Lecture Summaries
Lecture 1
Recall definitions and important facts: group, subgroup, normal subgroup, homomorphism, kernel of homomorphism, coset, quotient group
Examples: $(\mathbb{Z}, +)$ has (normal) subgroups $m\mathbb{Z}$ (for $m\in\mathbb{N}$) and quotients $\mathbb{Z}/m\mathbb{Z}$; $S_3$ has quotient $S_3/C_3$

Lecture 2
Recall Isomorphism Theorems; define subnormal series, composition series, solvable group; state and prove Jordan‒Hölder Theorem: In a finite group any two composition series hev the same length and (up to reordering) the same factors.

Lecture 3
Solvable groups: definition, closure under subgroups, quotients and extensions. List of very small finite groups.

Lecture 4
Prove that solvable groups are closed under extensions.
Direct product: $Q\!\times\!N$ with component-wise multiplication.
Semi-direct product: $Q\!\times\! N$ with multiplication $(q,n)(q',n')=(qq', n^{q'^\varphi}n')$, where $\varphi\colon Q\rightarrow \mathrm{Aut}(N)$ is a fixed homomorphism.

Lecture 5
More on semi-direct products, examples, conjugation action, (inner) automorphisms, centre.

Lecture 6
Characterisation: $G$ is a semi-direct product if and only if $Q\subset G\triangleright N$ with $G=QN$ and $Q\cap N=1$. Some more examples including mention of general and special linear groups, $GL_n(\mathbb{F})\supset SL_N(\mathbb{F})$, where $\mathbb{F}$ is any field. Special case: $\mathrm{Aut}(C_2\!\times\! C_2)\cong GL_2(\mathbb{F}_2)\cong S_3$.

Lecture 7
Commutator identities, commutator (derived) subgroup, characteristic subgroups, derived series.

Lecture 8
Prove that solvable $\Leftrightarrow$ derived series is finite and terminates in $1$. Sylow-$p$ subgroups and statement of Sylow's Theorem. Sylow-$p$ subgroups of symmetric groups.

Lecture 9
(Un)Restricted permutational/standard wreath products. Kaloujnine's characterisation of Sylow-$p$ subgroups of $S_n$.

Lecture 10
Formal notation for wreath products. Finite generation of wreath products.

Lecture 11
Discussion of Q1—Q3 on Problem Sheet 1.

Lecture 12
Discussion of Q3&Q4 on Problem Sheet 1. Definition of projective special linear groups; mention of projective geometry and simplicity of $PSL_n(\mathbb{F}_q)$ for $n\ge 3$, or $n=2$ and $q> 3 $.

Lecture 13
Definition of free groups, their existence and uniqueness for a given generating set. Reduced words. Definition of group presentations.

Lecture 14
Rank of free group, examples of presentations, Dehn's word problem, Novikov—Boone Theorem on finitely presented groups with unsolvable word problem.

Lecture 15
Proof that, for presentations $G=\langle X\mid R\rangle$ and $H=\langle Y\mid S\rangle$, a map $\alpha\colon X \rightarrow H$ extends to a homomorphism if $r^\alpha=1_H$ for all $r\in R$.

Lecture 16
Discussion of Q2—Q3 on Problem Sheet 1.

Lecture 17
Free products, definition, existence, uniqueness, normal forms and examples.

Lecture 18
Free products with amalgamation, existence, diagram, (rewriting into) normal form. Background: transversals, coset representatives.

Lecture 19
Definition of an HNN-extension as a certain subgroup of a suitably chosen free product with amalgamation.

Lecture 20
Normal form theorem for HNN-extensions and Britton's Lemma.

Lecture 21
Free products (with amalgamation) and HNN-extensions in the context of fundamental groups.

Lecture 22
Subgroups of free groups, embedding a countable group into a group with two generators.

Lecture 23
Discussion of Questions 2&4 of Problem Sheet 2. Signed permutations.

Lecture 24
Completion of the discussion of the automorphism group of the quaternion group.

Lecture 25
More on fundamental groups. Deck transformations, fundamental domain, space as quotient (i.e. orbit space) of a simply connected space by a discrete and free action of a group of isometries.

Lecture 26
Basic introduction to hyperbolic space. Definition of a hyperbolic group in terms of $\delta$-slim triangles.

Lecture 27
Cutting the surface of genus 2 to get an octagon with the boundary labelled by the relator of the fundamental group. Quasi-isometric embedding of the fundamental group into the universal covering space.

Lecture 28
Definition of interior and corresponding points of a geodesic triangle, the notion of a $\delta$-thin triangle and proof that for a geodesic metric space the existence of a $\delta$ such that every geodesic triangle is $\delta$-slim is equivalent to the existence of a (possibly different) $\delta$ such that every geodesic triangle is $\delta$-thin.

Lecture 29
Definition of Dehn presentation, how it is used to decide whether a word is trivial or not. Definition if Dehn function of a finitely generated group.

Lecture 30
Proof that a hyperbolic group has a Dehn presentation, and hence Dehn algorithm.

Lecture 31
Hyperbolic groups have only finitely many conjugacy classes of elements of finite order.

Lecture 32
Torsion elements in $G\ast H$ are conjugate to elements of finite order in $G\cup H$ (Q5 on Assignment 2). Review of wreath products: standard, permutational, restricted and unrestricted.
Research Problem: Is $G(9,1,4)=\langle x_0,x_1,\ldots, x_8\mid x_ix_{i+1}=x_{i+4},\, 0\le i<9,\,\text{indecees modulo 9}\rangle$ finite or infinite?

Lecture 33
Remarks on left versus right action and how one should adjust commutators and conjugation. Stabilisers of points of a set on which a group acts; centralisers of elements are stabilisers in the conjugation action.