Some maths

Group actions One of the most basic concepts of group theory is that of a group action . Informally we let a group act on a set by assigning a map (or function) from the set to itself to each element of the group. You may have heard that group theory is in some sense the study of symmetry; in order to unpack what this means we need to understand group actions. The formal definition of a group action is a follows: Let $G$ be a group and $\Omega$ be a set. Then $G$ acts on $\Omega$ if there exists some $\cdot : G \times \Omega \to \Omega$ such that: (1) $1 \cdot x = x$ for all $x \in \Omega$ (2)$(gh) \cdot x = g \cdot (h \cdot x)$ for all $g,h\in G$ and $x \in \Omega$. In a sense we are turning $G$ into a collection of functions from $\Omega$ to itself where the identity in $G$ becomes the identity function on $\Omega$, and the composition of two functions on $\Omega$ is just their product in $G$. We will sometimes write $g \cdot x$ as $x^g$. The first and most obvious example of a group...

The most simple (and unhelpful) definition of group theory is as follows: group theory is the study of algebraic objects called groups. When mathematicians use the world algebraic they are referring to the branch of mathematics known as abstract algebra - this is the study of collections of objects that have some kind of "structure". The integers To illuminate this definition we can give an example. Consider the collection of both positive and negative whole numbers i.e. the integers. We normally represent these using the symbol $\mathbb{Z}$.  When we have two integers, let's call them $m$ and $n$, we can add them together to get a third integer $m+n$. We call addition a binary operation on $\mathbb{Z}$ as it takes two inputs and gives a single output. We may sometimes write this formally as $+ : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$. This $+$ operation has a few interesting properties. The first property has to do with the integer $0$. For any integer $m$ (using...