9 Direct Products, Direct Sums, and Free Abelian Groups

Home , Category of groups, Direct product, Direct product of groups, Direct sum, Free product

9 Direct products, direct sums, and free abelian groups

9.1 Deﬁnition. A direct product of a family of groups Gi i I is a group { } ∈ i I Gi deﬁned as follows. As a set i I Gi is the cartesian product of the ∈ ∈ groups Gi.Givenelements(ai)i I , (bi)i I i I Gi we set ∈ ∈ ∈ ∈

(ai)i I (bi)i I := (aibi)i I ∈ · ∈ ∈

9.2 Deﬁnition. A weak direct product of a family of groups Gi i I is the { } ∈ subgroup of i I Gi given by ∈ w Gi := (ai)i I ai = ei Gi for ﬁnitely many i only { ∈ | ∈ } } i I ∈ w If all groups Gi are abelian then i I Gi is denoted i I Gi and it is called the ∈ ∈ direct sum of Gi i I . { } ∈

w 9.3 Note. If I is a ﬁnite set then i I Gi = i I Gi. ∈ ∈

9.4 Example.

Z/2Z Z/2Z = Z/2Z Z/2Z = (0, 0), (0, 1), (1, 0), (1, 1) × ⊕ { }

Note. Z/2Z Z/2Z is a the smallest non-cyclic group. It is called the Klein four group. ⊕

9.5 Example.

Z/2Z Z/3Z = (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2) ⊕ { } Note. Z/2Z Z/3Z is a cyclic group ((1, 1) is a generator), thus ⊕ Z/2Z Z/3Z = Z/6Z ⊕ ∼

27 9.6. Let S be a set. Denote by Fab(S) the set of all expressions of the form

kxx x S ∈ where kx Z and kx =0for ﬁnitely many x X only. ∈ ∈

Fab(S) is an abelian group with addition deﬁned by

kxx + lxx := (kx + lx)x x S x S x S ∈ ∈ ∈

9.7 Deﬁnition. The group Fab(S) is called the free abelian group generated by the set S.

In general a group G is free abelian if G ∼= Fab(S) for some set S.

9.8 Proposition. If S is a set then

Fab(S) ∼= Z x S ∈

Proof. The isomorphism is given by

f : Fab(S) Z,f( kxx)=(kx)x S → ∈ x S x S ∈ ∈

9.9 Note. We have a map of sets

i: S F (S),i(x)=1 x → ab ·

28 9.10 Theorem (The universal property of free abelian groups). Let S be a set and G be an abelian group. For any map of sets f : S G there exists a unique homomorphism f¯: F (S) G such that the following→ diagram commutes: → f S G i f¯ Fab(S)

Proof. Deﬁne f¯ by

f¯ kxx := kxf(x) x S x S ∈ ∈ Note: this is well deﬁned since k =0for almost all x S. x ∈

29 10 Categories and functors

10.1 Deﬁnition. A category C consists of

1) acollectionofobjects Ob(C)

2) for any a, b Ob(C) asetHomC(a, b) of morphisms from a to b ∈ 3) for any a, b, c Ob(C) afunction(“compositionlaw”) ∈

HomC(a, b) HomC(b, c) HomC(a, c) × → (f,g) g f → ◦ such that the following conditions are satisﬁed:

Associativity. • f (g h)=(f g) h ◦ ◦ ◦ ◦ for any morphisms f,g,h for which these compositions are deﬁned.

Identity. For any c Ob(C) there is a morphism idc HomC(c, c) such • that ∈ ∈ f id = f, id g = g ◦ c c ◦ for any f HomC(c, d),g HomC(b, c). ∈ ∈

10.2 Examples.

1) Set = the category of all sets.

Ob(Set)=the collection of all sets • HomS (A, B)= all maps of sets f : A B • et { → }

2) Gr = the category of all groups

Ob(Gr)=the collection of all groups • HomG (G, H)= all homomorphisms f : G H • r { → }

30 3) Ab = the category of all abelian groups

Ob(Ab)=the collection of all abelian groups • HomA (G, H)= all homomorphisms f : G H • b { → }

4) Top = the category of all topological spaces

Ob(Top)=the collection of all topological spaces • HomT (X, Y )= all continuous maps f : X Y • op { → }

5) Let G be a group. Deﬁne a category CG as follows: Ob(C )= • G {∗} HomC ( , )= elements of G • G ∗ ∗ { } composition of morphisms = multiplication in G •

6) AverysmallcategoryC: f c d Ob(C)= c, d • { } HomC(c, d)= f , HomC(d, c)=∅, HomC(c, c)=idc, HomC(d, d)= idd • { }

10.3 Deﬁnition. Amorphismf : c d in a category C is an isomorphism if there exists a morphism g : d c such→ that gf =id and fg =id. → c d If for some c, d C there exist an isomorphism f : c d then we say that the ∈ → objects c and d are isomorphic and we write c ∼= d.

10.4 Note. For an object c C deﬁne ∈ Aut(c):= all isomorphisms f : c c { → } Aut(c) with composition of morphisms is a group.

31 10.5 Deﬁnition. Let C, D be categories. A (covariant) functor F : C D consists of →

1) an assignment Ob(C) Ob(D),c F (c) → →

2) for every c, c C afunction ∈

HomC(c, c) HomD(F (c),F(c)),f F (f) → →

such that F (gf)=F (f)F (g) and F (idc)=idF (c).

10.6 Note. If F : C D is a functor and f : c c is an isomorphism in C → → then F (f): F (c) F (c) is an isomorphism in D. → In particular if c ∼= c in C then F (c) ∼= F (c) in D.

10.7 Examples.

1) U : Gr Set → If G Gr then U(G)= the set of elements of G ∈ { } If f : G H is a homomorphism then U(f): U(G) U(H) is the map of sets underlying→ this homomorphism. →

2) U : Ab Set, → deﬁned the same way as in 1).

Note. The functors U in 1), 2) are called forgetful functors.

3) Let G be a group. The commutator of a, b G is the element ∈ 1 1 [a, b]:=aba− b−

Note: [a, b]=e iﬀ ab = ba.

32 The commutator subgroup of G is the subgroup [G, G] G generated by the set S = [a, b] a, b G . ⊆ { | ∈ } Note.

(a) [G, G]= e iﬀ G is an abelian group. { } (b) [G, G] is a normal subgroup of G (check!). (c) G/[G, G] is an abelian group (check!). (d) If f : G H is a homomorphism then f([G, G]) [H, H]. → ⊆ (e) If f : G H is a homomorphism then f induces a homomorphism → f : G/[G, G] H/[H, H] ab →

given by fab(a[G, G]) = f(a)[H, H].

The abelianization functor Ab: Gr Ab is given by →

Ab(G):=G/[G, G], Ab(f):=fab

4) Recall: if S is a set then F (S) is the free group generated by S. Amapofsetsf : S T deﬁnes a homomorphism → f˜: F (S) F (T ) → λ given by f˜(xλ1 xλ2 x k )=f(x )λ1 f(x )λ2 f(x )λk . 1 2 ····· k 1 2 ····· k

Check: the assignment

S F (S), (f : S T ) (f˜: F (S) F (T )) → → → → Deﬁnes a functor F : Set Gr.Thisisthefree group functor. →

5) Similarly we have the free abelian group functor

F : Set Ab ab → where

33 F (S)=the free abelian group generated by the set S • ab if f : S T then F (f): F (S) F (T ) is given by • → ab ab → ab

Fab(f) kxx = kxf(x) x S x S ∈ ∈

34 11 Adjoint functors

11.1 Deﬁnition. Given two functors

L: C D and R: D C → → we say that L is the left adjoint functor of R and that R is the right adjoint functor of L if for any object c C we have a morphism ηc : c RL(c) such that: ∈ →

1) for any morphism f : c c in C the following diagram commutes: → f c c

ηc η c

RL(c) RL(c) RL(f)

2) for any c C and d D the map of sets ∈ ∈

HomD(L(c),d) HomC(c, R(d)) −→ f η R(f) (L(c) d) (c c RL(c) R(d)) → −→ → → is a bijection.

In such situation we say that (L, R) is an adjoint pair of functors.

11.2 Note.

1) The collection of morphisms ηc c C is called the unit of adjunction of (L, R). { } ∈

2) For any adjoint pair (L, R) we also have morphisms εd : LR(d) d d D { → } ∈ satisfying analogous conditions as ηc c C.Thiscollectionofmorphismsiscalled ∈ the counit of the adjunction. { }

35 11.3 Note. The morphism ηc is universal in the following sense. For any d D and any morphism f : c R(d) in C there is a unique morphism f¯: L(c) ∈ d in D such that the following→ diagram commutes: →

f c R(d) ηc R(f¯) RL(c)

This property is equivalent to part 2) of Deﬁnition 11.1.

11.4 Examples.

1) Recall that we have functors:

F : Set Gr, Gr Set: U → ← where F = free group functor, U = forgetful functor. The pair (F, U) is an adjoint pair. For S Set the unit of adjunction is given by the function ∈

i : S UF(S),i(x)=x S → S The universal property of free groups (9.10)saysthatforanyG Gr and any map of sets f : S U(G) there is a unique homomorphism∈ f¯: F (S) G such that we have→ a commutative diagram → f S U(G) iS U(f¯) UF(S)

36 2) We have functors

F : Set Ab, Set Ab: U ab → ←

where Fab = free abelian group functor, U = forgetful functor. Similarly as in 1) one can check that (Fab,U) is an adjoint pair.

3) Recall that we have the abelianization functor

Ab: Gr Ab, Ab(G)=G/[G, G] → This functor is left adjoint to the inclusion functor

J : Ab Gr, J(G)=G → (check!).

11.5 Note. It is not true that every functor has a left or right adjoint.

37 12 Categorical products and coproducts

12.1 Deﬁnition. Let ci i I be a family of objects in a category C.A(categor- { } ∈ ical) product of the family ci i I is an object p C equipped with morphisms ∈ π : p c for all i I that{ satisﬁes} the following∈ universal property. For any i → i ∈ object d C and a family of morphisms fi : d ci i I there exists a unique ∈ morphism∈f : d p such that π f = f for{ all i →I. } → i i ∈

f1 d f p c1 π1

f2 π2

12.2 Note. If a categorical product of ci i I exists then it is deﬁned uniquely ∈ up to isomorphism. We then write: { }

p = ci i I ∈

12.3 Examples.

1) In the category of groups Gr the categorical product of a family Gi i I { } ∈ is the direct product of groups i I Gi. ∈ Indeed, we have projection homomorphisms:

πi0 : Gi Gi0 ,πi0 ((gi)i I )=gi0 → ∈ i I ∈ Also, if for some group H we have homomorphisms fi : H Gi then this deﬁnes a homomorphism →

f : H Gi,f(h)=(fi(h))i I → ∈ i I ∈ Moreover, f is the unique homomorphism such that we have πif = fi.

38 2) By a similar argument if Gi i I is a family of abelian groups then the { } ∈ direct product Gi is the categorical product of the family Gi i I in i I ∈ the category Ab. ∈ { }

3) In the category Set the categorical product of a family of sets Ai i I is { } ∈ the cartesian product of sets i I Ai. ∈

12.4 Deﬁnition. Let ci i I be a family of objects in a category C.A(categori- { } ∈ cal) coproduct of the family ci i I is an object d C equipped with morphisms ∈ ε : c d for all i I that{ satisﬁes} the following∈ universal property. For any i i → ∈ object b C and a family of morphisms fi : ci b i I there exists a unique ∈ morphism∈f : d b such that fε = f for{ all i →I. } → i i ∈

ε1 f1

ε c 2 2 d f b f2

12.5 Note. If a categorical coproduct of ci i I exists then it is deﬁned uniquely ∈ up to isomorphism. We then write: { }

d = ci i I ∈

12.6 Examples.

1) In the category of sets Set the categorical coproduct of a family of sets

Ai i I is the disjoint union of sets i I Ai. { } ∈ ∈ 39 2) In the category of abelian groups Ab the categorical coproduct of a family

of abelian groups Gi i I is the direct sum i I Gi. { } ∈ ∈ The homomorphisms εi0 : Gi0 i I Gi are given by g (gi)i I where → ∈ → ∈ g if i = i g = 0 i e otherwise Gi

Given an abelian group H and homomorphisms fi : Gi H we have a homomorphism →

f : Gi H, f((gi)i I )= fi(gi) → ∈ i I i I ∈ ∈ This is the unique homomorphism satisfying fε = f for all i I. i i ∈

w 3) If Gi i I is a family of groups then i I Gi is not, in general, a coproduct { } ∈ ∈ of Gi i I .Takee.g.G1 = Z/2Z, G2 = Z/3Z.Wehavehomomorphisms { } ∈

f1 : Z/2Z GT ,f(1) = S1 → f2 : Z/3Z GT ,f(1) = R1 → However, there is no homomorphism f : Z/2Z Z/3Z GT such that ⊕ → fεi = f for i =1, 2.

12.7. Construction of coproducts in Gr.

Let Gi i I be a family of groups, and let S = Gi be the disjoint union of ∈ i I sets{ of elements} of these groups. A word in S is a∈ sequence

a1a2 ...ak

where k 0 and a1,a2,...,ak S.Considertheequivalencerelationofwords generated≥ by the following conditions:∈

1) if e is the trivial element in G for some i I then Gi i ∈ a ...a a ...a a ...a e a ...a 1 j j+1 k ∼ i j Gi j+1 k

40 2) if a ,a belong to the same group G for some i I then j j+1 i ∈ a ...a a ...a a ... (a a ) ...a 1 j j+1 k ∼ 1 j j+1 k product in Gi

Denote ∗G := equivalence classes of words i { } i I ∈ This set is a group with multiplication deﬁned by concatenation of words.

12.8 Deﬁnition. The group i I ∗Gi is called the free product of the family ∈ Gi i I { } ∈

12.9 Proposition. If Gi i I is a family of groups then i∗ I Gi is the coproduct { } ∈ ∈ of the family Gi i I in the category of groups. { } ∈

12.10 Note. The free product i∗ I Z is isomorphic to the free group generated by the set I. ∈