1.8 Direct Products and Direct Sums

Home , Direct product, Direct product of groups, Direct sum

1.8. DIRECT PRODUCTS AND DIRECT SUMS 25

We study products in the category of groups and coproducts in the category of abelian groups.

The Cartesian product of a family of groups {Gi | i ∈ I} is Y [ Gi = {f : I → Gi | f(i) ∈ Gi for i ∈ I}. i∈I i∈I Q Deﬁne the product operation in i∈I Gi by product of functions: Y (fg)(i) = f(i)g(i), for i ∈ I, f, g ∈ Gi. i∈I Q Then i∈I Gi becomes a group, called the direct product of groups {Gi | i ∈ I}. Q Q If we express f ∈ i∈I Gi as {f(i)}i∈I or simply {f(i)}, then the product in i∈I Gi is:

{ai}{bi} = {aibi}, ai, bi ∈ Gi, for all i ∈ I.

Thm 1.35. Let {Gi | i ∈ I} be a family of groups. For each k ∈ I, the map Y πk : Gi → Gk, f 7→ f(k)[or {ai} 7→ ak] i∈I is an epimorphism of groups. The πk is called the canonical projection of the direct product. Q Thm 1.36 ( i∈I Gi is a product in the category of groups). Let {Gi | i ∈ I} be a family of groups and {ϕi : H → Gi | i ∈ I} a family of group homomorphisms. Then there is a unique Q homomorphism ϕ : H → i∈I Gi such that πiϕ = ϕi for all i ∈ I and this properly determines Q i∈I Gi uniquely up to isomorphism. Q i∈I Gi ϕ ; πi

ϕi H / Gi

Q 0 Proof. Deﬁne ϕ : H → i∈I Gi by h 7→ {ϕi(h)}i∈I . Then ϕ is well-deﬁned and for h, h ∈ H: 0 0 0 0 ϕ(h)ϕ(h ) = {ϕi(h)}{ϕi(h )} = {ϕi(hh )} = ϕ(hh ). Q So ϕ is a homomorphism of groups. Obviously, πiϕ(h) = ϕi(h). So i∈I 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 {Gi | i ∈ I}, denoted Y w Y Gi, is the set of all f ∈ Gi such that f(i) = ei for all but a ﬁnite number of i ∈ I. i∈I i∈I Y w X If all Gi are abelian, Gi is called the (external) direct sum and is denoted Gi. i∈I i∈I

Thm 1.37. Let {Gi | i ∈ I} be a family of groups. Then

Y w Y 1. Gi is a normal subgroup of Gi; i∈I i∈I Q 2. for each k ∈ I, the map ιk : Gk → i∈I Gi, ιk(a) = {ai}i∈I , where ai = ei for i 6= k and ak = a, is a monomorphism; Q 3. for each i ∈ I, ιi(Gi) C i∈I 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 {Ai | i ∈ I} be a i∈I family of additive groups. If B is an additive group and {ψi : Ai → B | i ∈ I} a family of X homomorphisms, then there is a unique homomorphism ψ : Ai → B such that ψ ◦ ιi = ψi i∈I X for i ∈ I and this property determines Ai uniquely up to isomorphism. i∈I

ψi A / B i ;

ιi ψ P i∈I Ai

X X 0 Proof. Every b ∈ Ai can be uniquely written as b = ιi(ai) for ai ∈ Ai and I a ﬁnite i∈I i∈I0 X subset of I. Let ψ : Ai → B be deﬁned by i∈I ! X X ψ ιi(ai) := ψi(ai), ai ∈ Ai. i∈I0 i∈I0

Then ψ is a homomorphism of abelian groups, and ψ 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 {Ni | i ∈ I} be a family of normal subgroups of a group G. The subgroup [ h Nii consists of elements of the form i∈I

ni1 ni2 ··· nik for nij ∈ Nij , where i1, i2, ··· , ik are distinct elements of I. In particular, if I = {1, 2, ··· , n} has ﬁnite n [ cardinality, then h Nii = N1N2 ··· Nn. i=1

Thm 1.40. Let {Ni | i ∈ I} be a family of normal subgroups of a group G such that [ 1. G = h Nii; i∈I [ 2. for each k ∈ I, Nk ∩ h Nii = hei. i6=k

Y w Then G ' Ni. i∈I

−1 −1 Proof. If ni ∈ Ni and nj ∈ Nj for i 6= j, then ninjni nj ∈ Ni ∩ Nj = {e}. So ni and nj commute. Y w Every element of Ni is of the form {ai}i∈I where ai = e for all but finitely many i∈I Y w i ∈ I. Let I0 be the finite set {i ∈ I | ai 6= e}. Define the map ϕ : Ni → G, given i∈I Y by {ai}i∈I 7→ ai and {e}i∈I 7→ e. Then ϕ is an epimorphism of groups. If a noniden-

i∈I0 tity element {a } ∈ ker(ϕ), then I 6= ∅, Q a = e, and for any k ∈ I there is i i∈I 0 i∈I0 i 0 −1 Y [ ak = ai ∈ Nk ∩ h Nii = hei. It contradicts the assumption of I0. So ϕ is a monomor- i∈I0 i6=k i6=k phism and thus an isomorphism.

Def. If a family of normal subgroups {Ni | i ∈ I} satisﬁes the assumptions in Theorem 1.40, then G is said to be the internal weak direct product of {Ni | i ∈ I} (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 {Ni | i ∈ I} if and only if every element a of G can be uniquely written as a product

a = ni1 ni2 ··· nik for nij ∈ Nij , where i1, i2, ··· , ik are distinct elements of I.

Thm 1.42. Let {fi : Gi → Hi | i ∈ I} be a family of homomorphisms of groups and let Y Y Y f = fi : Gi → Hi given by {ai} 7→ {fi(ai)}. Then f is a group homomorphism and i∈I i∈I ! Y Y Y w Y w Ker f = Ker fi, Im f = Im fi, f Gi ⊂ Hi. i∈I i∈I i∈I i∈I

Cor 1.43. Let Ni C Gi for groups Gi, i ∈ I. Then Y Y Y Y Y 1. Ni C Gi and Gi/ Ni ' Gi/Ni. i∈I i∈I i∈I i∈I i∈I

Y w Y w Y w Y w Y w 2. Ni C Gi and Gi/ Ni ' Gi/Ni. i∈I i∈I i∈I i∈I i∈I