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9 Direct Products, Direct Sums, and Free Abelian Groups

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9 Direct Products, Direct Sums, and Free Abelian Groups 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.
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