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