Groups
In this note, I will briefly introduce groups, and some of their
properties. Group Action would be talked in another seperate blog.
I took Abstract Algebra in UW-Madison(MATH541) with professor
Chenxi Wu. This article is inspired by prof. Wu's lecture note, and many
formats, content, and structure of this blog are influenced by and
similar to it.
Groups, Subgroups
Let's first introduct some concepts and facts about group.
Definition 1. [Group, Abelian group, Trivial group, Permutation group]
A group
is a combination of a set and a map , also called group operation. The group operation are require to have: ("associativity") For any
, ; ("identity") There is an element
, such that for any , . This element is also written as ; ("inverse") For any
, there is a , such that .
Group option
can also be written as or . An abelian group has one more property:
- ("communitation") For any
, .
- ("communitation") For any
There are some special groups.
Definition 2. [Opposite Group, Trivial Group, Permutation Group]
An opposite group is
with ; A trivial group is
; The set of bijections from set
to itself forms a group under operation , which is called permutation group, denoted as . When , the permutation group can also be represented as ;
Groups are fundamentally set, which contains subsets. With the same operation as the original multiplication, the subsets can also form groups as they are "closed". Those subsets are called "subgroups".
Definition 3. [Subgroup]
Let
. , . , .
For any subgroup
Homomorphism, Isomorphism, Automorphism
Till now, we are only talking about relationships in groups. However, groups are not totally isolated: we can connect them as how we use functions to connect sets. However, these functions are special as groups are not simply sets. We call these functions "homomorphism".
Definition 4. [Homomorphism, Isomorphism, Endomorphism, Automorphism]
A homomorphism is a function
: and be two groups, and for any , . is the set of homomorphisms between two groups and .
An isomorphism is a bijective homomorphism. Specificly, if
is an isomorphism, is also a isomorphism. We say is isomorphic to if there is an isomorphism between and , with notation . An endomorphism is a homomorphism from
to itself. An automorphism is a isomorphism between a group and itself. Also, the set of all automorphisms of a group
forms a group, called automorphism group, written as .
Isomorphic and automorphism group are two concepts that we will use a lot in the following instructions and many problems in Abstract Algrebra. The reason is,
means the two groups are very "similar". You can shape every element in to an unique element in , and you can even apply multiplications on them, and get the corresponding product!
Till now, we have introduced almost every concept that we will be using in this blog. However, this is far not enough to end here - we only have some abstract, useless, non-sense definition. Why we need to define group? Why we need to define (homo)(iso)(auto)morphisms? It's impossible to answer, unless we can find some more "fun facts" and applications about them.
Isomorphism Theorem
First let's define two concepts kernal and image of a homomorphism.
Definition 5. [Kernal, Image]
Let
, or is called the kernal of . It can also be written as . Specificly, ; is called the image of , which can be written as . Also, .
Theorem 6. Let
can be written as the composition of with and with . Clearly, is a surjection and is an injection; iff . This is clear as ;
Theorem 6. Let
Definition 7. [Left coset]
Let
Definition 8. [Normal subgroup, Quotient group]
Let
For any subgroups that satiesfy the condition, we call then
normal subgroup, and the left cosets of
Theorem 9. Isomorphism Theorem Let
This theorem is powerful as for any
Semidirect Product
Theorem 7. Let
(Here, is a group under ); - There is an injective homomorphism
and a surjective homomorphism , such that , , and . - There exists two normal subgroups
, such that , , with , and for any , , such that ;
This theorem can be proved by
Definition 8. [Section, Semidirect product]
The injective homomorphism
The following part requires some basic knowledge of group actions. Please read the parts about "permutation representation".
- Title: Groups
- Author: Harry Huang (aka Wenyuan Huang, 黄问远)
- Created at : 2024-11-25 16:29:41
- Updated at : 2024-11-28 01:08:17
- Link: https://whuang369.com/blog/2024/11/25/Math/Abstract_Algebra/group_theory/
- License: This work is licensed under CC BY-NC-SA 4.0.