Download 9 Direct products, direct sums, and free abelian groups

9
Direct products, direct sums, and free abelian
groups
9.1
a family of groups {Gi }i∈I is a group
� Definition. A direct product of �
Gi is the cartesian product of the
i∈I Gi defined as follows. As a set
i∈I �
groups Gi . Given elements (ai )i∈I , (bi )i∈I ∈ i∈I Gi we set
(ai )i∈I · (bi )i∈I := (ai bi )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
If all groups Gi are abelian then
direct sum of {Gi }i∈I .
�w
9.3 Note. If I is a finite set then
is denoted
i∈I Gi
�
i∈I
Gi =
�
i∈I
Gi and it is called the
�w
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
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9.6. Let S be a set. Denote by Fab (S) the set of all expressions of the form
�
kx x
x∈S
where kx ∈ Z and kx �= 0 for finitely many x ∈ X only.
Fab (S) is an abelian group with addition defined by
�
�
�
kx x +
lx x :=
(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) ∼
=
Proof. The isomorphism is given by
�
f : Fab (S) →
Z,
�
f(
x∈S
Z
x∈S
�
kx x) = (kx )x∈S
x∈S
9.9 Note. We have a map of sets
i : S → Fab (S),
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i(x) = 1 · x
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
i
�
�
�
�
�
� f¯
�
�
�
�G
��
Fab (S)
Proof. Define f¯ by
f¯
�
�
x∈S
kx x
�
:=
�
kx f (x)
x∈S
Note: this is well defined since kx = 0 for almost all x ∈ S.
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10
Categories and functors
10.1 Definition. A category C consists of
1) a collection of objects Ob(C)
2) for any a, b ∈ Ob(C) a set HomC (a, b) of morphisms from a to b
3) for any a, b, c ∈ Ob(C) a function (“composition law”)
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 ◦ idc = f, idc ◦ g = g
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
• HomSet (A, B) = { all maps of sets f : A → B }
2) Gr = the category of all groups
• Ob(Gr) = the collection of all groups
• HomGr (G, H) = { all homomorphisms f : G → H }
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3) Ab = the category of all abelian groups
• Ob(Ab) = the collection of all abelian groups
• HomAb (G, H) = { all homomorphisms f : G → H }
4) Top = the category of all topological spaces
• Ob(Top) = the collection of all topological spaces
• HomTop (X, Y ) = { all continuous maps f : X → Y }
5) Let G be a group. Define a category CG as follows:
• Ob(CG ) = {∗}
• HomCG (∗, ∗) = { elements of G }
• composition of morphisms = multiplication in G
6) A very small category C:
c
f
�
d
• Ob(C) = {c, d}
• HomC (c, d) = {f }, HomC (d, c) = ∅, HomC (c, c) = idc , HomC (d, d) = idd
10.3 Definition. A morphism f : c → d in a category C is an isomorphism if
there exists a morphism g : d → c such that gf = idc and f g = idd .
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.
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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 a function
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
[a, b] := aba−1 b−1
Note: [a, b] = e iff ab = ba.
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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
fab : G/[G, G] → H/[H, H]
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.
A map of sets f : S → T defines a homomorphism
f˜: F (S) → F (T )
given by f˜(xλ1 1 xλ2 2 · · · · · xλk k ) = f (x1 )λ1 f (x2 )λ2 · · · · · f (xk )λk .
Check: the assignment
S �→ F (S),
(f : S → T ) �→ (f˜: F (S) → F (T ))
Defines a functor F : Set → Gr. This is the free group functor.
5) Similarly we have the free abelian group functor
Fab : Set → Ab
where
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• Fab (S) = the free abelian group generated by the set S
• if f : S → T then Fab (f ) : Fab (S) → Fab (T ) is given by
�
�
�
�
Fab (f )
kx x =
kx f (x)
x∈S
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x∈S
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Adjoint functors
11.1 Definition. Given two functors
L: C → D
R: D → C
and
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�
η c�
ηc
�
RL(c)
RL(f )
�
�
RL(c� )
2) for any c ∈ C and d ∈ D the map of sets
HomD (L(c), d) −→ HomC (c, R(d))
f
is a bijection.
ηc
R(f )
(L(c) → d) �−→ (c → RL(c) → R(d))
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 . This collection of morphisms is called
the counit of the adjunction.
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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
�
�
ηc
�
�
�
�
�
�
R(d)
��
�
� 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
iS : S → U F (S),
iS (x) = x
The universal property of free groups (9.10) says that for any G ∈ 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
�
�
iS
�
�
�
�
U F (S)
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�
�
� U (f¯)
�
U (G)
��
2) We have functors
Fab : Set → Ab,
Set ← Ab : U
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
(check!).
J : Ab → Gr,
J(G) = G
11.5 Note. It is not true that every functor has a left or right adjoint.
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12
Categorical products and coproducts
12.1 Definition. Let {ci }i∈I be a family of objects in a category C. A (categorical) product of the family {ci }i∈I is an object p ∈ C equipped with morphisms
πi : p → ci for all i ∈ I that satisfies the following universal property. For any
object d ∈ C and a family of morphisms {fi : d → ci }i∈I there exists a unique
morphism f : d → p such that πi f = fi for all i ∈ I.
f1
d�
�f
�
��
p
f2
π1
�
� c1
π2
� �
c2
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 πi f = fi .
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2) By a similar argument
if {Gi }i∈I is a family of abelian groups then the
�
direct product i∈I Gi is the categorical product of the family {Gi }i∈I in
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 (categorical) coproduct of the family {ci }i∈I is an object d ∈ C equipped with morphisms
εi : ci → d for all i ∈ I that satisfies the following universal property. For any
object b ∈ C and a family of morphisms {fi : ci → b}i∈I there exists a unique
morphism f : d → b such that f εi = fi for all i ∈ I.
c1
ε1
f1
c2
�
�d
ε2
f2
�
�f
�
�� �
�b
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 .
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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 = i0
gi =
eGi otherwise
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 f εi = fi for all i ∈ I.
�
3) If {Gi }i∈I is a family of groups then w
i∈I Gi is not, in general, a coproduct
of {Gi }i∈I . Take e.g. G1 = Z/2Z, G2 = Z/3Z. We have homomorphisms
f1 : Z/2Z → GT ,
f2 : Z/3Z → GT ,
f (1) = S1
f (1) = R1
However, there is no homomorphism f : Z/2Z ⊕ Z/3Z → GT such that
f εi = f for i = 1, 2.
12.7. Construction of coproducts in Gr.
�
Let {Gi }i∈I be a family of groups, and let S = i∈I Gi be the disjoint union of
sets of elements of these groups. A word in S is a sequence
a1 a2 . . . ak
where k ≥ 0 and a1 , a2 , . . . , ak ∈ S. Consider the equivalence relation of words
generated by the following conditions:
1) if eGi is the trivial element in Gi for some i ∈ I then
a1 . . . aj aj+1 . . . ak ∼ ai . . . aj eGi aj+1 . . . ak
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2) if aj , aj+1 belong to the same group Gi for some i ∈ I then
a1 . . . aj aj+1 . . . ak ∼ a1 . . . (aj aj+1 ) . . . ak
� �� �
product in Gi
Denote
�∗
i∈I
Gi := { equivalence classes of words }
This set is a group with multiplication defined by concatenation of words.
12.8 Definition. The group
{Gi }i∈I
�
i∈I
∗
Gi is called the free product of the family
12.9 Proposition. If {Gi }i∈I is a family of groups then
of the family {Gi }i∈I in the category of groups.
12.10 Note. The free product
by the set I.
�∗
i∈I
�∗
i∈I
Gi is the coproduct
Z is isomorphic to the free group generated
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