Introduction to Abstract Algebra: Group Theory
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This is an advanced level course of Introduction to Abstract Algebra with majors in Group Theory. The students who want to learn algebra at an advanced level, usually learn Introduction to Abstract Algebra: Group Theory. The course is offered for pure mathematics students in different universities around the world. However, the students who take the Introduction to Abstract Algebra: Group Theory course, are named super genius in group theory. Not so much difficult, but regular attention and interest can lead to the students in the right learning environment of mathematics. Many students around the world have their interest in learning Introduction to Abstract Algebra: Group Theory but they could’t find any proper course or instructor.
Abstract Algebra is comprised of one of the main topics which are also called Group theory. Group Theory or Group is actually the name of the fundamental four properties of mathematics that are frequently used in real analysis. We actually establish a strong background of Group Theory by defining different concepts. Proof of theorems and solutions of many examples is one of the interesting parts while studying Group Theory.
This course is filmed on a whiteboard (8 hours) and Tablet (2 hours). The length of this course is 10 hours with more than 15 sections and 100 videos. Almost every content of Group Theory has been included in this course. The students have difficulties in understanding the theorem, especially in Group Theory. Theorems have been explained with proof and examples in this course. A number of examples and exercises make this course easy for every student, even those who are taking this course the first time.
I assure all my students that they will enjoy this course. But however, if they have any difficulty then they can discuss it with me. I will answer your every question with a prompt response. One thing I will ask you is that you must see the contents sections and some free preview videos before enrolling in this course.
CONTENTS OF THIS COURSE
 Groups and related examples
 The identity element is the only element that is idempotent
 Cancellation law hold in a group G
 Definition of Subgroups and related examples
 H is a subgroup if ab^1 is contained in H
 The intersection of any collection of subgroups is a subgroup
 HuK is a subgroup if H is contained in Kor K is contained in H
 Cyclic group and related examples
 Every subgroup of a cyclic group is cyclic
 Definition of cosets and related examples
 Prove that the number of left or right cosets define the partition of a group G
 Statement and Proof of Lagrange’s Theorem
 Symmetric groups and related examples and exercises
 Group of querternian and Klein’s four group
 Normalizers, centralizers, and center of a group G and related theorem and examples
 Quotient or Factor groups
 Derived groups and related many examples
 Normal Subgroups, conjugacy classes, conjugate subgroups, and related examples and theorems
 Kernel of group
 Automorphism and inner automorphism
 P Group and related theorems and examples
 Relations in groups like homomorphism and isomorphism
 The centralizer is a subgroup of a group G
 The normalizers is a subgroup of a group G
 The Center of a group is a subgroup of a group G
 The relation of conjugacy is an equivalence relation
 Theorem and examples on quotient groups
 Double cosets and related examples
 Definition of automorphism
 What is an inner automorphism
 Every cyclic group is an abelian group
 Groups of residue classes on a different mode
 Examples of D_4 and D_5 groups
 Examples related to C_6 and V_4
 The first isomorphism theorem and its proof
 The 2nd isomorphism theorem and its proof
 The 3rd isomorphism theorem and its proof
 The direct product of cyclic group

2Defining GroupsVideo lesson

3Definition Notes SheetText lesson

4Examples of GroupsVideo lesson

5Quiz of Lecture 1 3Quiz
In this quiz, students can check their knowledge that they have gain in the previous 3 lectures

6Examples of Groups NotesText lesson

7Group of Cube Roots of UnityVideo lesson

8Group of Fourth and nth Roots of UnityVideo lesson

9Group of Set of Residue Classes Module 5 Under MultiplicationVideo lesson

10Group of Set of All 2 by 2 Non Singular MatricesVideo lesson

11Idempotent ElementVideo lesson

12The Cancellation Law Holds in a Group GVideo lesson

13TheoremVideo lesson

14Order of a Group G and Order of an Element in a Group GVideo lesson

15Group of Set of Residue Classes Module 8 Under MultiplicationVideo lesson

16TheoremVideo lesson

17Order of an Element and its Inverse are SameVideo lesson

18Group of Set of Residue Classes Module 9 Under MultiplicationVideo lesson

19ExampleVideo lesson

20If a Group G Has Three Elements Then It is AbelianVideo lesson

21Example of an Abelian GroupVideo lesson

22Example 2Video lesson

23Example 3Video lesson

24Example 4Video lesson

25Example 5Video lesson

26Defining SubgroupsVideo lesson

27Theorem (H is a Subgroup Iff ab^1 is an Element of G)Video lesson

28The Intersection of Any Collection of Subgroups is a SubgroupVideo lesson

29ExampleVideo lesson

30HuK is a Subgroup Iff H is Contained in K or K is Contained in HVideo lesson

31HuK is a Subgroup Iff H is Contained in K or K is Contained in H Part 2Video lesson

32ExampleVideo lesson

33Klein'S Four GroupVideo lesson

34ExampleVideo lesson

35Defining Cyclic GroupsVideo lesson

36Every Subgroup of a Cyclic Group is CyclicVideo lesson

37Every Cyclic Group is AbelianVideo lesson

38Group of QuerternianVideo lesson

39Caylay's Table for Cyclic GroupVideo lesson

40ExampleVideo lesson

41TheoremVideo lesson

42TheoremVideo lesson

43ExampleVideo lesson

44Example 2Video lesson

57Defining PermutationsVideo lesson

58Defining TranspositionsVideo lesson

59Multiplication of PermutationsVideo lesson

60Inverse and Disjoint Cycles OF PermutationsVideo lesson

61Permutations as Product of Transpositions + Even and Odd PermutationsVideo lesson

62Even or Odd PermutationsVideo lesson

63S_3 Symmetric Group or Group of Motion of TriangleVideo lesson

64D_4 Dihedral Group of MotionVideo lesson

65Diagonal Group D_5Video lesson

66Diagonal Group D_5 NotesText lesson
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