0. Prerequisites

We often just write $G$ for a group, as it is normally clear from the context which binary operation is being used for the definition. We will generally either use additive or multiplicative notation for the binary operation, i.e. $g+h$ or $gh$.

These are all Abelian! Some non-Abelian examples are as follows:

The group $S_{X}$ is called the permutation or symmetric group on $X$. In the special case $X=\{1,2,\ldots,n\}$, we simply write $S_{n}$.

A number of groups that will be important for us arise from linear algebra:

Observe that $GL_{n}(k)$ consists of invertible $n\times n$ matrices with entries from $k$. If $V$ has dimension $n$, we may identify $\operatorname{GL}(V)$ with $\operatorname{GL}_{n}(k)$ by choosing a basis of $V$ and using coordinate transformations.

A convenient way of defining a group is in terms of generators and relations; we write

here $g_{1},\ldots,g_{n}$ are called the generators and the $r_{1},\ldots,r_{m}$ are finite words in the generators, called relations. The elements of the group are all finite words in the generators modulo the equivalence relation that two words are equivalent if one can be obtained from the other by substituting in or out the relations; we say that a word is reduced if it cannot be made any shorter by using the relations.

Note that it follows from the definition that $\varphi(e_{G})=e_{H}$ and $\varphi(g^{-1})=\varphi(g)^{-1}$.

An invertible (bijective) homomorphism is called an isomorphism. If there exists an isomorphism from a group $G$ to a group $H$, then $G$ and $H$ are said to be isomorphic, and we write $G\cong H$. Isomorphic groups are essentially identical from a group-theoretic point of view.

In order to check whether a subset $H\subseteq G$ is a subgroup, one needs to check that $e_{G}\in H$, and that $H$ is closed under multiplication and when taking inverses.

A nice example of this is the following: $\operatorname{SL}_{n}(k)=\ker\det$ is the subgroup of $\operatorname{GL}_{n}(k)$ consisting of all matrices with determinant one.

Given a group $G$ with a subgroup $H$, one can form the coset space $G/H$. The elements of $G/H$ are called cosets; these are subsets of $G$ of the form

If a subgroup $N<G$ has the property that $gNg^{-1}=N$ for all $g\in G$, then $N$ is said to be normal. The coset space $G/N$ has a natural group structure, with multiplication given by $g_{1}N\cdot g_{2}N=g_{1}g_{2}N$. You can check this is a well-defined group operation due to $N$ being normal.

We normally don’t write out the function “$f$”; instead, we simply denote $f(g,x)$ by $g\cdot x$. In this notation, property (2) reads

Given a group action of a group $G$ on a set $X$, we let ${\mathcal{O}}_{G}(x)$ denote the orbit of a point $x\in X$ under $G$, that is ${\mathcal{O}}_{G}(x)=\{g\cdot x\,|\,g\in G\}$. The stabiliser of a point $x$ is defined as $\mathrm{Stab}_{G}(x)=\{g\in G\,|\,g\cdot x=x\}$.

In the special case of a group acting on itself by conjugation, the orbits are called conjugacy classes, and are instead denoted ${\mathcal{C}}_{g}$, i.e. ${\mathcal{C}}_{g}=\{hgh^{-1}\,|\,h\in G\}$.

Recall that a vector space $V$ over a field $k$ is a set combined with two operations:

1. (i)
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   vector addition: given two vectors ${\bf v},{\bf w}\in V$, we can “add” them together to obtain a new vector ${\bf v}+{\bf w}\in V$.

2. (ii)
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   scalar multiplication: given a vector ${\bf v}\in V$ and an element $\lambda\in k$, we can “multiply” ${\bf v}$ by $\lambda$ to obtain a new vector $\lambda{\bf v}\in V$.

These two operations must be compatible with each other and satisfy some natural properties, as listed in the axioms for vector spaces.

We will almost always just consider vector spaces over ${\mathbb{C}}$, however other examples that one might consider are $k={\mathbb{Q}},{\mathbb{R}},\mathbb{F}_{q}$ (here $q=p^{r}$ where $p$ is prime and $r\in{\mathbb{N}}$, and $\mathbb{F}_{q}$ denotes the field with $q$ elements).

Given two vector spaces $V,W$ (over the same field $k$), a function $T:V\rightarrow W$ is said to be linear if

for all ${\bf v},{\bf u}\in V$ and $\alpha,\beta\in k$. Two vector spaces are said to isomorphic if there exists an invertible linear map from one to the other.

The set of all linear maps from $V$ to $W$ is denoted $\operatorname{Hom}(V,W)$, and we write $\operatorname{Hom}(V)$ instead of $\operatorname{Hom}(V,V)$. These spaces carry a natural vector space structure inherited from $V$ and $W$; given $S,T\in\operatorname{Hom}(V,W)$ and $\lambda\in k$, we define $S+T\in\operatorname{Hom}(V,W)$ and $\lambda T\in\operatorname{Hom}(V,W)$ by

for all ${\bf v}\in V$.

Note that to show that a nonempty subset is a subspace, one simply needs to check that it is closed under vector addition and scalar multiplication.

A vector space $V$ is an Abelian group with respect to the vector addition operation. A subspace $U$ of $V$ is therefore a normal subgroup with respect to this operation, allowing us to consider the quotient group $(V/W,+,{\bf 0})$. We can then define a scalar multiplication on this quotient by the formula

for all $\lambda\in k$ and ${\bf v}+W\in V/W$. This definition gives the quotient group a vector space structure, and we call this the quotient space of $V$ with respect to $W$.

Given $T\in\operatorname{Hom}(V,W)$, define a new map $\widetilde{T}:V/\ker(T)\rightarrow W$ by

for all ${\bf v}+\ker(T)\in V/\ker(T)$. Then $\widetilde{T}:T/\ker(T)\rightarrow{\mathrm{im}}(T)$ is an isomorphism of vector spaces.

The external direct sum of two vector spaces $V,U$ is the set

with the addition and scalar multiplication operations being defined component-wise.

If $V$ and $U$ are subspaces of a vector space $W$ such that $V\cap U=\{{\bf 0}\}$ and every element of $W$ may be written as the sum of an element of $V$ and an element of $U$, then we say that $W$ is the internal direct sum of $V$ and $U$, and the map $({\bf v},{\bf u})\mapsto{\bf v}+{\bf u}$ is an isomorphism from $V\oplus U$ to $W$.

One can define sums $V_{1}\oplus V_{2}\oplus V_{3}\oplus\ldots$ of multiple vector spaces in a similar way.

A basis ${\bf v}_{1},{\bf v}_{2},\ldots$, of $V$ is said to be an eigenbasis of $T\in\operatorname{Hom}(V)$ if every element of the basis is an eigenvector of $T$. If $T$ has an eigenbasis, then $T$ is said to be diagonalisable. We will make use of the following result at a few key moments in the module: