1, these statements are true because s and hence every element in the centre of a loop Li G £ t , i = 2 , 4 , 6 , is a square, while ^1 is not a square in a loop of £ 1 , £3 or £ 5 . If m i = 1, Li G £1 and L2 G £ 2 , then Li ^ Mi6(Q8,2) and L2 = ^leiQs)')
so Li ^ //2. (The loop Mie(Qs)
while there are nine elements of order 2 in
has a unique element of order 2 Mie{Qs^2).)
Suppose mi = 1, L3 G £ 3 and L4 G £4. Then 2(Z/4) = (ti) X (^1^2)- In Z/4 = L4/{tit2),
we have x'^ = u'^ = y'^ = ti and hence L4 = Mie{Qs)-
On
the other hand, all nonabelian quotients of order 16 of L3 are isomorphic to Mie{Qs',2)
because, in any such quotient, we have ^'^ = u'^ = 1 while, in
^leCQs)? the only element of order 2 is central. Similar arguments apply in the last case, m i = 1. If Le G £ 5 , then Le = Le/{{tit2)
x (^1^3)) — ^wiQs)
while, in any quotient of L5 G £5 by a
central subloop, u has order 2.
D
4. F i n i t e R A l o o p s of s m a l l order Table 3 provides a means of determining all indecomposable RA loops of a given order, up to isomorphism. Those RA loops which are not indecomposable are, as we have seen, direct products of an indecomposable RA loop with an abelian group. As an illustration, we enumerate all the RA loops of order 64 or less. In what follows, we assume previous notation. T h u s , when we write an RA loop L as M(G^ *?yo)9 we understand t h a t G is a nonabelian group with
144
V. THE CLASSIFICATION OF FINITE RA LOOPS
a unique nonidentity commutator, 5, go is a central element of G and * is the involution on G defined by
\
if sg
geZ{G)
otherwise.
If L is indecomposable, then G = D x A^ where D is an indecomposable group with D/Z{D)
^ C2 X C2 and A is abelian. If \L\ < 64, then \Z(G)\
0, m 2 , m 3 > 0. The unique commutator of G is s = tf^^
, which is in the first factor. Therefore, reordering of the second
and third factors leads to isomorphic groups. Once again, we employ the notation of M. Hall Jr. and J. K. Senior [HS64] to denote the groups involved in our construction and label the Moufang loops which arise with the notation of 0 . Chein [ C h e 7 8 ] . T h e smallest possible order for an RA loop is 16. In this case, \G\ = 8, so G = D4 OT G = Qs' There are two loops, both indecomposable: 1. M ( D 4 , * , l ) = M i 6 ( g 8 , 2 ) a n d 2. M ( g 8 , * , ^ i ) = MieiQs).
the Cayley loop.
If an RA loop has order 32, its centre, being of order 4, must be either C4 or C2 X C2. If Z{G) \i x^ = y^ = tuG
= C4 and x^ = y^ = 1, we have G = 16r26, while
= I6r2d.
If Z{G) = C2 x C2, the choice x^ = l,y^
gives G = 16r2Ci, while the choice x^ — t\^y^
= t2
— ^2 gives G — 16r2C2. The
corresponding indecomposable RA loops are 1. M ( 1 6 r 2 6 , * , l ) = M32(£^t,16), 2. M ( 1 6 r 2 ( i , * , / i ) - M 3 2 ( 5 , 5 , 5 , 2 , 2 , 4 ) , 3. M(16r2Ci, *, 1) - M32(16r2C2,16r2C2,16r2Ci, 16r2Ci) and
4. M(i6r2C2,*,/i) = M32(i6r2C2, i6r2C2, i6r24. i6r24). T h e only other RA loops of order 32 are direct products. 5. M32(C,9) = Mie{Qs) X C2 = M ( g 8 , * , ^ i ) X C2 = M{QsX C2,*,(/i,l))and 6. M32(g8 X C2,2) = M32{Q8 X C 2 , * , ( l , l ) ) = M{Qs,^,l) Mi6(g8,2)xC2.
XC2
=
4. FINITE RA LOOPS OF SMALL ORDER
145
4. The indecomposable RA Loops of order less than or equal to 64 TABLE
\w1
p"
y'
G
1
1
D4
z
1 161
i(G) C2
16 j
C2
h
h
Qs
32
C4
1
1
32
C4
h
h
16126 l6T2d
M{lQT2d,*,ti)
1
t2
16r2Ci
M(16r2Ci,*,l)
h
t. An algebra ^ G -4 is called 72-semisimple if TZ{A) = {0}. In the class of alternative rings, we consider three radicals: P , the prime (or Baer) radical, A/i7, the upper nil (or Kothe) radical and J^ the Jacobson radical, which is also known in the literature as the Smiley, Zhevlakov or Kleinfeld radical. We recall the definitions and some basic properties of these radicals, referring the reader to [Div65] and [ZSSS82] for proofs. The prime radical V{A) of an alternative algebra A is the smallest ideal of A such that A/V{A) is semiprime. It turns out that V{A) is the intersection of all prime ideals of A. The upper nil radical Mil{A) of A is the largest nil ideal of A. It is a basic fact that V{A) = J^il{A) is the set of nilpotent elements if A is commutative and associative. An element a £ A\s called quasi-regular if there exists an element b £ A, called the quasi-inverse of a, such that ab = ba = a + b. An ideal is said to be quasi-regular if aU its elements are quasi-regular. A left ideal / of A is said to be modular if there exists an element e £ A such that ae — a £ I for all a G A. The Jacobson radical J{A) of A is the largest quasi-regular ideal of A. This ideal is also the intersection of all the maximal modular left (or right) ideals of A. If A has a unity, then any left ideal is modular, so J{A) is the intersection of all maximal left ideals. The three above-mentioned radicals are related as follows: V{A) C Mil[A) C
J{A),
1. AUGMENTATION IDEALS
149
Furthermore, these radicals are hereditary; that is, TZ{I) = Ir\TZ{A) for any (two-sided) / of A, where TZ is any of the three radicals P , A/zV, J, (See, for example, [ZSSS82, Theorem 9, p. 167 and Corollary, p. 207].) Let L be an RA loop and R a commutative and associative ring. To determine a radical of the alternative loop ring RL^ we may often assume that R has a unity. We proceed to show why. By R^ we denote the natural extension of i2 to a ring with unity. Thus R^ = Rif R already has a unity; otherwise, R^ = RxZ = {{T^Z) \ r e R^ Z E Z } , with componentwise addition and multiplication defined by (^i, Zi)(r2,>2^2) = (^1^*2 +-^^1^2 + >2:2ri,>2ri2r2), for ^1? ^2 € R and zi^Z2 G Z. In particular, R^/R = Z. Since Z is semisimple for any of the three above-mentioned radicals 72., it follows that TZ{R) = TZ{R^),
1. Augmentation ideals Let i2 be a commutative and associative ring and let L be a (possibly associative) Moufang loop. For a normal subloop TV of L, the canonical map L -^ L/N lifts to an i2-linear ring homomorphism €f\j : RL -^ R[L/N] whose kernel we denote by An{L^N). If R is clear from the context, then we denote this ideal simply A(Z/,7V). In the special case that N = L^ the homomorphism €L:
is called the augmentation map
RL^
R
and denoted simply €. Thus, for a =
The scalar €(a) is called the augmentation of a. The kernel of €, which is called the augmentation ideal^ is denoted A(Z/) rather than A(L,Z/). For rings R without unity, we will sometimes abuse notation by writing rx — ry as r{x — y), and r{xy^ as {rx)y^ where x^y £ L and r ^ R. This should not create any confusion as both notations agree in R^L, We refer the reader to §11.5.2 where the notation M(G, *,^o) was introduced and recall that RA loops are precisely loops of this form (see Theorem IV.3.1).
150
VI. THE JACOBSON AND PRIME RADICALS
1.1 Lemma. Let L be a (possibly associative) Moufang loop whose loop ring RL is alternative and let N be a normal subloop of L. Then
neN
neN
IfL = M(G^ *55^o) i^ «^ R^ loop and N is a normal subloop of L contained in G, then A(L, N) = A(G, TV) + A(G, N)u. Let a = Y^i^iOtii G AR(L,N)^ where each a£ G R. Denote by a and I the images of a and ^, respectively, in the loop ring R[L/N]. Writing L = U ^ e T ^ ^ ' ^^^ some transversal T of TV in L, we have PROOF.
and so, for each x E: T,
£eL/=x
Since (ix~^)x — i for any x^i ^ L (recall that a Moufang loop is an inverse property loop—Theorem IL3.1), it follows that V^
OLii —
i^Lj=x
2_.
{o.iix~^)x
—
£eL,iz=x
2_.
OLiX —
£eL/=x
2_\
OLi{^ix~^ — X)X
£eL,I=x
and, for similar reasons, that y^
a^t, =
£eL,i=x
2_.
a£x{x'~^f,)—
£eL/=x
2_\
OLix — 2_.
£eL,£=x
OLix[x~^t — V),
£eL,£=x
So
^^^
££L/=x
^^^
£eL/=x
Since N is normal, ix~^ and x~^i e N fori =x^ thus aG^(l-n)iiL
and
a G ] ^ i?L(l - n).
neN
So A(L,iV) C Y.neNRH'^ - ") and AiL,N) reverse inclusions are obvious.
neN
C EneAr(l " " ) ^ ^ - The
1. AUGMENTATION IDEALS
151
For the last statement, suppose that L = M(G^ *5 5'o) is an RA loop and that a subset TV of G is normal in L, Since L = G U Gu^ for some u £ L, it is clear that RL = RG + RGu^ and so A(L,7V)= ^{l-n)RL=
^ ( l ~ n ) i ? G + ^{l
neN
neN
= A(G,N)
+
-
n)RGu
neN
A{G,N)u,
The result follows.
D
Now assume that R has a unity. For any finite subloop TV of a Moufang loop L with I TV I invertible in i2, we define the elements TV and TV of RL by TV = y
n
and
TV = r^TV.
neN
'
'
1.2 Lemma. Let R be a commutative and associative ring with unity and let L be a group or an RA loop. If N is a finite subloop of L, then (i) T V ^ ^ |TV|TV;
(ii) TV is central if and only if TV is normal in L; thus, if L is an RA loop with L' = {l,'^}, then N is central if and only if s £ N or N is central. Furthermore, 2/|TV| is invertible in R, (iii) N'^ = N is an idempotent in RL; (iv) if N is normal in L, then {RL)N = R[L/N]; (v) if TV is normal in L, then RL = {RL)N @ RL{\ - N) and RL{1-N) PROOF.
=
AR(L,N),
Clearly nTV = TV for any neN.
Therefore TV^ = |TV|TV, which
is (i). Recall that if L is an RA loop, the map I i-^ i'^ given by i
if ^ is central
si
otherwise
=
0, such that ad £ R\, Since each such a is invertible in 7?2? it follows that I = R2{I r\ Ri) foT every ideal I of R2. Also note that, if M is a maximal ideal of /?i, then R2M is a maximal ideal of R2 if M does not contain any of the variables Xi] otherwise, R2M = i?2Let d G J{R2)' Then, for any maximal ideal M of /?i, there exists a monomial a = X'!^^ • • • X^"^, each rrii > 0, so that ad G RiXi • • - Xn. Clearly ad G M if M contains one of the variables. On the other hand, if M does not contain any variable, then ad G R2M 0 R\. Since M is also prime, R2M H Ri = M, so ad G M. Since M is an arbitrary maximal ideal of /2i, we have shown that ad ^ J{R\), Thus ad e J{Ri)
= V{R)[X^,,.,
,Xnl
2. RADICALS OF ABELIAN GROUP RINGS
155
by Proposition 2.2, so
de-PiR)[X^,X{\...,Xn,X-']. Since V(R)[Xi^X^^^...
J(R2) =
,Xn,X^^] is a sum of nilpotent ideals, we obtain
ViR)[Xi,Xr\...,Xn,X-'] = PiR[Xr,X{\...,Xn,X-']).
D
To determine the radical of an arbitrary group ring RA of an abelian group A, we first consider finite groups. We need several results which we state as propositions. Recall that a commutative and associative ring T is said to be an integral extension of a subring i2, if each element x G T is integral over R in the sense that x satisfies a monic equation X^ + n x ^ - ^
+ • • • + Vn-lX + rn = 0
with each coefficient n G R. 2.4 Proposition. Let A be a finite abelian group. Then J{R) R andV{R) = V{RA)nR.
—
J{RA)^
P R O O F . We may assume that R has a unity. Since J{RA) fl /2 is an ideal of R and because an element of R is invertible in RA if and only if it is invertible in 72, it is clear that J{RA) 0 R C J'(R). To prove the reverse inclusion, it is sufficient to show that, if M is a maximal ideal of RA^ then J{R) C M D R, For this, note that each element a £ A satisfies al^l = 1. It follows that the field F = RA/M is an integral extension of the domain D = R/{R 0 M). Consequently, if 0 7^ cf 6 /?, then, for some d i , . . . ,dn e D, n> I,
{d-^Y + d^{d-^Y-^ + . . . + dn-l{d-^) + rfn = 0. Hence {di +--' + dn-id''-^+ dnd^-^)d ^ -l.sod-^ e Di Thus i) is a field, so i? n M is a maximal ideal of R. Consequently J{R) C M r\ R, Since V{R) is a nil ideal, so is V{R)A. Hence V{R) C V{R)A H R C V{RA) n R. The inclusion V[RA) DRC V{R) is also clear since V{RA) n R is a nil ideal of R, D
156
VI. THE JACOBSON AND PRIME RADICALS
2.5 Proposition. Let A be a finite abelian group. If r = ^aeA^a^ ^ TZ{RA)j then \A\ra G TZ{RA) for each a ^ A, where Tt is either J or V. In particular \A\ra G IZ{R), Again we may assume that R has a unity and first consider the Jacobson radical. Because of Proposition 2.4, it is sufficient to prove the statement for {R/M)A where M is a maximal ideal of /?; thus we may assume that 72 is a field. Let p be the characteristic of R, li p divides |^|, then \A\r = 0 for any r ^ R and the result is obvious. So we assume p ]^ |^|; hence, either p = 0^ or p > 0 and A does not contain elements of order p. We consider the latter case first. Assume p > 0 and that A does not contain elements of order p. We have to show that J(RA) = {0}. Since RA is a finite dimensional algebra over the field R, J{RA) = V{RA), hence it is sufficient to show that V{RA) = {0}. Let a = YlaeA '^^a e V{RA), Then a^ = 0 for some ^ > 1 and thus a^^ = 0 for some q with p^ > L Since pR — {0}, PROOF.
0 = a^' =
Y,{raay\ aeA
Because a^"^ ^ 1 for a / 1, it follows that r^ is nilpotent and thus r\ — 0. Since aa~^ is also nilpotent, what we have just shown yields that r^ — ^ also. Thus a = 0, so V{RA) = {0}. Now assume that p = 0 and let R denote the algebraic closure of R. Clearly
V{RA) =
V(RA)r\RA,
so, to prove V{RA) = {0}, we may assume that R is algebraically closed. Since char i? = 0, it is known (see, for example, [Coh77, Exercise 8, p. 271]) that R is obtained from a real closed field F by adjoining an element i with i'^ = —1. For any element r = / i + i/2 6 R^ /15/2 € F,, write r = fi — if2. It is then easily verified that the map RA —^ RA defined by
a^A
a^A
is a ring involution. Now the coefficient of 1 in aa is ^^eA ^^^a > 0. Since F is formally real, it follows that a = 0 if and only if a a = 0. Suppose fi e RA and /J^ =:. 0. Let a = /3^. Then a^ = a a = 0 too, so a = 0.
2. RADICALS OF ABELIAN GROUP RINGS
Therefore /3 = 0. Thus RA that is, V{RA) = {0}. Finally, we consider the prove the result for {R/P)A^ assume that i2 is a domain. V{RA) C V(KA) = J{KA),
157
does not contain nonzero nilpotent elements; prime radical and note that it is sufficient to where P is any prime ideal of R, Thus we may Let K denote the field of fractions of R. Since the result follows from the first part. D
For an ideal I oi R and a positive integer n, let In be the set of all elements r E R with nr G / : In = {r e R\nr
e /}.
Clearly In is an ideal of R and, if / is semiprime, then In^ = In for any m > 1. We denote by P the set of (positive) prime numbers. For any Moufang loop L and p G P, by Lp, we mean the set of elements in L whose order is a power of p and, by Lp/, the set of elements of order relatively prime to p. Clearly, if L = yl is an abelian group, both Ap and Apt are subgroups of A, In what follows, we shall make use of the following obvious, but useful observation. If a Moufang loop L is the direct product H X K^ then the loop RL of L over a ring R can be viewed as {RH)K^ the loop ring of K over the coefficient ring RH. 2.6 P r o p o s i t i o n . Let A he a finite abelian group. Then
n{RA) = n{R)A + Y^ n{R)pA{A, Ap), where TZ — J orV.
Furthermore^ IZ{RA)/TZ{R)A
is nilpotent.
Let TZ = J or V. We prove the result by induction on the number of primes dividing the order of A, If A is the trivial group, then the result is obvious. Assume that yl is a (nontrivial) p-group. Because of Proposition 2.4, 7^(i^) C n{RA). Since RA/TZ{R)A ^ {R/n{R))A and because TZ(R/TZ{R)) = {0}, we may assume that TZ{R) = {0}. Write Z = {0}. By Proposition 2.5 and because R is semiprime, TZ{RA) C ZpA. Clearly ZpA is an ideal of RA. Hence, since 72. is hereditary, TZ{RA) — n{ZpA). Now notice that for any a e A, (Zp(l - a))P" C pZpA = {0} for some n > 1. It follows that Azp{A) is nilpotent and thus Azp{A) C PROOF.
158
VI. THE JACOBSON AND PRIME RADICALS
7Z(ZpAp), As ZpA/Azp(A) = Zp and because TZ = {0} and thus also TZ{Zp) = {0}, we obtain that TZ{RA) = Azp{A), as required. For the general case, assume j9 is a prime dividing n = |A|. Since A is finite, we have A = Ap x Api, so RA = {RAp/)Ap is the group ring of Ap over the ring RApf, The induction hypothesis yields 7^(i^A) = n{RAp,)Ap
+ {n{RAp,))^
ARA^MP).
By Lemma 1.1, we know that ARA ,{Ap) = AR{A^AP). Furthermore, if a e {TZ{RAp>)) , then a = /3i + /32, where /Ji = v\Apf\a and /?2 = wpa for some integers v and w. Then /?2 € 7Z(RApf) and pPi G TZ{R)A by the induction hypothesis. Hence a G TZ(RApt) + {TZ{R))pA. The induction hypothesis, applied to RApf^ yields the result. D 2.7 T h e o r e m . Let A be an abelian group. Define V{R) = J{R) torsion and V(R) = V{R) otherwise. Then
if A is
1. (May [May76]; J{RA) = V{R)A + ZpeP V(/2)pA(^, Ap). 2. (Karpilovsky [Kar82]^ V{RA) = V{R)A + Ep6P nR)v^{A,
Ap).
Let TZ be either J or V and first consider the case that A is finitely generated. Write A = At x Aj., where At is the (finite) torsion subgroup of A and Aj is a free abelian subgroup of A., say of rank n. If n = 0, then TZ{RA) has the required form by Proposition 2.6. Now assume n > 1. Since RA = {RAt)Af^ we have PROOF.
RA^(RAt)[XuX^\...,Xn,X-']. Corollary 2.3 implies that TZ{RA) = V{RAt)Af which, by Proposition 2.6, has the required form. For an arbitrary abelian group A, the foregoing implies that TZ{RAi) C TZ{RA2) for finitely generated subgroups Ai C A2 of A. Since A = [jAi^ where the union runs through the finitely generated subgroups Ai of ^ , we have 7Z{RA) — [jTZ(RAi). This fact is easily verified for the prime radical and, for the Jacobson radical, it follows from J{RA)r\RAi C J(RAi). (See the proof of Lemma 3.2.) The result follows. D
3. RADICALS OF LOOP RINGS
159
3, R a d i c a l s of l o o p rings Throughout this section, L = M(G, *9^o) is an RA loop with centre 2 and i2 is a commutative and associative ring, not necessarily with unity. Recall that L can be written L — G \J Gu and that G — {Z^x^y) is generated by Z and two elements x and y, (See §11.5.2 and Theorem IV.3.1.) 3.1 Lemma. Let TZ be either the Jacobson or prime radical and let H be L or G. Then n{R[Z(H)]) C 11{Z{RH)) C n{RH). The result is obvious for the prime radical. The Jacobson radical is hereditary. Furthermore, R[Z[H)\ is an ideal in R^[Z{H)], Z{RH) is an ideal in Z(R^H) and RH is an ideal in R^H. It follows that J{R[Z(H)]) ^ J(R^[Z{H)]) 0 R[Z{H)], J{Z{RH)) = J{Z{R^H)) n Z{RH) and J{RH) = J{R^H) n RH, Hence, to prove the result, we may assume that R has a unity. Let a e J{Z{RH)) and a G RH. Let t{a) rr a + a* and n{a) = aa^ denote the trace and norm of a (see §111.5) and recall that both t{a) and n(a) are central in RH. Then t{a)a — n{a)a^ is in J{Z{RH))^ so this element is quasi-regular. Thus 1 — t{a)a + n(a)a"^ is an invertible element in 2 ( / 2 / r ) , with inverse 6, say. Let d = ba'^n^a) (so that d + b — bt(a)a = 1) and / = —baa + d. Remembering that a^ — t[a)a — n{a)\ and that a, b and d are in the centre of RH ^ we have PROOF.
(aa)f — —b(j?o? + aad — aad — ba^t(a)a + d = aa{d — t{a)ba) + d = aa{l — b) + d = aa + f. Thus aa is quasiregular, with quasi-inverse / . Hence aRH is a nonzero quasi-regular ideal of RH, so aRH C J{RH) and a G J{RH). Thus J{Z{RH))C J{RH). If, in the above, a G J{R[Z{H)]) and a G Z{RH), then aa has quasiinverse / G Z{RH). So J{R[Z{H)]) C J{Z{RH)). U 3.2 Lemma. Le^ H = L or G, let Z = Z{H) and let S be a subloop of H. Then 1. J{RH)nRS 2. J{RH)nRZ 3. ViRH)nRZ
CJ{RS). = J{RZ), = V{RZ).
160
VI. THE JACOBSON AND PRIME RADICALS
Suppose a 6 RS is quasi-regular in RH^ say with quasi-inverse /?. Write /? = /?! + /J2, where supp(/?i) C S and supp(/32) 0 5 = 0. Then PROOF.
ap = a + p and, extracting only the terms with support in 5 , we obtain a/3i z= a + f3i.
Thus a is quasi-regular in RS. Hence a G i?*? is quasi-regular in RS if and only if a is quasi-regular in RH. Since J{RH) H JR5 is an ideal in RS, it follows that J{RH) n RS C J{RS). This proves the first part. By Lemma 3.1, the second part is now also clear. We now turn to the prime radical. Because of Lemma 3.1, V{RZ) C V{RH) n RZ, To establish the reverse inclusion, we show that if a G V{RH) n RZ, then a ^ P, for any prime ideal P of RZ, For this, it is sufficient to show that if P is a prime ideal of RZ, then there exists a prime ideal Q of RH with Q n RZ = P, So let P be a prime ideal of RZ, Then {RL)P = Yl/teT^^-) where T is a transversal for Z m L and 1 G T. A support argument yields that
C^tP)
n RZ = P,
teT
and thus {RL)P D RZ = P. Hence there exists an ideal Q of RL which is maximal with respect to the condition Q D RZ = P. We claim that Q is a prime ideal, therefore completing the proof of 3. Indeed, suppose / and J are ideals of RL properly containing Q. By construction, there exist a, 6 G RZ \ P with a E: I and b £ J, Since P is prime in RZ, ab ^ P, thus abiQ, So IJ % Q. D 3.3 Lemma. Let L = M(G,*,po) f>^ «^ R^ loop with centre Z. L = GuGu andG = {Z, x, y). Let TZ be J or V. If
Write
7 = (ao + OLix -\-ay + a^xy) -f (/?o + P\x -\- /?22/ + fi3xy)u G TZ{RL), with ai,f3i G RZ, then 8a,-,8/3,- G n{RZ)
for i = I,,..
,4. If
7 = ao + a i x + a2y + OL^xy G TZ{RG), thenAai G n{RZ)
for i = 1 , . . . ,4.
3. RADICALS OF LOOP RINGS PROOF.
161
Consider the map / i : RL -» RL defined by a -^ I3u \-^ a — fin
where a^/S ^ RG. It is readily verified that f\ is a ring isomorphism. By Theorem IV.3.1, we also have L ~ M(G2,*,y^) = G2 ^ G2y and L = M(G'3,*,a;^) = G3UG3X, where G2 = {Z{L),x,u) and G3 = {Z{L),y,u). So we also obtain the following isomorphisms RL —> RL: f2'- 7 + Sy^
J - 6y
h'- V + C,x \-^ T] - C,x where 7, ^ G RG2 and ?/, C ^ RG3. Let 7 = (ao + Oi\x + a2y + a^xy) + (/3o + Aa: + ^2y + P3xy)u G Tl{RL), where a^-, A G i22, i = 1 , . . . ,4. Since both the Jacobson and prime radicals are invariant under isomorphisms, it follows that 7 + / i ( 7 ) = 2(ao + c^ix + a2y + a^xy) G Tl{RL) and also that 7 ^ " ' + h{iu-^)
= 2(Po + Pix + f32y + p3xy) G TZ(RL).
Using a similar argument with the map /2, we obtain 4(ao + a i x ) G 7^(i^L)
and
4(^2 + aax) G 7^(i^L)
4(/Jo + Pix) G 7e(i2L)
and
4(/32 + /?3a:) G 7^(i^L).
and also
Repeating the above argument with /a, it follows that 8ai,8/Jt G TZ{RL) for i = 1 , . . . ,4. By Lemma 3.2, 8a,-,8/3, G n{RZ). The final statement, concerning TZ{RG)^ is proved similarly. D Let Z/ be an RA loop or a nonabelian group contained in an RA loop. By Proposition V.1.1, the set Lp of elements of L of p-power order, is a normal subloop of L; in fact, with p odd, Lp is even central. 3.4 Theorem. Let V{R) — J{R) V{R) otherwise. Then
if G (and thus L) is torsion and V{R) —
1. J{RL) = V{R)L + Y:^^^^{R)V^{L,L^). 2. J ( f l G ) = V(il)G+EpGpV(i?)pA(G,Gp). 3. V{RL) = mi{RL) = V{R)L + EpeP^(^)pA(Z,, L^).
162
VI. THE JACOBSON AND PRIME RADICALS
4. r{RG)
= m{RG)
In particular^ if R/V{R) n{RG) + n{RG)u.
= TiR)G
+^^^pV{R)j>A{G,Gp).
has no 2-torsion or if L2 = G2, then 1Z{RL) =
Let H = L or G. By Theorem 2.7, J(R) C J{R[Z{H)]), (Note that H is torsion if and only if Z{H) is torsion since squares in H are central.) Therefore, by Lemma 3.2, we have V{R) C J[RH), Replacing RH by RH/V{R)H ^ (R/V{R))H, if necessary, it foUows that, to prove (1) or (2), we may assume V{R) = {0}. For (3) and (4), we may assume that R is semiprime, because V(R) C V(RH) and (R/V(R))H ^ RH/V(RH). Let 7^ = J or V. Assume that V{R) = {0} if 7^ = J and V(R) = {0} if TZ = V. To prove both parts, it is now sufficient to show that, under these assumptions, n{RH) C EpGP ^ ( ^ ) P ^ ( ^ ' ^ p ) ^ V{RH). By Lemma 3.3, 8n{RH) C n{RZ)H and therefore, by Theorem 2.7, STl^RH) C T{R)H, where T{R) is the ideal of R consisting of the elements of finite additive order. Since R is semiprime, for any p € P, PROOF.
{r e R\ p^r - 0 for some n > Q] = {r e R \ pr - G] = Z^y, where Z = {0}. Since T{R)H is an ideal of RH^ it follows that TZiRH) = 'R(T{R)H)
=
^^^^TliZ^H).
We may therefore also assume that R = Zp. Consider the case p = 2 first. (Thus 1 + 3 = 1 — 5.) Since (1 + sy = 2(1 + 5), {RH{1 + s)y = {0}. Hence RH{1 + s) = RH{\ - s) C V{RH), Clearly RH{\+s)C AR{H,H2). Note that RH/RH{l + s)^ /2[///7/'] and A H ( / / , H2)/RH{l + s) ^ AR{H/H', {H/H')2). Because H/H' is an abelian group. Theorem 2.7 implies that n(RH/RH{l
+ s)) ^ AR{H, H2)/RH{1 + s) = V{RH/RH{1
+ s)).
Hence n{RH)
= RH{\ + 5) + AR{H, H2) = AR{H, H2) =
V{RH),
as required. Consider now an odd prime p. Let 7 = (ao + otix + ay + a^xy) + {(5Q + (5ix + /?2y + P3xy)u
3. RADICALS OF LOOP RINGS
163
be an element of 7Z{RH)^ each ai,/?^ G R2. Of course we assume that each /3i = 0 in case H = G. By Lemma 3.3, 8a^ and 8f3i are each in J{RZ). Since 8 and p are relatively prime, we obtain that each ai and each /J^ is in J{RZ). Because of Theorem 2.7, ai and j3i are in lS.R{Z^Zp), Hence, by Lemma 3.2 and Theorem 2.7, n{RH) C R^HAR(Z,ZP) C V{RH). Since Hp = Zp and because AR(H^HP) = R^HAR(Z^ZP)^ it follows that n(RH) = AH(if, Hp) = V{RH), again as required. Finally, suppose that either R/V{R) has no 2-torsion or that L2 = 02It is then easily seen that, for any prime p, V(R)pA{L,Lp)
C V{R)pA{G,Gp)
+
V{R)pA{G,Gp)u.
By what we have shown, it then follows that n{RL)
= n{RG) + n{RG)u,
3.5 Corollary. Let H = L or G. Then J{RH) if one of the following conditions is satisfied, 1. J(R) = p(^R),
D is a nil ideal if and only
2. H is not torsion. Furthermore, if either of these conditions is satisfied, J{RH)
=
V{RH).
Suppose J{RH) is nil and H is torsion. By Theorem 3.4, V{R) = J{R) ^ J{RH). Hence J{R) is nil and, because R is commutative, J{R) — V{R), This proves one implication. For the converse, assume that J{R) — V{R) or that H is not torsion. In either case. Theorem 3.4 implies that PROOF.
J{RH)
= V{R)H + Y, VWpA(7y, Hp) = V{RL).
D
peP
Parts of the corollaries which follow were proved in [GP86]. Their proofs are almost immediate applications of the description of the prime and Jacobson radicals of alternative loop rings obtained in Theorem 3.4. According to this theorem, for instance, the prime radical of RL is {0} if and only i{V{R) = {0} and, whenever V{R)p ^ 0 (that is, whenever, pr = 0 for some prime p and nonzero r E R)^ then Lp = {1} (that is, L contains no elements of order p). Thus Corollary 3.6 can most easily be verified by recognizing the equivalence of each statement with statement 4.
164
VI. THE JACOBSON AND PRIME RADICALS
3.6 Corollary. Let L = M(G', *,^o) be an RA loop. Then the following conditions are equivalent. 1. 2. 3. 4.
5. 6. 7. 8.
9.
Z{RG) is semiprime, R[Z{L)] = R[Z(G)] is semiprime. RG is semiprime. R is semiprime and, if R contains an element of additive order n > \, then G does not contain central elements of order n (and thus no elements of order n and no normal subgroups of order n). RG has no nonzero nil ideals. Z{RL) is semiprime. RL is semiprime. R is semiprime and, if R contains an element of additive order n > \, then L does not contain central elements of order n (and thus no elements of order n and no normal subgroups of order n). RL contains no nonzero nil ideals.
3.7 Corollary. Let L = M{G,^,go) be an RA loop. Let V{R) = J{R) if G (and thus also L) is torsion and V{R) = V{R) otherwise. Then the following conditions are equivalent. 1. 2. 3. 4. 5. 6.
J{Z{RL))^Q. J{Z{RG)) = Q. J{RG) = 0. J ( i ? L ) = 0. J{RZ) = 0. V{R) = 0 and, if R G does not contain 7. V{R) = 0 and, if R L does not contain
contains an element of additive order n > I, then central elements of order n. contains an element of additive ordern > I, then central elements of order n.
Since, by Corollary IV. 1.9, the order of a subloop of a finite RA loop L divides the order of L, the next corollary, which extends Maschke's Theorem for group algebras, is clear. 3.8 Corollary. Let L = M(G,*,^o) be a finite RA loop, let F be a field and let H — L or G. Then FH is Jacobson semisimple if and only if char F = 0 or char F = p > 0 and H does not contain elements of order p.
3. RADICALS OF LOOP RINGS
165
We have shown t h a t there is strong relationship between the radical of the loop ring and the radical of its centre. In this context the following proposition is of interest [ G P 8 7 ] . 3.9 P r o p o s i t i o n . Let L = M ( G , *,po) be an RA loop and assume that R has no elements
of additive order 2. Let H = L or G. Then every
ideal of RH contains hence central also in PROOF.
a nonzero central element
nonzero
which is central in RH
and
RL.
The last statement follows from Proposition III.4.1 which im-
plies t h a t Z{RG)
C
Z{RL).
Let J be a nonzero ideal of RH and let r be a nonzero element of J. Multiplying by the inverse of a loop element, if necessary, we may assume t h a t 1 G s u p p ( r ) . Write r in the form (1)
r = z + x,
O ^ ^ G Z{RH),
x = ^ a d i
Z{RH)
where T is some finite subset of s u p p ( a ) . We prove by induction on \T\ t h a t J contains a nonzero central element. If \T\ = 0, then r = 2: is the desired element. Now assume t h a t \T\ > 0 and t h a t J contains a nonzero central element whenever it contains a nonzero element which, when written in the form (1), has fewer t h a t | r | elements in the support of its noncentral term. Let r be a nonzero element of J (with 1 in its support) and write r in the form (1). If r G
Z[RH)^
there is nothing to prove. Otherwise, there exist elements to £ T and IQ £ H such t h a t toio ^ 4^0• Write T = To U Tj where To = {t e T \ tio 9^ 4 0 and Ti = T \ TQ. Note t h a t |Ti| < \T\.
Letting s denote the nonidentity
commutator of H^ we have, for any t G To^ t + l^^tio Corollary III. 1.5, is an element oi Z{RH). ri = r + q^rio
where zi = "^^-^^teT
— t -\- st which, by
Therefore J contains the element
= 2^ + ^ at{t + 5t) + 2 ^ a^^ = ^1 + 2 ^ att, ten teTi teTi (^t{t + st) G Z{RH).
Since 72 has no 2-torsion, ri has
1 in its support. Thus r\ is not 0, and central by the induction hypothesis, so the proof is complete.
D
166
VI. THE JACOBSON AND PRIME RADICALS
4. T h e s t r u c t u r e of a s e m i s i m p l e a l t e r n a t i v e a l g e b r a Let F be a field and L a finite RA loop whose order is relatively prime to the characteristic of F, In this section, we apply a theorem of J. Dieudonne about general nonassociative algebras to show that the loop algebra FL is the direct sum of fields and Cayley-Dickson algebras. 4.1 Theorem (Dieudonne). Let A be a nonassociative algebra with unity of finite dimension over a field F. Suppose A contains no nonzero ideals with square 0 and that f: Ax A -^ F is a nondegenerate symmetric bilinear form on A which is associative in the sense that f{xy^z) == f{x^yz) for all x^y^z £ A. Then A is the direct sum of unique ideals A i , . . . , A^ which are simple rings. Any ideal of A is the direct sum of a subset of { ^ i , . . . , An}^ First we show that A can be written as stated. If A is simple, there is nothing to prove; otherwise, let B be an ideal of A of minimal (positive) dimension and note that B-^ = {a G ^ | f{0",b) — 0 for aU 6 G 5 } is an ideal of A., by symmetry and associativity of / . Since B fl B^ C 5 , either B (^ B^ = {{)] or B f\ B^ = B, We show Bn B^ ^ B. By hypothesis, B^ ^ {0}, so the ideal of A generated by B^ must equal B. Thus any b £ B can be written in the form 6 = YL{^t^j)^ij'> where bi^bj G B and each Uij is a product of R(x)s and/or L{x)s for x £ A^ where, as always, R{x): a \-^ ax and L{x): a \-^ xa denote the right and left translation maps on A. Now PROOF.
f{yR{x),z)
= f{yx,z)
= f{y,xz)
=
f{y,zL{x))
and f{yL{x),z)
= f{xy,z)
= f{z,xy)
= f{zx,y)
=
f{y,zR{x))
together imply that, for any product 11 of maps of the form R{x) and Z/(a:), we can write /(yn,2r) = / ( y , ^ n ' ) , where 11' is another product of R{x)s and L{x)s. Suppose that BOB-^ = B. Let b = Y^{bibj)nij be a nonzero element of B and let a be any element of A, Then, for some IIJ-, / ( 6 , a ) = 5 ] / ( ( M , ) n i „ a ) = 5 ] M 6 „ a n ^ ) = Y^{bi,b,{an'-^^)) = 0
4. THE STRUCTURE OF A SEMISIMPLE ALTERNATIVE ALGEBRA
167
since bi ^ B C B^ and hj{a\l\-) G 5 . By nondegeneracy, 6 = 0, a contradiction. We conclude that B fl B^ = {0} and hence, by Proposition 1.4.4, that A = B Q B^ is the direct sum of the ideals B and B^, Also, by Corollary 1.4.6, the restriction of / to B^ is nondegenerate. Since BB^ C 5 n 5-^ = {0} and, similarly, B^B = {0}, any ideal of B is an ideal of A (so 5 is a simple ring) and any ideal of B-^ is an ideal of A. The existence of the required decomposition of A now follows by induction. We have shown that A = Ai Q - • - Q An is the direct sum of simple ideals Ai, Let / be an ideal of A. Since A has a unity, we can write / = (/ n yli) © • • • © ( / n An)' Since / D Ai is an ideal of Ai, each / fl Ai is {0} or Ai, Thus / is the direct sum of certain of the Ai and, if / is a simple ring, exactly one. If A can also be written ^ = 5 i © • • • ® Bm, where the Bi are ideals and simple rings then, by what we have just seen, Bi is one of the Ai, Without loss of generality, Bi = Ai, Similarly, B2 = Ai for some z > 1 (since B^ D B2 = {0}). It follows that n = m and that the sets { A i , . . . , An} and { 5 i , . . . , Bn} are identical. D When an algebra A is written as the direct sum ^ i ffi • • •ffiyl^ of simple ideals, as in the Theorem of Dieudonne, the ideals Ai are called the simple components of A, The following theorem is a consequence of Corollary 3.6; however, there is an easier argument that includes finite groups which we wish to present. Let F be a field and L a finite group or RA loop. For a G FL, the right translation map R{a): FL -^ FL is a linear transformation of FL and the map R: FL -^ Ej\d(FL) is known as the right regular representation of FL. Representing R{a) as a matrix relative to the basis L (with entries in F) and denoting the trace of R(a) by tr R(a), note that 1, Ul j^ ie L, then tr R{i) = 0 and 2. tTR(l)= \L\, 4.2 Theorem (Maschke). Let L be a finite group or RA loop and let F be a field. If the characteristic of F does not divide the order of L, then FL has zero Jacobson radical; in particular^ FL is semiprime. Since FL is a finite dimensional algebra, its Jacobson radical is nilpotent [ZSSS82, Theorem 12.2.2]. Let 0 7^ a := Er=i c^^iA G
PROOF.
J{FL)
168
VI. THE JACOBSON AND PRIME RADICALS
J{FL),
Since J{FL)
is an ideal of F L , multiplying by an appropriate i~^
if necessary, we may assume t h a t a i 7^ 0. By Artin's Theorem, R{aY
= R{a'^) so, since a is nilpotent, R(a)
is
a nilpotent linear transformation. Hence 0 = tr R{a) = S ^ = i Oii tr R((,i) = ai\L\.
Since a i 7^ 0, it follows t h a t the characteristic of F divides |/>|, a
contradiction.
D
4 . 3 C o r o l l a r y . Let H be a finite group or a finite RA loop and let F be a field of characteristic
relatively prime to \L\, Then FH is the direct sum of
ideals which are simple PROOF.
rings.
By Theorem 4.2, FH contains no nonzero ideals with square
0. For X = ^ OLii and y — ^^d
in FE,, define
/(^.2/) ^ X ! «A:/?^ Then / is a bilinear form on FL which is symmetric because ki — \ m L if and only \{ ik — 1, and associative because hk -1 — \ m L \{ and only if h'ki
— 1 too. Moreover, / is nondegenerate because, if x = ^ a^t / 0 and
i G s u p p ( x ) , we have f{x,i-^)
= a^ 7^ 0. The result now follows by the
Theorem of Dieudonne.
D
4 . 4 R e m a r k . In the case of a group ring of a finite group G over a field of characteristic p not dividing |G|, there is a more "classical" proof of Corollary 4.3. Here is a sketch. Suppose / is an ideal of KG.
As a A^-linear m a p , the natural homomor-
phism TT: KG -^ KG 11 splits, so there is a A'-linear m a p ^ : KGjI such t h a t TT o 1/; is the identity m a p on KGjl. A'G-modules and define Q\ KG/1
for X G KG/1.
Consider KGjl
—^ KG
and KG as
—> KG by
It is routine to verify t h a t 0 is a KG-mod\i\e
that TT o 0 is the identity m a p on KG/1,
Hence
KG = ker TT © (^ 0 7r){KG) ^ / ©
KA/I
morphism such
4. THE STRUCTURE OF A SEMISIMPLE ALTERNATIVE ALGEBRA
169
as rings. Since KG has a unity, it follows t h a t / is generated by a central idempotent. Since / is an arbitrary ideal, and because KG is finite dimensional, it follows t h a t KG is a direct sum of simple rings. In particular, J {KG) is trivial. In characteristic different from 2, any alternative loop algebra which is not associative comes equipped with an involution * and a nondegenerate quadratic form n defined by n ( r ) = rr"". (See Sections III.4 and III.5.) 4.5 T h e o r e m . Let L — M ( G , *,po) be a finite RA loop and let F be a field of characteristic
which does not divide \L\. Let H = L or G.
that B is a simple subalgebra of FH, Z{B)
C Z(FH),
multiplicative
Then the map n: B -^ Z(B)
nondegenerate
tion algebra. In particular,
quadratic form
PROOF.
simple component For any a G Z{B)
Thus B is a of FL is a algebra.
of FG is a quaternion
algebra.
composition Each
non-
a'^n{b),
and a* = a for any a € Z{FH).
Since f{x^y)
composi-
and any b ^ B., we have
bilinear form associated with / ; thus, f{x^y) for x , y G FL.
on B.
is a Cayley-Dickson
n{ab) = (a6)(6*a*) = since 66* G Z{FH)
such that B* C B and
defined by n(b) = bb"" is a
each simple component
algebra which, if not associative, commutative
with unity,
Suppose
Let / denote the
= n{x -\- y) — n{x) — n{y)
= xy* + yx*^ f is bilinear, so n is quadratic.
T h a t n permits composition follows from the fact t h a t 66* is central for any 6 G FH.
We show t h a t n is nondegenerate. Thus, let W = BnB^
= {beB\
f{b, x) = 0 for aU x G B}.
We must prove t h a t W = 0. To achieve this, we first use the definition of / and n(pr + qr) = n{p + q)n{r) to obtain n{pr) + n{qr) + f{pr, qr) = [n{p) + n{q) + / ( p , q)]n{r). Thus /(/^r, qr) — f{p^ ^)^(^) for any p^q,r ^ B. Replacing r by r + / gives / ( p r , qr) + f{pr, qt) + f{pt, qr) + f{pt,
qt) = f{p.q)[n{r)
+ n{t) +
f{r,t)]
170
VI. THE JACOBSON AND PRIME RADICALS
and hence f{pr, qt) + f{pt, qr) = f{p, q)f{r, t), for all p,q,r,t G B, With p = w e W and ^ = e, the unity of 5 , we get f{wr^ q) — 0, showing that W^ is a right ideal of B, With r = w ^W and g = e, we see that f(pw,t) = 0, which shows that W is also a left ideal of B. IfW^ 0, then W = B, so e e W, Then 0 = / ( e , e ) = 2e^ = 2e, a contradiction. It follows that W = 0 and that B is indeed a composition algebra. For the final statements of the theorem, observe first that if FH = Ai®'''®Anis the direct sum of simple ideals Ai, then certainly the centre of each Ai is contained in the centre of FH, Writing I = ^ei as the sum of orthogonal central idempotents e^, clearly €{ is a unity for Ai. Moreover e* = €{ implies that Ai 0 A* is a nonzero ideal of Ai. Hence Ai 0 A* = Ai^ so Ai = A*. By the first part, each Ai is a composition algebra. Hence the result follows from Hurwitz's Theorem. D 4.6 Proposition. Let L = M(G, *,po) be an RA loop with V = G' = {1,5} and let F be a field of characteristic different from 2. 1. If FL = A 1 ® • • • © An is the direct sum of simple ideals Ai with unities Ci, then the following statements are equivalent. (i) Ai is commutative. (ii) Ai is associative. (iii) Si = Ci, where Si = sci is the component of s in Ai. 2. / / FG = Ai Q ''' Q An is the direct sum of simple ideals Ai with unities Ci then the following statements are equivalent. (i) Ai is commutative. (ii) Si = Ci, where Si is the component of s in Ai. P R O O F . We prove only the first statement, the proof of 2 being almost identical. The set {ei, 6 2 , . . . , e^} consists of central pairwise orthogonal idempotents whose sum is 1. Let B denote one of the Ai and let TT : FL -^ B denote the natural projection. Thus 7r(s) = sci = Si. Since s is central in FL, Si is in the field which is the centre of B. Since s'^ = 1 and Ci is the unity of B, we have Si = ie^. Now B is generated by {7r(^) | g 6 L}. For g^h^k £ L, the
4. THE STRUCTURE OF A SEMISIMPLE ALTERNATIVE ALGEBRA
171
(ring) associator [^{g)',7r{h)^7r{k)] = 7r([^,/i, A;]) and the (ring) c o m m u t a t o r [7r(^),7r(/i)] = 7r([^,/i]) are of the form (e^- — c)7r{gh • k) and (e^- — d)7r(gh) respectively, where c and d are €{ or Si^ according as (g^h^k)
and (g^h) are
1 or 5, respectively. Now the result follows easily.
D
If Ai is a noncommutative component of FL (and hence ^^ = —e^), if ^ and h are elements of L which do not commute, and if gi and hi are, respectively, the components in Ai of these elements, then gh = shg implies gihi = Sihigi = —higi. Thus gi and hi do not commute in Ai. With a similar argument with respect to elements g^h^k of L which do not associate, we obtain the following rather curious result. 4 . 7 C o r o l l a r y . With F, L and FL as in Proposition g^h £ L commute noncommutative
4-6, two
elements
if and only if their images gi and hi commute
in some
simple component
Ai. Three elements g^h^k ^ L
if and only if their images associate in some simple component
associate
which is not
associative. 4.8 C o r o l l a r y . Let L — M{G^^^go) field of characteristic
^ F[H/H']
PROOF.
with F / / ^ and ^^s
Then
FH^^QFH^
is a direct sum of fields and FH^
is a direct sum of Cay ley-Dickson sum of quaternion
RA loop and let F be a
which does not divide \L\. Let H — L or G. FH =
where FH^^
be a finite
= A ( L , L')
algebras if H = L and otherwise
a direct
algebras.
Because of Lemma 1.2,
^
F[HIH']
FH = FH^^
+
and FH^
=
FH^ AF{L,L').
Since ^^s
=
^
— — ^ ^ , the result now follows from Corollary 4.3, Theorem 4 . 1 ,
Theorem 4.5 and Proposition 4.6.
D
Chapter VII
Loop Algebras of Finite Indecomposable R A Loops
In this chapter, we construct the primitive central idempotents of the rational loop algebra QL of a finite RA loop L and provide an exphcit description of the decomposition of QL into simple rings. This work is mainly based on [JLM945 J L M ] and will be critical to our determination of the loop of units in the integral loop ring ZL in Chapter XII. We need the following two lemmas. 0.1 L e m m a . Let G and L be (possibly associative) Moufang loops and let F be a field. Then F[G x L]^ FG ® F FL, PROOF.
The m a p s / G : FG ^ /^[GxL] a n d / L : FL -> FfGx L] defined
by
are algebra homomorphisms, so there exists an algebra homomorphism FG®FL-* F[G X L] satisfying
(see §1.5.8). Since this map sends g ® t to {g^t) (and hence a basis to a basis), it is an isomorphism. D If F is a field of characteristic diflPerent from 2 and C2 = {0) is cyclic of order 2, then FC2 = F^^ ® F ^ ^ F ® F. Hence the group algebra F[C2 X C2 X • • • X C2] of the direct product of n copies of C2 is isomorphic 173
174
VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS
to the direct sum of 2^ copies of F , which we denote 2^F. This obvious remark will be useful since, for any RA loop L, L/Z{L) = C2 X C2 X C2. (See Corollary IV.2.3.) 0.2 Lemma. Let L be a loop and let F C E be a field extension. Then E®FFL = EL; in particular, if F has characteristic Q, then F®QQL = FL. We note that the maps fp'. FL -^ EL and fE- E ^ EL defined by fri^) = 2;, fE{oL) = ctfl 3're F-algebra homomorphisms and so there exists an F-algebra homomorphism / : E ®/r FL —> EL mapping a ® x to ax. Since f{l®£)=zi and since {I ® i \ i e L} is dm F-basis for E ® F FL (by Corollary 1.5.6), / is bijective. D PROOF.
Because of Lemma 0.2, a description of QL yields a description of FL over any field F of characteristic 0. When an RA loop L = M(G,*,^o) is indecomposable, we shall import notation from §V.3. Thus, we write G = {Z{L)^x^y) and Z(L) — {t\) x (^2) X (^3) X {w), where • 0(^1) > 2 and 0(^2),0(^3),o(it;) > 1; • 6 = {x,y) e {ti); • x^ e {ti) X (^2) and y^ e {ti) X (^2) x (^3). 1. P r i m i t i v e i d e m p o t e n t s of c o m m u t a t i v e rational group algebras A nonzero (central) idempotent in a ring A with unity is primitive if it is not the sum of two nonzero orthogonal (central) idempotents. If A is the direct sum of simple subrings, then a central idempotent e is primitive if and only if Ae is a simple ring, in which case Ae is one of the simple components. In order to determine the primitive central idempotents of the rational loop algebra of a finite RA loop, we first need to solve this problem for rational group algebras of finite abelian groups. For this, we require some preliminary results. 1.1 Lemma. Suppose H and K are normal subloops of a finite Moufang loop (or group) L with K C H C L, Let f = H and e = K. Then dim(QL)(e - f) = dim(QL)e - dim(QL)/.
1. COMMUTATIVE GROUP ALGEBRAS
175
P R O O F . Note that e and / are central idempotents, by Lemma VI.1.2. Since K C H, we have ef = / . Clearly (QL)e = ( Q L ) / + {QL){e - f) and the sum here is direct because / ( e — / ) = 0. Hence the result is clear. D
If F and E are fields, F C E, we denote by [E: F] the degree of the field extension; that is, the dimension of £^ as a vector space over F, An element ^ G C is said to be a root of unity if ^^ = 1 for some n > 1. A root of unity ^ is a primitive mth root of unity if m is the smallest positive integer with the property that ^^ = 1. The mth cyclotomic polynomial is
the product taken over all primitive mth roots of unity. It is well known that ^rn{x) has coefficients in Q and that this polynomial is irreducible over Q (see, for example, [Coh77, Theorem 2, p. 192]). The degree of ^m{x) is (t>{m)^ where / denotes the Euler function; that is, (m) is the number of integers A:, 1 < A: < m, which are relatively prime to m. Let ^rn be a primitive mth root of unity. Then the mapping
($„(x)) '^^^'^> defined by x + ($7^(x)) \-^ (^ is an isomorphism and so
is a field extension of Q of degree (j){m,) known as a cyclotomic field. Since, for n > 1, the polynomial x'^ — I factors
x" - 1 = J] $^(x), the rational group algebra of the cyclic group Cn of order n is
Q c . - - ^ ^ - - e I ^ M ^ - 0 I Q(e^). (x^-i) ^r ($m(^)) ^r 1.2 Lemma. Let p be a rational prime and let A = (a) be a cyclic group of order p'^, m>l. Let A = AoDAiD^''DAm
= {l}
176
VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS
be the descending chain of all subgroups of A; thus Ai = {a^ ). Then the primitive idempotents of the rational group algebra QA are Co = A
and ei = Ai — A^-i,
I < i < m.
Furthermore^ (QA)e^ = Q(^pi), where ^^i denotes a primitive pHh root of unity, and, for any primitive idempotent e of QA, we have e(l)p^ = [{QA)e: Q], where e{\) denotes the coefficient of 1 in e. PROOF.
Since Q ^ = e™oQ(^p-),
the algebra QA contains precisely m-\-\ primitive idempotents. Since e^^Ai — Ai^ for z > 1, we have coCi = 0 for i > 1. Also, if I < i < i , then AiAj = Ai^ so e^eJ = (A^ - A^.i){Aj
- A^_i) ^^ /I2
/ii/ij
—l
A.'l — l -j- /\i — \ ^^ Jii
Jli/ij — l
which is 0 if i < j and e^ if i = j . It follows that eo, e i , . . . , e^^ are m + 1 pairwise orthogonal (central) idempotents; hence they are primitive. Note that they sum to 1. To prove the last statement of the lemma, we use Lemma VI. 1.2 to observe that {QA)eo = {QA)A = Q and, for 1 < z < m, that (1)
{QA)J,^Q[A/A^]^QB,
where B is cyclic of order p\ By Lemma 1.1, the degree of the cyclotomic field (QA)ei over Q is [{QA){Ji - A^,): Q] = [{QA)'Ar. Q] - [ ( Q ^ ) ^ : Q]
Hence {QA)e^ = Q(^px) for z = 1 , . . . ,m. Also, for i = 1 , . . . , m.
while eo(l)p™ = 1 = [{QA)eo: Q], since (QA)eo = Q- Thus, for any primitive idempotent e of QA, e(l)p'" is as indicated. D
1. COMMUTATIVE GROUP ALGEBRAS
177
1.3 Remark. The proof of Lemma 1.2 strongly depends on the fact that cyclotomic polynomials are irreducible in Q[X]; thus, it can be extended only to group algebras over fields with the same property. The lemma is not true in general; in fact, if F = Fj is the field of seven elements, then X^ - 1 = (X - 1)(X - 2)(X - 4), so the group algebra over F of the cyclic group of order 3 is z.^ ^ nX] ^ F[X] ^ F[X] ^ F[X] ^ - (X3 - 1) " (X - 1) "^ (X - 2) "^ (X - 4)' On the other hand, QC3 ^ Q ® Q ( 6 ) where ^3 denotes a primitive cube root of unity. Thus the rational group algebra of C3 over Q contains only two nontrivial primitive central idempotents while the group algebra of the same group over F contains three, so they cannot be obtained as in Lemma 1.2.
In what follows, we will often use without comment the fact any finite subgroup of a field is cyclic [Coh77, Theorem 1, p. 190]. Also, for r > 1 and unless otherwise stated, the symbol ^r always denotes a primitive rth root of unity. 1.4 Theorem.
1. Let p be a prime and G a finite abelian p-group.
Then the primitive idempotents of QG are G and H — {H^CP"^'^), where H runs through those subgroups of G such that G = {H^c) for some c £ G and G/ H is cyclic of order p ^ > 1. Furthermore, (a) iQG)G^Q. (b) iQG)iH-{H,cP-'))^Qi^,rr.). (c) The number of primitive idempotents e with {QG)e = Qi^d) equals the number of subgroups H with G/H cyclic of order d. 2. [P W50] Let G be a finite abelian group and write G = GiX- - -xGe as the direct product of finite pi-groups Gi, for distinct primes pi^... ^p£. Then the primitive idempotents of QG are of the form ei62 • • *e^, where each Ci is a primitive idempotent of QGi. Furthermore, (a) (QG)eie2 • • -e^ ^ Q{^d) for some d with d\\G\.
178
VII. LOOP ALGEBRAS OF FINITE INDECOMPOSABLE RA LOOPS
(b) The number
of primitive
equals the number
idempotents
of subgroups
e with {QG)e
=
Q(^d)
H of G with G/ H cyclic of
order d. (c) QG = ®Ai^i(^dQ{^d)f
where ad is the number of subgroups
H
a | \(jr\
of G with G/H PROOF.
cyclic of order d.
1. Suppose G is a finite abelian p-group for some prime p. By
Lemma VI.1.2, {QG)G = Q[G/G] = Q, so the idempotent G is primitive. Let e be a primitive idempotent of QG and assume e ^ G, By Maschke's Theorem and commutativity, the semisimple artinian ring QG is the direct sum of
fields.
In particular, (QG)e is a field and it contains the
finite
subgroup Ge = {ge \ g G G} which is necessarily cyclic, say of order p ^ , m > 0. Let K = {g e G \ ge = e } . Then Ge ^ G/K c e G^ where c denotes the coset Kc.
= (c) for some
Since, again by Lemma VL1.2,
QG = (QG)A' e (QG)(1 - A^) and because Ke = e, it follows t h a t (QG)e = (QG)Ke
is actually a simple component of ( Q G ) A \ Thus e is also a primitive
idempotent of ( Q G ) / ? ^ Q[G/A']. Let (fi: QG -^ Q[G/A^] be the homomorphism induced by the natural m a p G —^ G/K.
Recall t h a t e — Ke »-> ^{e) under the isomorphism
(QG)A' -^ Q[G/A']. Thus ^{e) is a primitive idempotent in Q[G/A']. Writing e — YlfgeG^{d)9'> ^id) ^ Q' ^ ^ woie^ t h a t some e{g) ^ 1/|G| because e 7^ G. Since ke = e for any k G A', it follows t h a t e{g) = ^{kg) for any g E G and k ^ K^ so t h a t the coefficients of e are constant on cosets of A'. Thus
(2)
^= J2 ^^•^^'^' i=0
with each ai G Q and, for some i, ai/\K\ ai ^ l/\G/K\.
^
1/|G|; t h a t is, for some i,
Since
(3)
V(e) = Yl "«^' t=0
it must be t h a t (p{e) ^ G/K.
Hence Lemma 1.2 implies t h a t 1, Let e = H - (7J,CP*"'). Then
^'
'
heH 0<j 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 p-group 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 pi-subgroups 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 — ei---e^, 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 Gi-i], 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 2-loop 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 ley-Dickson 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 Cayley-Dickson 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 2-loop 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 2-groups. 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 Cayley-Dickson 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 Cayley-Dickson 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 ley-Dickson 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 2-group). 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 Cayley-Dickson 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 Cayley-Dickson 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 ley-Dickson 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 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 Cayley-Dickson 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 2-group 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 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 Cayley-Dickson 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,
/l-exy\ ^12 =
/ l + 0x2/\ 2/-0a: ;:; e = —-—e
y
T:
1 + Qxy ^ e,
622 =
and ^21 = I where 0 = tf^^ let
1 + Qxy\ /l-0xy\ 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 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 Cayley-Dickson algebra and that {QD)e = M2(Q(6-i)). Noting that mi > 1, let 0 = tf~\. Then 0^ = - e and {Qxy'^y = e. It is readily verified that PROOF.
1 - exy-'^ en
=
l-Qxy-^\
e ,
€22 =
/ l + 0 x y - i '\
1 + 0xy-i 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(6-i ))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 so-called 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 commutator-associator 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 torsion-free PROOF.
197
abelian group is an ordered
group.
Let A be a torsion-free 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 torsion-free, 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 commutator-associator 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 commutator-associator 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, 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 + {l-s)P2
=
[±^+{l-s)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 g-th £ 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) 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 ^^* = ^2m-2 _^ ^ ^ - 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'^^ = g2-2m
^ p ^ - m ^ g^ g2i _ g-2
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 2-group. Recall t h a t a hamiltonian 2-group is (isomorphic to) Qs X Ey where Qs is the quaternion group of order eight and E is a, (possibly trivial) elementary abelian 2-group. 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) =
2-group or a hamiltonian
Moufang 2-loop.
±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 commutator-associator 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^.
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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 + ali-al
+ 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) - n-2.
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 l-A. 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 +