1.8. DIRECT PRODUCTS AND DIRECT SUMS 25 1.8 Direct Products and Direct Sums We study products in the category of groups and coproducts in the category of abelian groups. The Cartesian product of a family of groups fGi j i 2 Ig is Y [ Gi = ff : I ! Gi j f(i) 2 Gi for i 2 Ig: i2I i2I Q Define the product operation in i2I Gi by product of functions: Y (fg)(i) = f(i)g(i); for i 2 I; f; g 2 Gi: i2I Q Then i2I Gi becomes a group, called the direct product of groups fGi j i 2 Ig. Q Q If we express f 2 i2I Gi as ff(i)gi2I or simply ff(i)g, then the product in i2I Gi is: faigfbig = faibig; ai; bi 2 Gi; for all i 2 I: Thm 1.35. Let fGi j i 2 Ig be a family of groups. For each k 2 I, the map Y πk : Gi ! Gk; f 7! f(k)[or faig 7! ak] i2I is an epimorphism of groups. The πk is called the canonical projection of the direct product. Q Thm 1.36 ( i2I Gi is a product in the category of groups). Let fGi j i 2 Ig be a family of groups and f'i : H ! Gi j i 2 Ig a family of group homomorphisms. Then there is a unique Q homomorphism ' : H ! i2I Gi such that πi' = 'i for all i 2 I and this properly determines Q i2I Gi uniquely up to isomorphism. Q i2I Gi ' ; πi 'i H / Gi Q 0 Proof. Define ' : H ! i2I Gi by h 7! f'i(h)gi2I . Then ' is well-defined and for h; h 2 H: 0 0 0 0 '(h)'(h ) = f'i(h)gf'i(h )g = f'i(hh )g = '(hh ): Q So ' is a homomorphism of groups. Obviously, πi'(h) = 'i(h). So i2I Gi is a product in the category of groups. The uniqueness is evident. 26 CHAPTER 1. GROUPS Def. The (external) weak direct product of a family of groups fGi j i 2 Ig, denoted Y w Y Gi, is the set of all f 2 Gi such that f(i) = ei for all but a finite number of i 2 I. i2I i2I Y w X If all Gi are abelian, Gi is called the (external) direct sum and is denoted Gi. i2I i2I Thm 1.37. Let fGi j i 2 Ig be a family of groups. Then Y w Y 1. Gi is a normal subgroup of Gi; i2I i2I Q 2. for each k 2 I, the map ιk : Gk ! i2I Gi, ιk(a) = faigi2I , where ai = ei for i 6= k and ak = a, is a monomorphism; Q 3. for each i 2 I, ιi(Gi) C i2I Gi. The maps ιk above are called the canonical injections. X Thm 1.38 ( Ai is a coproduct in the category of abelian groups). Let fAi j i 2 Ig be a i2I family of additive groups. If B is an additive group and f i : Ai ! B j i 2 Ig a family of X homomorphisms, then there is a unique homomorphism : Ai ! B such that ◦ ιi = i i2I X for i 2 I and this property determines Ai uniquely up to isomorphism. i2I i A / B i ; ιi P i2I Ai X X 0 Proof. Every b 2 Ai can be uniquely written as b = ιi(ai) for ai 2 Ai and I a finite i2I i2I0 X subset of I. Let : Ai ! B be defined by i2I ! X X ιi(ai) := i(ai); ai 2 Ai: i2I0 i2I0 Then is a homomorphism of abelian groups, and makes the above diagram commutative. 1.8. DIRECT PRODUCTS AND DIRECT SUMS 27 Remark. The theorem is false if the word abelian/additive is omitted. The external weak direct product is not a coproduct in the category of all groups. Thm 1.39. Let fNi j i 2 Ig be a family of normal subgroups of a group G. The subgroup [ h Nii consists of elements of the form i2I ni1 ni2 ··· nik for nij 2 Nij ; where i1; i2; ··· ; ik are distinct elements of I. In particular, if I = f1; 2; ··· ; ng has finite n [ cardinality, then h Nii = N1N2 ··· Nn. i=1 Thm 1.40. Let fNi j i 2 Ig be a family of normal subgroups of a group G such that [ 1. G = h Nii; i2I [ 2. for each k 2 I, Nk \ h Nii = hei. i6=k Y w Then G ' Ni. i2I −1 −1 Proof. If ni 2 Ni and nj 2 Nj for i 6= j, then ninjni nj 2 Ni \ Nj = feg. So ni and nj commute. Y w Every element of Ni is of the form faigi2I where ai = e for all but finitely many i2I Y w i 2 I. Let I0 be the finite set fi 2 I j ai 6= eg. Define the map ' : Ni ! G, given i2I Y by faigi2I 7! ai and fegi2I 7! e. Then ' is an epimorphism of groups. If a noniden- i2I0 tity element fa g 2 ker('), then I 6= ;, Q a = e, and for any k 2 I there is i i2I 0 i2I0 i 0 −1 Y [ ak = ai 2 Nk \ h Nii = hei. It contradicts the assumption of I0. So ' is a monomor- i2I0 i6=k i6=k phism and thus an isomorphism. Def. If a family of normal subgroups fNi j i 2 Ig satisfies the assumptions in Theorem 1.40, then G is said to be the internal weak direct product of fNi j i 2 Ig (or internal direct sum if G is abelian). 28 CHAPTER 1. GROUPS Thm 1.41. A group G is the internal weak direct product of a family of its normal subgroups fNi j i 2 Ig if and only if every element a of G can be uniquely written as a product a = ni1 ni2 ··· nik for nij 2 Nij ; where i1; i2; ··· ; ik are distinct elements of I. Thm 1.42. Let ffi : Gi ! Hi j i 2 Ig be a family of homomorphisms of groups and let Y Y Y f = fi : Gi ! Hi given by faig 7! ffi(ai)g. Then f is a group homomorphism and i2I i2I ! Y Y Y w Y w Ker f = Ker fi; Im f = Im fi; f Gi ⊂ Hi: i2I i2I i2I i2I Cor 1.43. Let Ni C Gi for groups Gi, i 2 I. Then Y Y Y Y Y 1. Ni C Gi and Gi= Ni ' Gi=Ni. i2I i2I i2I i2I i2I Y w Y w Y w Y w Y w 2. Ni C Gi and Gi= Ni ' Gi=Ni. i2I i2I i2I i2I i2I.