Abstract Algebra: Theory of Groups and Vector Spaces

Video Lectures

Displaying all 38 video lectures.
Lecture 1
Review: Linear algebra & Definition of groups
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Review: Linear algebra & Definition of groups
Week 1: Review of linear algebra. Groups. Examples of groups. Basic properties and constructions.This video: Introduction to the course; Review: Linear algebra; Definition of groups
Lecture 2
Symmetric groups, Subgroups of ℤ & Cyclic subgroups
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Symmetric groups, Subgroups of ℤ & Cyclic subgroups
Topics: Administrative notes; Generalities on groups; Symmetric groups on n letters; A stabilizer subgroup; The subgroups of ℤ; Cyclic subgroups gen by elementNotes for this lecture:
Lecture 3
Isomorphisms & Homomorphisms
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Isomorphisms & Homomorphisms
Topics: The story so far; Isomorphisms; Homomorphisms; Images
Lecture 4
Kernels, Normality, Centers and Inner Autos
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Kernels, Normality, Centers and Inner Autos
Week 2: Permutations. Cosets, Z/nZ. Topics in this video: Review, kernels, normality; Examples; Centers and inner autos
Lecture 5
Equivalence Relations & Cosets
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Equivalence Relations & Cosets
Topics: Equivalence relations; Cosets; Examples
Lecture 6
Congruence mod n
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Congruence mod n
Topics: Congruence mod n; (Z/nZ)*
Lecture 7
Quotients
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Quotients
Week 3: Quotient groups, first isomorphism theorem. Abstract fields, abstract vectorspaces. Construction and invariants of vectorspaces.
Lecture 8
More on Quotients & Vectorspaces
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More on Quotients & Vectorspaces
Lecture 9
More on Vectorspaces
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More on Vectorspaces
Week 3: Quotient groups, first isomorphism theorem. Abstract fields, abstract vectorspaces. Construction and invariants of vectorspaces.This video: ...ContinuedNotes for this lecture: http://www.extension.harvard.edu/sites/default/files/openlea... lectures are from the Harvard Faculty of Arts and Sciences course Mathematics 122, which was offered as an online course at the Extension School. View Complete course (Syllabus, Notes, Problem Sets, etc) at: http://www.extension.harvard.edu/open-learning-initiative/ab...
Lecture 10
Bases and vectorspaces; Matrices and linear transfer
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Bases and vectorspaces; Matrices and linear transfer
Week 4: Abstract linear operators and how to calculate with them. Properties and construction of operators.
Lecture 11
Bases & Matrices
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Bases & Matrices
Lecture 12
Eigenvalues and Eigenvectors
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Eigenvalues and Eigenvectors
Lecture 13
Review for midterm & Orthogonal groups
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Review for midterm & Orthogonal groups
Lecture 14
Orthogonal Groups & Geometry
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Orthogonal Groups & Geometry
Lecture 15
Finite groups of motions
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Finite groups of motions
Lecture 16
Discrete groups of motions
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Discrete groups of motions
Lecture 17
Abstract group actions
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Abstract group actions
Week 6: Isometries of plane figures.
Lecture 18
Group actions
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Group actions
Week 6: Isometries of plane figures. Cyclic and dihedral groups. Finite and discrete subgroups of symmetry groups.This video: Group actionsNotes for this lecture: http://www.extension.harvard.edu/sites/default/files/openlea... lectures are from the Harvard Faculty of Arts and Sciences course Mathematics 122, which was offered as an online course at the Extension School. View Complete course (Syllabus, Notes, Problem Sets, etc) at: http://www.extension.harvard.edu/open-learning-initiative/ab...
Lecture 19
Group actions II
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Group actions II
Lecture 20
Basic properties and constructions of group actions
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Basic properties and constructions of group actions
Week 7: Group actions. Basic properties and constructions. Groups acting on themselves by left multiplication. Groups acting on themselves by conjugation.
Lecture 21
Groups acting on themselves by left multiplication
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Groups acting on themselves by left multiplication
Lecture 22
Groups acting on themselves by conjugation
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Groups acting on themselves by conjugation
Lecture 23
Alternating group structure
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Alternating group structure
Week 8: A5 and the symmetries of an icosahedron. Sylow theorems. Study of permutation groups
Lecture 24
Ring Theory
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Ring Theory
Week 8: A5 and the symmetries of an icosahedron. Sylow theorems. Study of permutation groups
Lecture 25
Ring Theory II
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Ring Theory II
Week 8: A5 and the symmetries of an icosahedron. Sylow theorems. Study of permutation groups
Lecture 26
Examples of Rings
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Examples of Rings
Week 9: Rings. Examples of rings. Basic properties and constructions.
Lecture 27
Examples of Rings II
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Examples of Rings II
Lecture 28
Basic properties and constructions of Rings
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Basic properties and constructions of Rings
Lecture 29
More on Rings
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More on Rings
Lecture 30
Extensions of Rings: Quotient rings
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Extensions of Rings: Quotient rings
Lecture 31
Extensions of Rings: Integral domains
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Extensions of Rings: Integral domains
Lecture 32
Extensions of Rings: Fields of fractions
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Extensions of Rings: Fields of fractions
Lecture 33
Gauss’ lemma
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Gauss’ lemma
Lecture 34
Eisenstein’s criterion
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Eisenstein’s criterion
Week 12: Euclidean domains, PIDs, UFDs. Gauss' lemma. Eisenstein's criterion. Algebraic integers.
Lecture 35
Algebraic integers
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Algebraic integers
Week 12: Euclidean domains, PIDs, UFDs. Gauss' lemma. Eisenstein's criterion. Algebraic integers.
Lecture 36
Dedekind domains & Ideal class groups
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Dedekind domains & Ideal class groups
Week 13: Structure of ring of integers in a quadratic field. Dedekind domains. Ideal class groups.
Lecture 37
Review 1
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Review 1
Lecture 38
Review 2
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Review 2