## A description of topics covered each class:

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

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