Lecture 1: Basics of Group Theory

Definitions, Examples: $\mathbb{Z}, \mathbb{R}, \mathbb{C}, \mathbb{Q}, \mathbb{R}_{>0}, \mathbb{Q}_{>0}, Sym(X), S_n$

Cycle notation

Homomorphisms and isomorphisms and examples.

Examples of Classification Theorems

Lecture 2: Symmetric Groups, group actions and cyclic groups

Symmetric groups: cycle notation, multiplication and cycle decompositions

Direct Products of groups

Group Actions: Definition, Examples: trivial action, $\mathbb{Z}$ acting on $\mathbb{R}, \mathbb{Z}^2$ acting on $\mathbb{R},$ left multiplication and conjugation.

Permutation representations.

Faithful actions and Examples

Subgroups: Defintion and definition of subgroup generated by a set $\langle S\rangle$.

Cyclic Groups and orders of elements.

Lecture 3: Cosets

Cosets of a subgroup, partitions of the group by cosets.

Normal Subgroups

Kernels

Quotient Groups

First Isomorphism Theorem

Lecture 4: Isomorphism theorems

Second, Third and Fourth Isomorphism Theorems

Lecture 5: Parity of Permutations

Transpositions, generators for the symmetric group.

Even and odd permutations, definitions

Alternating Group: Definition, examples in small cases.

Group Actions: Definitions of Kernel, Stabiliser, Orbit

Orbit Stabiliser Theorem

Lecture 6: $A_5$ is simple

Conjugacy Classes and centralisers. Examples in Abelian groups and symmetric groups

The class equation. Proof that $p$-groups have non-trivial centre.

Normal subgroups are unions of conjugacy classes.

Conjugacy classes in $S_5$, complete computation.

Conjugacy classes in $A_5$ and proof that $A_5$ is simple.

Lecture 7: Groups acting on subgroups and cosets

$G$ acts on $G/H$ by left multiplication.

Kernel is the normal core of $H$: the largest normal subgroup contained in $H$.

$G$ acting on subgroups by conjugation.

Automorphism groups, Inner automorphisms and outer automorphism groups

Fixed point properties for $p$-groups

Proof of Cauchy's theorem

Statement of Sylow's theorem

Lecture 8: Proof and Applications of Sylow's theorem

Definition of normaliser of a subgroup

Proof of Sylow's theorem using fixed point properties of $p$-groups.

Soluble groups, examples ($p$-groups)

Applications of Sylow's theorems: Groups of order 30, $pq$, $p^2q^2$, 105 are soluble.

Lecture 9: Direct Products, Commutators and Abelianisations

Definition of Direct products, commutators, commutator subgroups and abelianisation.

Proof that $G$ is normal in $G\times H$ and $G\times H/G \cong H$.

Proof that the abelianisation is the largest abelian quotient of $G$.

Discussion of the fundamental theorem of finitely generated abelian groups.

Lecture 10: Detecting direct products and semi-direct products

Lecture 11: Introduction to Rings

Definition and examples of rings

Examples: $\mathbb{Z}$, Matrix rings, Polynomial rings, Group rings

Lecture 12: Ideals

Properties of ideals, intersections, generation, prime ideals, maximal ideals

Prime ideal iff $R/I$ is an integral domain

Maximal ideal iff $R/I$ is a field

Lecture 13: Euclidean domains and Principal Ideal Domains

Definition of Euclidean domain (ED)

Examples: Fields, $\mathbb{Z}, F[x]$

$\mathbb{Z}[x]$ is not an ED

Definition of Principal ideal domain (PID)

ED implies PID

Definition of divisors and uniques of greatest common divisor using ideals

Lecture 14: Unique Factorisation Domains

Definition of prime and irreducible elements

Definition of unique factorisation domains

$r$ is prime iff $r$ is irreducible in a UFD

Principal ideal domains are unique factorisation domains

Lecture 15: Field of fractions and quadratic integer rings

Definition of quadratic integer rings

Proof units are elements with norm 1

Definition of polynomial rings in many variables

$F[x]$ is a unique factorisation domain

Definition of localisations

Lecture 16: Polynomial rings are unique factorisation domains

Irreducible polynomials over $R[x]$ are irreducible over $F[x]$

A polynomial over $F[x]$ is irreducible over $R[x]$ if the gcd of the coefficients is 1

Proof that the polynomial ring over a unique factorisation domain is a unique factorisation domain

Lecture 17: Modules and Module homomorphisms

Definition of module

Examples: $\mathbb{Z}$ modules are abelian groups, modules over a field are vector spaces

Definition of a submodule

Definition of homomorphism and sets of homomorphisms

Proof the set of module homomorphisms is a module

Lecture 18: Quotient Modules and Free Modules

Definition of Quotient module

Statement of Isomorphism theorems for modules

Finite sums of submodules

Generating sets for modules

Direct products

Definition of Free module and proof of existence

Lecture 19: Free modules, Direct Sum, Direct Prodcut, Zorn's Lemma

Proof of Universal Property

$F(A)\cong M$ for any free module on $A$

Definition of Direct Sum and Direct product

Proof they are different

Discussion of Zorn's Lemma

Proof that every ring has a maximal ideal

Lecture 20: Modules over Principal Ideal Domains

3 Equivalent definitions of Noetherian ring

PIDs are Noetherian

Definition of rank of a module

Rank decreases for submodules of free modules over integral domains

Statement of that any sub module of a free module over a PID is free and has a basis that can be extended

Proof of fundamental theorem of finitely generated modules over PIDs

Lecture 21: Modules over PIDs and Tensor Products

Proof of submodules of free modules over a PID are free.

Discussion of how this fails in general

Discussion of extending scalars

Definition of Tensor product

Several examples

Discussion of universal property of $S\otimes_R M$ as an $S$-module

Lecture 22: Tensor products and universal properties

Proof of universal property of tensor product of $S\otimes_R M$ as an $S$-module

Examples of this universal property

Proof that the tensor product is generated by basic tensors of generators

Definition of bilinear maps

Proof of the universal property that any bilinear map from $M\times N$ gives a unique homomorphism from $M\otimes_R N$

Lecture 23: Associativity of Tensor products

Proof that tensor product is associative and commutative

Proof that tensor product is distributive over direct sum

Proof that tensor products of free modules are free

Definition of the tensor product of two maps

Lecture 24: Exact sequences

Definition of exact sequences

Rephrasing of injective and surjective in terms of exact sequences

Definition and examples of short exact sequences

Statement of the 5-lemma

Lecture 25: The 5-lemma and the functors ${\rm{Hom}}_R(D, -)$ and $D\otimes_R-$

Proof of the 5-lemma for two short exact sequences

Definition of projective modules

Definition and discussion of ${\rm{Hom}}_R(D, -)$ for an $R$-module $D$

Definition and discussion of $D\otimes_R-$ for an $R$-module $D$

Examples to show that they do not take exact sequences to exact sequences

Statement that ${\rm{Hom}}_R(D, -)$ preserves exactness on the left

Statement that $D\otimes_R-$ preserves exactness on the left

Lecture 26: Exactness of ${\rm{Hom}}_R(D, -)$ and $D\otimes_R-$

Proof that ${\rm{Hom}}_R(D, -)$ is left exact and $D\otimes_R-$ is right exact

Equivalence of definitions of projective modules