Algebra 1

Introduction - Algebra 1.

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.
Problem solving.
The Orbit Counting Theorem.
Fixed points of group actions.
Second application: Fixed points of group actions.
Sylow Theorems - a preliminary proposition.
Sylow Theorem I.
Problem solving I.
Problem solving II.
Sylow Theorem II.
Sylow Theorem III.
Problem solving I.
Problem solving II.
Free Groups I.
Free Groups IIa.
Free Groups IIb.
Free Groups III.
Free Groups IV.
Problem Solving/Examples.
Generators and relations for symmetric groups – I.
Generators and relations for symmetric groups – II.
Definition of a Ring.
Euclidean Domains.
Gaussian Integers.
The Fundamental Theorem of Arithmetic.
Divisibility and Ideals.
Factorization and the Noetherian Condition.
Examples of Ideals in Commutative Rings.
Problem Solving/Examples.
The Ring of Formal Power Series.
Fraction Fields.
Path Algebra of a Quiver.
Ideals In Non Commutative Rings.
Product of Rings.
Ring Homomorphisms.
Quotient Rings.
Problem solving.
Tensor and Exterior Algebras.
Modules : definition.
Modules over polynomial rings $K[x]$.
Modules: alternative definition.
Modules: more examples.
Submodules.
General constructions of submodules.
Problem solving.
Quotient modules.
Homomorphisms.
More examples of homomorphisms.
First isomorphism theorem.
Direct sums of modules.
Complementary submodules.
Change of ring.
Problem solving.
Free Modules (finitely generated).
Determinants.
Primary Decomposition.
Problem solving.
Finitely generated modules and the Noetherian condition.
Counterexamples to the Noetherian condition.
Generators and relations for Finitely Generated Modules.
General Linear Group over a Commutative Ring.
Equivalence of Matrices.
Smith Canonical Form for a Euclidean domain.
solved_problems1.
Smith Canonical Form for PID.
Structure of finitely generated modules over a PID.
Structure of a finitely generated abelian group.
Similarity of Matrices.
Deciding Similarity.
Rational Canonical Form.Jordan Canonical Form.