......Contents
Preface Chapter 1 Groupst1
1.1 Semigroups, monoids and groupst1

1.2 Subgroupst6

1.3 Theactionofagrouponasett11

1.4 The Sylow theoremt18

1.5 Homomorphisms and normal subgroupst20

1.6 Directproductsanddirectsumst29

1.7 Simple groupst36

1.8 Nilpotent groups and solvable groupst39
Chapter 2 Modules t44

2.1 Rings and ring homomorphismst44
Modules and free modulest55

2.2
2.3 Projectivemodulesandinjectivemodulest67

2.4 Homologicaldimensionsandsemisimpleringst75

2.5 Tensor product and weak dimensiont83
Localizationt95

2.6 Noetherianmodules and UFDt104
2.7
2.8 Finitely generated modules over a PIDt115

Chapter 3 Fields and Galois Theory of Equationst129
Extensionsof .eldst129

3.1
3.2 Splitting .elds, and normalityt137

3.3 ThemaintheoremofGaloistheoryt147
Radical extensionst155

3.4
3.5 Constructionwithstraight-edgeandcompasst157
The Hilbert Nullstellensatzt160

3.6 Chapter 4 Introduction of Various Algebras t167
4.1 Associativealgebrast167

4.2 CoassociativecoalgebrasandHopfalgebrast178

4.3 Nonassociative algebrast182

Chapter 5 Category t193

5.1 Category: Direct limits and colimitst193

5.2 Functors and natural transformationst198

5.3 Abelian categories and homological groupst207

Bibliography t217
Index t219Chapter 1 Groups
The concept of a group is of fundamental importance in algebra and other subjects. We say two groups are the same if they are isomorphic. Just as classi.cations of .nite-dimensional vector spaces over a number .eld, one of the fundamental question in group theory is to classify all groups up to isomorphism of groups, which means to .nd a necessary and su.cient condition for two groups to be isomorphic. This is a very complicated question. However, a larger amount of miscellaneous information on structure of a group has been explored in this chapter.
1.1 Semigroups, monoids and groups
Let’s .rst recall some known binary operations. For example, for any two n × n matrices A, B over the complex number .eld C, we can de.ne binary operations of A and B,suchas A + B, A . B and AB. For any two mappings f : X → Y and
g : Y → Z,where X, Y and Z are nonempty sets, we can de.ne the composition of f and g by g . f : X → Z, x
→ g(f(x)). A binary operation on a nonempty set S is a mapping from S × S to S,where S × S := {(a, b)|a, b ∈ S} is the Cartersian product of S. Under this map, there is only one element in S corresponding to each (a, b) ∈ S × S. The unique element is usually denoted by a · b, simply denoted by ab sometimes.
De.nition 1.1.1 A binary operation on a set G is to be associative if (a · b) · c = a · (b · c) for any a, b, c ∈ G.A semigroup is a nonempty set G together with an associative binary operation · on G. The binary operation of a semigroup G is usually called the product, or multiplication of G.
Example 1.1.1 Let N be the set of all natural numbers. Then (N, +) with addition of numbers and (N, ×) with multiplication of numbers are semigroups.
f gh
It is well known that h · (g · f)=(h · g) · f for any mappings X → Y → Z → W . For any nonempty set X, XX := {f|f is a mapping from X to X} is a semigroup with composition of mappings.
Let Mn(P)be the set of all n×n matrices over a number .eld P.Then (Mn(P), +) with usual matrix addition and (Mn(P), ·) with usual matrix multiplication are semi-groups. (Mn(P), .) with usual matrix subtraction is not a semigroup.
√√
a + b .1 c + d .1
Let H := √ √|a, d, c, d ∈R .Then H is a semigroup
.c + d .1 a .b .1 with matrix addition. It is also a semigroup with matrix multiplication. H is called a quaternion division. Suppose Ω is an open subset of R2 and x0 ∈Ωisa .xed point. A loop with a .xed point x0 in Ω is a continuous mapping . :[0, 1] →Ω such that .(0) = .(1) = x0. Let L be the set of all loops with a .xed point x0 in Ω. De.ne φ1 ·φ2(t)= φ1(2t)if 11
0 : t :

, φ1 ·φ2(t)= φ2(2t .1) if

: t : 1. It is easy to check that L is not a
22 semigroup with this binary operation ·.
Example 1.1.2 Given any nonempty set A.Let S(A)be the set of all .nite sequences (or strings) of elements from A.Then elements in S(A)are also called words over A, or words with alphabets in A.Then S(A) becomes a semigroup with the string concatenation.
De.nition 1.1.2 An element e of a semigroup S is called an identity of S provided that ea = ae = a for all a ∈S.A monoid is a semigroup with an identity.
Suppose e1,e2 are identities of a monoid M.Since e2 is an identity, e1 = e1e2. Similarly, e2 = e1e2. Hence e1 = e2. Thus a monoid has a unique identity. We denote the unique identity of a monoid by e in this chapter unless otherwise speci.ed. If there are several monoids, we usually use eM to emphasis that it is the identity of M.
Example 1.1.3 (N, +) is a monoid with identity 0 and (N, ·)is a monoid with identity 1. The set 2Z of all even numbers is not a monoid with the multiplication of numbers. It is only a semigroup.
Example 1.1.4 Let S be a semigroup and choose an element e/∈S. De.ne a binary operation on S+ := S ∪{e}as follows. If a, b ∈S,then ab is the product of a and b in S.Otherwise ae = ea = a for any a ∈ S+ . It is easy to check that S+ is a monoid with the identity e. In particular, for any given nonempty set A, M(A):= S(A)+ is a monoid, where S(A) is the semigroup de.ned in Example 1.1.2. The identity of M(A) is also called an empty word.
De.nition 1.1.3 Let M be a monoid with identity e.An
element
a ∈M is invertible in M if there is an element b ∈M such that ab = ba = e.A group is a monoid such that every element is invertible.
We know that an invertible matrix has only one inverse matrix. Similarly, every invertible element in a monoid has only one element b satisfying ab = ba = e.In fact, if there are two elements b, c such that ab = ba = ac = ca = e,then b = eb = (ca)b = c(ab)= ce = c. This unique element b is called the inverse of a, denoted
.1 .1 .1).1
by a. For any invertible element a,its inverse ais invertible and (a= a by De.nition 1.1.3.
n.1
Let a be an element in a semigroup G. De.ne a1 := a, a2 := aa,and an := aa for n>
1. Further, de.ne a1a2 ···an := (a1 ···an.1)an inductively for a1, ··· , .1)n
an ∈G.Let a0 := e if a is in a monoid G. For any invertible a, de.ne a.n := (afor any n . 1...................