MTH 411 Lecture Notes Based on Hungerford, Abstract Algebra

151 Pages·2006·0.64 MB·English

MTH 411
Lecture Notes
Based on Hungerford, Abstract Algebra
Ulrich Meierfrankenfeld
Department of Mathematics
Michigan State University
East Lansing MI 48824
[email protected]
August 28, 2014
2
Contents
1 Groups 5
1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Functions and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Basic Properties of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7 Lagrange’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8 Normal Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.9 The Isomorphism Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2 Group Actions and Sylow’s Theorem 53
2.1 Group Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 Sylow’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3 Field Extensions 79
3.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2 Simple Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.4 Separable Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.5 Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A Sets 113
A.1 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
A.2 Bijections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
A.3 Cardinalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
B List of Theorems, Definitions, etc 121
B.1 List of Theorems, Propositions and Lemmas . . . . . . . . . . . . . . . . . . 121
B.2 Definitions from the Lecture Notes . . . . . . . . . . . . . . . . . . . . . . . 139
B.3 Definitions from the Homework . . . . . . . . . . . . . . . . . . . . . . . . . 147
3
4 CONTENTS
Chapter 1
Groups
1.1 Sets
Naively a set S is collection of object such that for each object x either x is contained in S
or x is not contained in S. We use the symbol ’∈’ to express containment. So x ∈ S means
that x is contained in S and x ∈/ S means that x is not contained in S. Thus we have
For all objects x : x ∈ S or x ∈/ S.
You might think that every collection of objects is a set. But we will now see that this
cannot be true. For this let A be the collection of all sets. Suppose that A is a set. Then A
iscontainedinA. ThisalreadyseemslikeacontradictionButmaybeasetcanbecontained
in itself. So we need to refine our argument. We say that a set S is nice if S is not contained
in S. Now let B be the collection of all nice set. Suppose that B is a set. Then either B is
contained in B or B is not contained in B.
Suppose that B is contained in B. Since B is the collection of all nice sets we conclude
that B is nice. The definition of nice now implies that that B is not contained in B, a
contradiction.
Suppose that B is not contained in B. Then by definition of ’nice’, B is a nice set. But
B is the collection of all nice sets and so B is contained in B, again a contradiction.
This shows that B cannot be a set. Therefore B is a collection of objects, but is not set.
What kind of collections of objects are sets is studied in Set Theory.
The easiest of all sets is the empty set denote by {} or ∅. The empty set is defined by
For all objects x : x (cid:54)∈ ∅.
So the empty set has no members.
Given an object s we can form the singleton {s}, the set whose only members is s:
For all objects x : x ∈ {s} if and only if x = s
If A and B is a set then also its union A∪B is a set. A∪B is defined by
5
6 CHAPTER 1. GROUPS
For all objects x : x ∈ A∪B if and only if x ∈ A or x ∈ B.
The natural numbers are defined as follows:
0 := ∅
1 := 0∪{0} = {0} = {∅}
2 := 1∪{1} = {0,1} = {∅,{∅}}
3 := 2∪{2} = {0,1,2} = {∅,{∅},{∅,{∅}}}
4 := 4∪{4} = {0,1,2,3} = {∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}}
. . . . . . .
. . . . . . .
. . . . . . .
n+1 := n∪{n} = {0,1,2,3,...n}
One of the axioms of set theory says that the collection of all the natural numbers
{0,1,2,3,4,...}
is set. We denote this set by N.
Addition on N is defined as follows: n+0 := n, n+1 := n∪{n} and inductively
n+(m+1) := (n+m)+1.
Multiplication on N is defined as follows: n·0 := n, n·1 := n and inductively
n·(m+1) := (n·m)+n.
1.2 Functions and Relations
We now introduce two important notations which we will use frequently to construct new
sets from old ones. Let I ,I ,...I be sets and let Φ be some formula which for given
1 2 n
elements i ∈ I ,i ∈ I ,...,i ∈ I allows to compute a new object Φ(i ,i ,...,i ). Then
1 1 2 2 n n 1 2 n
{Φ(i ,i ,...,i ) | i ∈ I ,...,i ∈ I }
1 2 n 1 1 n n
is the set defined by
x ∈ {Φ(i ,i ,...,i ) | i ∈ I ,...,i ∈ I }
1 2 n 1 1 n n
if and only
there exist objects i ,i ,...,i with i ∈ I ,i ∈ I ,...,i ∈ I and x = Φ(i ,i ,...,i ) .
1 2 n 1 1 2 2 n n 1 2 n
1.2. FUNCTIONS AND RELATIONS 7
In Set Theory it is shown that {Φ(i ,i ,...,i ) | i ∈ I ,...,i ∈ I } is indeed a set.
1 2 n 1 1 n n
Let P be a statement involving a variable t. Let I be set. Then
{i ∈ I | P(i)}
is the set defined by
x ∈ {i ∈ I | P(i)} if and only if x ∈ I and P is true for t = x.
Under appropriate condition it is shown in Set Theory that {i ∈ I | P(i)} is a set.
Let a and b be objects. Then the ordered pair (a,b) is defined as (a,b) := {{a},{a,b}}.
We will prove that
(a,b) = (c,d) if and only if a = c and b = d.
For this we first establish a simple lemma:
Lemma 1.2.1. Let u,a,b be objects with {u,a} = {u,b}. Then a = b.
Proof. We consider the two cases a = u and a (cid:54)= u.
Suppose first that a = u. Then b ∈ {u,b} = {u,a} = {a} and so a = b.
Suppose next that a (cid:54)= u. Since a ∈ {u,a} = {u,b}, a = u or a = b. But a (cid:54)= u and so
a = b.
Proposition 1.2.2. Let a,b,c,d be objects. Then
(a,b) = (c,d) if and only if a = c and b = d.
Proof. Suppose (a,b) = (c,d). We need to show that a = c.
We will first show that a = b. Since
{a} ∈ {{a},{a,b}} = (a,b) = (c,d) = {{c},{c,d}},
we have
{a} = {c} or {a} = {c,d}.
In the first case a = c and in the second c = d and again a = c.
Froma = cweget{{{a},{a,b}} = {{c},{c,d} = {{a},{a,d}. Soby1.2.1{a,b} = {a,d}
and applying 1.2.1 again, b = d.
If I and J are sets we define I ×J := {(i,j) | i ∈ I,j ∈ J}.
A relation on I and J is triple r = (I,J,R) where R is a subset I×J. If i ∈ I and j ∈ J
we write irj if (i,j) ∈ R.
For example let R := {(n,m) | n,m ∈ N,n ∈ m} and let < be the triple (N,N,R). Let
n,m ∈ N. Then n < m if and only if n ∈ m. Since m = {0,1,2,...,m−1} we see that
n < m if and only if n is one of 0,1,2,3,...,m−1.
8 CHAPTER 1. GROUPS
A function from I to J is a relation f = (I,J,R) on I and J such that for each i ∈ I
there exists a unique j ∈ J with (i,j) ∈ R. We denote this unique j by f(i). So for i ∈ I
and j ∈ J the following three statements are equivalent:
ifj ⇐⇒ (i,j) ∈ R ⇐⇒ j = f(i).
We denote the function f = (I,J,R) by
f : I → J, i → f(i).
So R = {(i,f(i)) | i ∈ I}.
For example
f : N → N, m → m2
denotes the function (N,N,{(m,m2) | n ∈ N})
Informally, a function f from I to J is a rule which assigns to each element i of I a
unique element f(i) in J.
A function f : I → J is called 1-1 if i = k whenever i,k ∈ I with f(i) = f(k).
f is called onto if for each j ∈ I there exists i ∈ I with f(i) = j. Observe that f is 1-1
and onto if and only if for each j ∈ J there exists a unique i ∈ I with f(i) = j.
If f : I → J and g : J → K are functions, then the composition g◦f of g and f is the
function from I to K defined by (g◦f)(i) = g(f(i)) for all i ∈ I.
1.3 Definition and Examples
Definition 1.3.1. Let S be a set. A binary operation is a function ∗ : S ×S → S. We
denote the image of (s,t) under ∗ by s∗t.
Let I be a set. Given a formula φ which assigns to each pair of element a,b ∈ I some
objectφ(a,b). Then φ determines a binary operation ∗ : I×I → I,(a,b) → φ(a,b) provided
for all a,b ∈ I:
(i) φ(a,b) can be evaluated and φ(a,b) only depends on a and b; and
(ii) φ(a,b) is an element of I.
If (i) holds we say that ∗ is well-defined. And if (ii) holds we say that I is closed under
∗.
Example 1.3.2.
(1) + : Z×Z,(n,m) → n+m is a binary operation.
(2) · : Z×Z,(n,m) → nm is a binary operation.
(3) · : Q×Q,(n,m) → nm is a binary operation.
1.3. DEFINITION AND EXAMPLES 9
(4) Let I = {a,b,c,d} and define ∗ : I ×I → I by
∗ a b c d
a b a c a
b a b c d
c d b a a
d a d a b
Here for x,y ∈ I, x∗y is the entree in row x, column y. For example b∗c = c and
c∗b = b.
Then ∗ is a binary operation.
(5)
(cid:3) a b c d
a a a a a
b a a a a
c a a a a
d a a a a
(cid:3) is a binary operation on I.
(6)
∗ a b c d
a b a c a
b a e c d
c d b a a
d a d a b
is not a binary operation. Indeed, according to the table, b∗b = e, but e is not an
element of I. Hence I is not closed under ∗ and so ∗ is not a binary operation on I.
(7) Let I be a set . A 1-1 and onto function f : I → I is called a permutation of I.
Sym(I) denotes the set of all permutations of I. If f and g are permutations of I then
by A.2.3(c) also the composition f ◦g is a permutation of I. Hence the map
◦ : Sym(I)×Sym(I),(f,g) → f ◦g
is a binary operation on Sym(I).
10 CHAPTER 1. GROUPS
(8) (cid:5) : Z ×Z ,([a] ,[b] ) → [ab2+1] , where [a] denotes the congruence class of a modulo
3 3 3 3 3 3
3, is not a binary operation. Indeed we have [0] = [3] but
3 3
[(−1)02+1] = [(−1)1] = [−1] (cid:54)= [1] = [(−1)10] = [(−1)32+1]
3 3 3 3 3 3
and so (cid:5) is not well-defined.
(9) ⊕ : Q×Q → Q,(a,b) → a is not a binary operation. Since 1 is not defined, ⊕ is not
b 0
well-defined.
Definition 1.3.3. Let ∗ be a binary operation on a set I. Then ∗ is called associative if
(a∗b)∗c = a∗(b∗c) for all a,b,c ∈ I
Example 1.3.4.
We investigate which of the binary operations in 1.3.2 are associative.
(1) Addition on Z is associative.
(2) Multiplication on Z is associative.
(3) Multiplication on Q is associative.
(4) ∗ in 1.3.2(4) is not associative. For example
a∗(d∗c) = a∗a = b and (a∗d)∗c = a∗c = c.
(5) (cid:3) in 1.3.2(5) is associative since x∗(y∗z) = a = (x∗y)∗z for any x,y,z ∈ {a,b,c,d}.
(7) Composition of functions is associative: Let f : I → J, g : J → K and h : K → L be
functions. Then for all i ∈ I,
((f ◦g)◦h)(i) = (f ◦g)(h(i)) = f(g(h(i)))
and
(f ◦(g◦h))(i) = f((g◦h)(i)) = f(g(h(i))).
Thus f ◦(g◦h) = (f ◦g)◦h.
Definition 1.3.5. Let I be a set and ∗ a binary operation on I. An identity of ∗ in I is a
element e ∈ I with e∗i = i and i = i∗e for all i ∈ I.
Example 1.3.6.