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Introduction - Algebra 1
2
Permutations
3
Group Axioms
4
Order and Conjugacy
5
Subgroups
6
Problem solving
7
Group Actions
8
Cosets
9
Group Homomorphisms
10
Normal subgroups
11
Qutient Groups
12
Product and Chinese Remainder Theorem
13
Dihedral Groups
14
Semidirect products
15
Problem solving
16
The Orbit Counting Theorem
17
Fixed points of group actions
18
Second application: Fixed points of group actions
19
Sylow Theorems - a preliminary proposition
20
Sylow Theorem I
21
Problem solving I
22
Problem solving II
23
Sylow Theorem II
24
Sylow Theorem III
25
Problem solving I
26
Problem solving II
27
Free Groups I
28
Free Groups IIa
29
Free Groups IIb
30
Free Groups III
31
Free Groups IV
32
Problem Solving/Examples
33
Generators and relations for symmetric groups – I
34
Generators and relations for symmetric groups – II
35
Definition of a Ring
36
Euclidean Domains
37
Gaussian Integers
38
The Fundamental Theorem of Arithmetic
39
Divisibility and Ideals
40
Factorization and the Noetherian Condition
41
Examples of Ideals in Commutative Rings
42
Problem Solving/Examples
43
The Ring of Formal Power Series
44
Fraction Fields
45
Path Algebra of a Quiver
46
Ideals In Non Commutative Rings
47
Product of Rings
48
Ring Homomorphisms
49
Quotient Rings
50
Problem solving
51
Tensor and Exterior Algebras
52
Modules : definition
53
Modules over polynomial rings $K[x]$
54
Modules: alternative definition
55
Modules: more examples
56
Submodules
57
General constructions of submodules
58
Problem solving
59
Quotient modules
60
Homomorphisms
61
More examples of homomorphisms
62
First isomorphism theorem
63
Direct sums of modules
64
Complementary submodules
65
Change of ring
66
Problem solving
67
Free Modules (finitely generated)
68
Determinants
69
Primary Decomposition
70
Problem solving
71
Finitely generated modules and the Noetherian condition
72
Counterexamples to the Noetherian condition
73
Generators and relations for Finitely Generated Modules
74
General Linear Group over a Commutative Ring
75
Equivalence of Matrices
76
Smith Canonical Form for a Euclidean domain
77
solved_problems1
78
Smith Canonical Form for PID
79
Structure of finitely generated modules over a PID
80
Structure of a finitely generated abelian group
81
Similarity of Matrices
82
Deciding Similarity
83
Rational Canonical Form
84
Jordan Canonical Form
Description:
PRE-REQUISITES: BSc-level linear algebra INTENDED AUDIENCE: Any interested learners COURSE OUTLINE: Foundational PG level course in Algebra, suitable for M.Sc and first-year Ph.D. students in Mathematics. Learn Permutations. Group Axioms. Order and Conjugacy. Subgroups. Problem solving. Group Actions. Cosets. Group Homomorphisms. Normal subgroups. Qutient Groups. Product and Chinese Remainder Theorem. Dihedral Groups. Semidirect products. ABOUT INSTRUCTOR: Prof. S. Viswanath is a faculty at The Institute of Mathematical Sciences, Chennai. His research interest is in representation theory. Prof. Amritanshu Prasad is a faculty at The Institute of Mathematical Sciences, Chennai. His research interest is in representation theory.

Algebra 1

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Introduction - Algebra 1