Some maths

Minimal base sizes In this post I will talk about the concept of a base of a group action. This gives rise to a property of the action which is often interesting to calculate: the minimal base size. Firstly let $G$ be a group acting transitively on a set $\Omega$. We say that $G$ acts transitively if for every $x,y \in \Omega$ there exists $g \in G$ such that $x^g = y$. Now let $B$ be a subset of $X$. We denote by $G_B$ the pointwise stabiliser of $B$. This is the set of $g \in G$ such that $x^g = x$ for all $x \in B$, i.e. the collection of all $g \in G$ that fix or "stabilise" every element of $B$. We call such a subset $B$ a base  for the action if its pointwise stabiliser is trivial, that is, $G_B = {1}$. This means that only the identity fixes all elements of $B$. The main use for bases is in the field of computational group theory. The main reason for this is that the action a group on a base gives a lot of information about the full action of the group. To see this let

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