Lecture Notes
- Lecture 0 (August 21)
Definition of a group. Examples: symmetric group, dihedral group,
general linear group, free group.
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Lecture 1 (August 22)
Subgroups. Order of a group, an element. Generating sets.
Cyclic groups. Generators of symmetric group.
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Lecture 2 (August 23)
Groups given by generators and relations. Presentation of
dihedral group.
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Lecture 3 (August 24)
Groups given by generators and relations continued: free group,
symmetric group again.
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Lecture 4 (August 27)
Left and right cosets. Index of a subgroup.
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Lecture 5 (August 28)
Normal subgroups. Quotient group.
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Lecture 6 (August 29)
Group homomorphisms. Isomorphic groups. Kernel and Image.
First isomorphism theorem.
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Lecture 7 (August 30)
Some applications of the first isomorphism theorem. Precise definition of
a presentation of a group.
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Lecture 8 (August 31)
Proof of the presentation of the symmetric group. Second isomorphism
theorem.
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Lecture 9 (September 4)
A quick review. Definition of group actions. Orbits, stabilizers and
fixed points.
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Lecture 10 (September 5)
Transitive actions. Some counting results. Burnside's theorem on
counting number of orbits.
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Lecture 11 (September 6)
Action via conjugation. Conjugacy classes, centralizers.
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Lecture 12 (September 7)
Some cool proofs... optional!
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Lecture 13 (September 10)
p-groups. Sylow theorem.
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Lecture 14 (September 11)
An application of Sylow theorems. Groups of size = square of a prime.
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Lecture 15 (September 12)
Direct product. Abelian p-groups. Classification of finite abelian groups.
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Lecture 16 (September 13)
Applications of Sylow theorems - proving something isn't simple!
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Lecture 17 (September 14)
Sylow theorems continued - examples of Sylow p-subgroups for
some interesting groups (dihedral, linear over finite field,
symmetric).
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Lecture 18 (September 19)
Semidirect products. Examples.
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Lecture 19 (September 20)
Semidirect products via group actions. Automorphism group.
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Lecture 20 (September 21)
Revisiting finite abelian groups - another classification theorem.
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Lecture 21 (September 24)
Automorphisms of cyclic groups.
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Lecture 22 (September 25)
Semidirect products cntd. Example.
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Lecture 23 (September 26)
Semidirect products end. Simple groups. Alternating groups.
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Lecture 24 (September 27)
Basic properties of composition series.
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Lecture 25 (September 28)
Jordan Holder series. Zassenhaus lemma.
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Lecture 26 (October 1)
Commutator series. Solvable groups - definition and examples.
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Lecture 27 (October 2)
Equivalent characterizations of solvable groups. p-groups are
solvable.
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Lecture 28 (October 3)
Central series, nilpotent groups. Basic properties of nilpotent groups.
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Lecture 29 (October 4)
Nilpotent if and only if direct product of p-groups.
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Lecture 30 (October 5)
Aut(dihedral group). General lemmas for Jordan-Holder series etc.
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Lecture 31 (October 15)
Definition of rings. Examples. Invertible elements. Zero divisors.
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Lecture 32 (October 16)
Ring homomorphisms. Subrings. Ideals.
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Lecture 33 (October 17)
More on ideals. Coprime ideals. First isomorphism theorem for ring homomorphisms.
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Lecture 34 (October 18)
Operations with ideals: intersection, sum and product. Chinese remainder theorem.
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Lecture 35 (October 19)
Ideals in polynomial ring K[X], ring of Gaussian integers Z[i].
Analogues of other isomorphism theorem.
- Lecture 36 (October 22)
Principal ideal domains. Prime and Maximal ideals.
- Lecture 37 (October 23)
Existence of maximal ideals. Primes vs Maximal.
- Lecture 38 (October 24)
Local rings. Examples: quotient of polynomial ring, power series ring,
p-adic.
- Lecture 39 (October 25)
Ring of fractions - definition. Some examples.
- Lecture 40 (October 26)
Field of fractions; localization at a prime ideal. Ideals in the ring
of fractions.
- Lecture 41 (October 29)
Ideals in the ring of fractions continued. Nilradical.
- Lecture 42 (October 30)
Jacobson radical. Landscape of rings.
- Lecture 43 (October 31)
Noetherian rings. Definitions and examples.
- Lecture 44 (November 1)
Noetherian rings. Hilbert Basis Theorem.
- Lecture 45 (November 2 and 5)
Noetherian rings - Primary decomposition theorem of ideals.
- Lecture 46 (November 6)
Primary decomposition of ideals when R is a PID.
- Lecture 47 (November 7)
Euclidean domains. Examples of rings of quadratic integers.
- Lecture 48 (November 8)
More examples. Unique factorization domains.
- Lecture 49 (November 9)
UFD's continued. Gauss' lemma. R UFD implies R[x] is UFD.
- Lecture 50 (November 13)
Polynomials - factorization. Eisenstein's criterion.
- Lecture 51 (November 14)
Towards Grobner bases I.
- Lecture 52 (November 15)
Towards Grobner bases II.
- Lecture 53 (November 16)
Grobner bases III - Buchberger's algorithm.