Groups and Group Actions
Proposition 4.18. Let S be a nonempty set of abelian groups, and let G s be
the group corresponding to the members of S. If(G,{is})and(G0,{is0}) are
two direct sums, then the homomorphismsis : Gs → G andis0 : Gs → G0 are
one-one, there exists a unique homomorphism8:G→G0such thati0
s =8◦is
for alls ∈S, and8is an isomorphism.
PROOF. In Figure 4.5 let H = G0 and ϕs = i0
s. If 8 : G → G0 is the
homomorphism produced by the fact thatGis a direct sum, then we have8◦is
= i0
s for alls. Reversing the roles ofG andG0, we obtain a homomorphism
80:G0→Gwith80◦i0
s =is for alls. Therefore(80◦8)◦is =80◦is0 =is.
In Figure 4.5 we next letH =G andϕs =is for alls. Then the identity 1G
onGhas the same property 1G◦is =is relative to allis that80◦8has, and the
uniqueness says that80◦8=1G. Reversing the roles ofG andG0, we obtain
8◦80=1G0. Therefore8is an isomorphism.
For uniqueness suppose that9 : G → G0 is another homomorphism with
i0
s =9◦isfor alls ∈S. Then the argument of the previous paragraph shows that
80◦9=1G. Applying8on the left gives9=(8◦80)◦9 =8◦(80◦9)=
8◦1G =8. Thus9=8.
Finally we have to show that thesthmapping of a direct sum is one-one on Gs. It is enough to show thatis0is one-one onGs. TakingGas the external direct
sumLs∈SGs withis equal to the embedding mapping, form the isomorphism
80:G0→Gthat has just been proved to exist. This satisfiesis =80◦i0
s for all
s∈S. Sinceis is one-one,is0 must be one-one. §
EXAMPLE. The groupQ×is the direct sum of copies ofZ, one for each prime,
plus one copy ofZ/2Z. If pis a prime, the mappingip : Z → Q× is given
byip(n) = pn. The remaining coordinate gives the sign. The isomorphism
results from unique factorization, only finitely many primes being involved for any particular nonzero rational number.
4. Rings and Fields
In this section we begin a two-section digression in order to develop some more number theory beyond what is in Chapter I and to make some definitions as new notions arise. In later sections of the present chapter, some of this material will yield further examples of concrete groups and tools for working with them.
We begin with the additive groupZ/mZof integers modulo a positive integer m. We continue to write [a] for the equivalence class of the integerawhen it is helpful to do so. Our interest will be in the multiplication structure thatZ/mZ inherits from multiplication inZ. Namely, we attempt to define
[a][b]=[ab].
To see that this formula is meaningful inZ/mZ, we need to check that the same equivalence class results on the right side if the representatives of [a] and [b] are changed. Thus let [a] = [a0] and [b] = [b0]. Thenm dividesa−a0 and
b−b0and must divide the sum of products(a−a0)b+a0(b−b0)=ab−a0b0.
Consequently [ab]=[a0b0], and multiplication is well defined. Ifxandyare in Z/mZ, their product is often denoted byx ymodm.
The same kind of argument as just given shows that the associativity of multi- plication inZand the distributive laws imply corresponding facts aboutZ/mZ. The result is thatZ/mZis a “commutative ring with identity” in the sense of the following definitions.
Aringis a set Rwith two operations R× R → R, usually calledaddition
andmultiplicationand often denoted by(a,b)7→a+band(a,b)7→ab, such that
(i) Ris an abelian group under addition,
(ii) multiplication is associative in the sense thata(bc)=(ab)cfor alla,b,c in R,
(iii) the two distributive laws
a(b+c)=(ab)+(ac) and (b+c)a=(ba)+(ca)
hold for alla,b,cinR.
The additive identity is denoted by 0, and the additive inverse ofais denoted by −a. A suma+(−b)is often abbreviateda−b. By convention when parentheses are absent, multiplications are to be carried out before additions and subtractions. Thus the distributive laws may be rewritten as
a(b+c)=ab+ac and (b+c)a=ba+ca.
A ringRis called acommutative ringif multiplication satisfies the commutative law
A ring Ris called a ring with identity6 if there exists an element 1 such that
1a=a1=afor allain R. It is immediate from the definitions that
• 0a = 0 anda0= 0 in any ring (since, in the case of the first formula, 0=0a−0a=(0+0)a−0a=0a+0a−0a=0a),
• the multiplicative identity is unique in a ring with identity (since 10 =
101=1),
• (−1)a = −a=a(−1)in any ring with identity (partly since 0=0a =
(1+(−1))a=1a+(−1)a=a+(−1)a).
In a ring with identity, it will be convenient not to insist that the identity be different from the zero element 0. If 1 and 0 do happen to coincide inR, then it readily follows that 0 is the only element ofR, and Ris said to be thezero ring. The setZof integers is a basic example of a commutative ring with identity. Returning toZ/mZ, suppose now thatmis a prime p. If [a] is inZ/pZwitha
in{1,2, . . . ,p−1}, then GCD(a,p)=1 and Proposition 1.2 produces integers
r andswithar+ ps =1. Modulop, this equation reads [a][r]=[1]. In other words, [r] is a multiplicative inverse of [a]. The result is thatZ/pZ, whenpis a prime, is a “field” in the sense of the following definition.
AfieldFis a commutative ring with identity such thatF6=0 and such that (v) to eacha6=0 inFcorresponds an elementa−1inFsuch thataa−1
=1. In other words,F×=F− {0}is an abelian group under multiplication. Inverses
are necessarily unique as a consequence of one of the properties of groups. When p is prime, we shall write Fp for the fieldZ/pZ. Its multiplicative
groupF×
p has orderp−1, and Lagrange’s Theorem (Corollary 4.8) immediately
implies thatap−1
≡1 mod pwheneveraand pare relatively prime. This result is known asFermat’s Little Theorem.7
For generalm, certain members ofZ/mZhave multiplicative inverses. The product of two such elements is again one, and the inverse of one is again one. Thus, even thoughZ/mZneed not be a field, the subset(Z/mZ)×of members ofZ/mZwith multiplicative inverses is a group. The same argument as whenm is prime shows that the class ofahas an inverse if and only if GCD(a,m)=1. The number of such classes was defined in Chapter I in terms of the Eulerϕ
function asϕ(m), and a formula forϕ(m)was obtained in Corollary 1.10. The 6Some authors, particularly when discussing only algebra, find it convenient to incorporate the existence of an identity into the definition of a ring. However, in real analysis some important natural rings do not have an identity, and the theory is made more complicated by forcing an identity into the picture. For example the space of integrable functions onRforms a very natural ring, with convolution as multiplication, and there is no identity; forcing an identity into the picture in such a way that the space remains stable under translations makes the space large and unwieldy. The distinction between working with rings and working with rings with identity will be discussed further in Section 11.
conclusion is that(Z/mZ)×is an abelian group of orderϕ(m). Application of
Lagrange’s Theorem yields Euler’s generalization of Fermat’s Little Theorem, namely thataϕ(m)
≡ 1 modm for every positive integerm and every integera relatively prime tom.
More generally, in any ringRwith identity, aunitis defined to be any element asuch that there exists an elementa−1withaa−1
=a−1a
=1. The elementa−1
is unique if it exists8and is called themultiplicative inverseofa. The units ofR
form a group denoted byR×. For example the groupZ×consists of+1 and−1,
and the zero ringRhasR×= {0}. If Ris a nonzero ring, then 0 is not inR×.
Here are some further examples of fields. EXAMPLES OF FIELDS.
(1)Q,R, andC. These are all fields.
(2)Q[θ]. This was introduced between Examples 8 and 9 of Section 1. It is assumed thatθ is a complex number and that there exists an integern > 0 such that the complex numbers 1, θ, θ2, . . . , θn are linearly dependent overQ. The setQ[θ] is defined to be the linear span overQof all powers 1, θ, θ2, . . . of
θ, which is the same as the linear span of the finite set 1, θ, θ2, . . . , θn−1. The
setQ[θ] was shown in Proposition 4.1 to be a subset ofCthat is closed under the arithmetic operations, including the passage to reciprocals in the case of the nonzero elements. It is therefore a field.
(3) A field of 4 elements. LetF4= {0,1, θ, θ+1}, whereθis some symbol not
standing for 0 or 1. Define addition inF4and multiplication inF×4 by requiring
thata+0=0+a=afor alla, that
1+1=0, 1+θ =(θ+1), 1+(θ+1)=θ, θ+1=(θ+1), θ+θ =0, θ+(θ+1)=1, (θ+1)+1=θ, (θ+1)+θ =1, (θ+1)+(θ+1)=0, and that 11=1, 1θ =θ, 1(θ+1)=(θ+1), θ1=θ, θ θ =(θ+1), θ (θ+1)=1, (θ+1)1=(θ+1), (θ+1)θ =1, (θ+1)(θ+1)=θ.
The result is a field. With this direct approach a certain amount of checking is necessary to verify all the properties of a field. We shall return to this matter in Chapter IX when we consider finite fields more generally, and we shall then have a way of constructingF4that avoids tedious checking.
8In fact, ifb andcexist withab
= ca = 1, thena is a unit witha−1
= b = cbecause
In analogy with the theory of groups, we define asubringof a ring to be a nonempty subset that is closed under addition, negation, and multiplication. The set 2Zof even integers is a subring of the ringZof integers. Asubfieldof a field is a subset containing 0 and 1 that is closed under addition, negation, multiplication, and multiplicative inverses for its nonzero elements. The setQof rationals is a subfield of the fieldRof reals.
Intermediate between rings and fields are two kinds of objects—integral do- mains and division rings—that arise frequently enough to merit their own names. The setting for the first is a commutative ring R. A nonzero elementa of R is called azero divisorif there is some nonzerobin R withab = 0. For example the element 2 in the ringZ/6Zis a zero divisor because 2·3 = 0. Anintegral domainis anonzerocommutative ring with identity having no zero divisors. Fields have no zero divisors since ifaandbare nonzero, thenab =0 would forceb = 1b = (a−1a)b = a−1(ab) = a−10 = 0 and would give a
contradiction; therefore every field is an integral domain. The ring of integers
Zis another example of an integral domain, and the polynomial ringsQ[X] and
R[X] andC[X] introduced in Section I.3 are further examples. A cancellation law for multiplication holds in any integral domain:
ab=ac with a6=0 implies b=c.
In fact,ab=acimpliesa(b−c)=0; sincea6=0,b−cmust be 0.
The other object with its own name is adivision ring, which is a nonzero ring with identity such that every nonzero element is a unit. The commutative division rings are the fields, and we have encountered only one noncommutative division ring so far. That is the setHof quaternions, which was introduced in Section 1. Division rings that are not fields will play only a minor role in this book but are of great interest in Chapters II and III ofAdvanced Algebra.
Let us turn to mappings. A functionϕ : R → R0 between two rings is an isomorphismof rings ifϕis one-one onto and satisfiesϕ(a+b)=ϕ(a)+ϕ(b)
andϕ(ab) = ϕ(a)ϕ(b) for all a andb in R. In other words, ϕ is to be an isomorphism of the additive groups and to satisfyϕ(ab) = ϕ(a)ϕ(b). Such a mapping carries the identity, if any, inR to the identity ofR0. The relation “is
isomorphic to” is an equivalence relation. Common notation for an isomorphism of rings is R ∼= R0; because of the symmetry, one can say that Rand R0 are isomorphic.
A functionϕ : R→ R0between two rings is ahomomorphismof rings ifϕ
satisfiesϕ(a+b) = ϕ(a)+ϕ(b)andϕ(ab) = ϕ(a)ϕ(b)for allaandbin R. In other words,ϕis to be a homomorphism of the additive groups and to satisfy
ϕ(ab)=ϕ(a)ϕ(b).
EXAMPLES OF HOMOMORPHISMS OF RINGS.
(2) The evaluation mappingϕ:R[X]→Rgiven byP(X)7→ P(r)for some fixedr inR.
(3) Mappings with the direct productZ×Z. The additive groupZ×Zbecomes a commutative ring with identity under coordinate-by-coordinate multiplication, namely(a,a0)+(b,b0) = (a+b,a0+b0). The identity is(1,1). Projection (a,a0)7→ato the first coordinate is a homomorphism of ringsZ×Z→Zthat
carries identity to identity. Inclusiona7→(a,0)ofZinto the first coordinate is a homomorphism of ringsZ→Z×Zthat does not carry identity to identity.9