9 Direct products, direct sums, and free abelian groups
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1 9 Direct products, direct sums, and free abelian groups 9.1 Definition. A direct product of a family of groups {G i } i I is a group i I G i defined as follows. As a set i I G i is the cartesian product of the groups G i.givenelements(a i ) i I, (b i ) i I i I G i we set (a i ) i I (b i ) i I := (a i b i ) i I 9.2 Definition. A weak direct product of a family of groups {G i } i I is the subgroup of i I G i given by w Gi := {(a i ) i I a i = e i G i } for finitely many i only} i I If all groups G i are abelian then w i I G i is denoted i I G i and it is called the direct sum of {G i } i I. 9.3 Note. If I is a finite set then i I G i = w i I G i. 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
2 9.6. Let S be a set. Denote by F ab (S) the set of all expressions of the form k x x x S where k x Z and k x =0for finitely many x X only. F ab (S) is an abelian group with addition defined by k x x + l x x := x + l x )x x S x S x S(k 9.7 Definition. The group F ab (S) is called the free abelian group generated by the set S. In general a group G is free abelian if G = F ab (S) for some set S. 9.8 Proposition. If S is a set then F ab (S) = x S Z Proof. The isomorphism is given by f : F ab (S) Z, f( k x x)=(k x ) x S x S x S 9.9 Note. We have a map of sets i: S F ab (S), i(x) =1 x 28
3 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: S G i f F ab (S) f Proof. Define f by f k x x := k x f(x) x S x S Note: this is well defined since k x =0for almost all x S. 29
4 10 Categories and functors 10.1 Definition. A category C consists of 1) acollectionofobjects Ob(C) 2) for any a, b Ob(C) asethom C (a, b) of morphisms from a to b 3) for any a, b, c Ob(C) afunction( compositionlaw ) Hom C (a, b) Hom C (b, c) Hom C (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 id c Hom C (c, c) such that f id c = f, id c g = g for any f Hom C (c, d),g Hom C (b, c) Examples. 1) Set = the category of all sets. Ob(Set) =the collection of all sets Hom Set (A, B) ={ all maps of sets f : A B } 2) Gr = the category of all groups Ob(Gr) =the collection of all groups Hom Gr (G, H) ={ all homomorphisms f : G H } 30
5 3) Ab = the category of all abelian groups Ob(Ab) =the collection of all abelian groups Hom Ab (G, H) ={ all homomorphisms f : G H } 4) Top = the category of all topological spaces Ob(Top) =the collection of all topological spaces Hom Top (X, Y )={ all continuous maps f : X Y } 5) Let G be a group. Define a category C G as follows: Ob(C G )={ } Hom CG (, ) ={ elements of G } composition of morphisms = multiplication in G 6) AverysmallcategoryC: c f d Ob(C) ={c, d} Hom C (c, d) ={f}, Hom C (d, c) =, Hom C (c, c) =id c, Hom C (d, d) = id d 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 c and fg =id 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 Note. For an object c C define Aut(c) :={ all isomorphisms f : c c } Aut(c) with composition of morphisms is a group. 31
6 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 Hom C (c, c ) Hom D (F (c),f(c )), f F (f) such that F (gf) =F (f)f (g) and F (id c )=id F (c) 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 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 [a, b] :=aba 1 b 1 Note: [a, b] =e iff ab = ba. 32
7 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 ab : G/[G, G] H/[H, H] given by f ab (a[g, G]) = f(a)[h, H]. The abelianization functor Ab: Gr Ab is given by Ab(G) :=G/[G, G], Ab(f) :=f ab 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 ) given by f(x λ 1 1 x λ 2 2 x λ k k )=f(x 1) λ 1 f(x 2 ) λ2 f(x k ) λ 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 ab : Set Ab where 33
8 F ab (S) =the free abelian group generated by the set S if f : S T then F ab (f): F ab (S) F ab (T ) is given by F ab (f) k x x = k x f(x) x S x S 34
9 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 η c : c RL(c) such that: 1) for any morphism f : c c in C the following diagram commutes: c f c η c η c RL(c) RL(f) RL(c ) 2) for any c C and d D the map of sets Hom D (L(c),d) Hom C (c, R(d)) is a bijection. (L(c) f d) (c ηc RL(c) R(f) R(d)) In such situation we say that (L, R) is an adjoint pair of functors Note. 1) The collection of morphisms {η c } c C is called the unit of adjunction of (L, R). 2) For any adjoint pair (L, R) we also have morphisms {ε d : LR(d) d} d D satisfying analogous conditions as {η c } c C.Thiscollectionofmorphismsiscalled the counit of the adjunction. 35
10 11.3 Note. The morphism η c 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) η c R( f) RL(c) This property is equivalent to part 2) of Definition 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 : S UF(S), i S (x) =x 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) i S U( f) UF(S) 36
11 2) We have functors F ab : Set Ab, Set Ab: U where F ab = free abelian group functor, U = forgetful functor. Similarly as in 1) one can check that (F ab,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 (check!). J : Ab Gr, J(G) =G 11.5 Note. It is not true that every functor has a left or right adjoint. 37
12 12 Categorical products and coproducts 12.1 Definition. Let {c i } i I be a family of objects in a category C. A(categorical) product of the family {c i } i I is an object p C equipped with morphisms π i : p c i for all i I that satisfies the following universal property. For any object d C and a family of morphisms {f i : d c i } i I there exists a unique morphism f : d p such that π i f = f i for all i I. d f p f 1 π 1 c 1 f 2 π 2 c Note. If a categorical product of {c i } i I exists then it is defined uniquely up to isomorphism. We then write: p = i I c i 12.3 Examples. 1) In the category of groups Gr the categorical product of a family {G i } i I is the direct product of groups i I G i. Indeed, we have projection homomorphisms: π i0 : i I G i G i0, π i0 ((g i ) i I )=g i0 Also, if for some group H we have homomorphisms f i : H G i then this defines a homomorphism f : H i I G i, f(h) =(f i (h)) i I Moreover, f is the unique homomorphism such that we have π i f = f i. 38
13 2) By a similar argument if {G i } i I is a family of abelian groups then the direct product i I G i is the categorical product of the family {G i } i I in the category Ab. 3) In the category Set the categorical product of a family of sets {A i } i I is the cartesian product of sets i I A i Definition. Let {c i } i I be a family of objects in a category C. A(categorical) coproduct of the family {c i } i I is an object d C equipped with morphisms ε i : c i d for all i I that satisfies the following universal property. For any object b C and a family of morphisms {f i : c i b} i I there exists a unique morphism f : d b such that fε i = f i for all i I. c 1 c 2 ε 2 f 2 ε 1 d f b f Note. If a categorical coproduct of {c i } i I exists then it is defined uniquely up to isomorphism. We then write: d = i I c i 12.6 Examples. 1) In the category of sets Set the categorical coproduct of a family of sets {A i } i I is the disjoint union of sets i I A i. 39
14 2) In the category of abelian groups Ab the categorical coproduct of a family of abelian groups {G i } i I is the direct sum i I G i. The homomorphisms ε i0 : G i0 i I G i are given by g (g i ) i I where g g i = e Gi if i = i0 otherwise Given an abelian group H and homomorphisms f i : G i H we have a homomorphism f : i I G i H, f((g i ) i I )= i I f i (g i ) This is the unique homomorphism satisfying fε i = f i for all i I. 3) If {G i } i I is a family of groups then w i I G i is not, in general, a coproduct of {G i } i I.Takee.g.G 1 = Z/2Z, G 2 = Z/3Z. Wehavehomomorphisms f 1 : Z/2Z G T, f(1) = S 1 f 2 : Z/3Z G T, f(1) = R 1 However, there is no homomorphism f : Z/2Z Z/3Z G T fε i = f for i =1, 2. such that Construction of coproducts in Gr. Let {G i } i I be a family of groups, and let S = i I G i be the disjoint union of sets of elements of these groups. A word in S is a sequence a 1 a 2...a k where k 0 and a 1,a 2,...,a k S. Considertheequivalencerelationofwords generated by the following conditions: 1) if e Gi is the trivial element in G i for some i I then a 1...a j a j+1...a k a i...a j e Gi a j+1...a k 40
15 2) if a j,a j+1 belong to the same group G i for some i I then a 1...a j a j+1...a k a 1... (a j a j+1 )...a k product in G i Denote Gi := { equivalence classes of words } i I This set is a group with multiplication defined by concatenation of words Definition. The group i I G i is called the free product of the family {G i } i I 12.9 Proposition. If {G i } i I is a family of groups then i I G i is the coproduct of the family {G i } i I in the category of groups Note. The free product i I Z is isomorphic to the free group generated by the set I. 41
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