The basics of bases

Recently, I was very lucky to win a Discovery Early Career Researcher Award (DECRA). These grants give three years of funding to early career researchers from any discipline, from anywhere in the world, to work in Australia.

As you can see from the screenshot, I had to apply for funding for a specific project – mine is entitled “The existence and abundance of small bases of permutation groups”.

Today, I want to tell you about what these words mean, and why people care about bases for permutation groups. Here we go!

Consider a regular n-sided polygon. What are its symmetries? For simplicity’s sake, let’s consider a square. Well, first we can rotate it. We can think of every rotation we do as a clockwise rotation, since rotating anticlockwise by i vertices is the same as rotating clockwise by n-i vertices.

We also have reflections – choose any line of mirror symmetry through the polygon and reflect the corners through it.

The collection of all of these rotations and reflections forms a group. Groups have some special properties. For example, we can combine any two symmetries in the group to get a new symmetry.

The groups of symmetries of regular n-sided polygon are very well studied (as you might imagine). They are called the dihedral groups.

We can consider the dihedral groups as permutation groups, since they permute (or mix around) the vertices of the n-sided polygon they are acting on. We usually use a Greek omega ($\Omega$) to denote the set of elements that we permute.

Permutation groups are important in lots of areas of science – they can distinguish molecules or crystal structures in chemistry, and can also help you to solve Rubik’s cubes!

So that’s permutation groups – how about bases? Well a base is just a subset of $\Omega$ such that the only symmetry that fix the points in our base is the one that does nothing at all.

Let’s look at an example with the square. If I fix one of the vertices of the square (say, the vertex labelled 1), then I can no longer rotate it, but I can still reflect through a line of symmetry through that corner. So if we want to build a base, then we need another vertex.

We could take the vertex opposite the one we’ve already chosen (here the vertex labelled 3), but that wouldn’t be very helpful, since we can still reflect through that line of symmetry.

So let’s take a vertex adjacent to our original vertex, maybe vertex 2. Now we can’t reflect anymore, so those two vertices form a base.

Bases are useful because they can tell us something about the associated permutation group. For example, in the 1800s and early 1900s, finding a base could allow you to say something about the size of your permutation group.

More recently, bases have been used in algorithms that allow us to work with permutation groups computationally.

In both of these applications, being able to find small bases is advantageous, because it makes the size results stronger, and the algorithms more efficient. However, small bases can be very difficult (and sometimes impossible!) to find.

Even though mathematicians have been studying bases for many years, there’s still so much we don’t know. So my aim for this project is to make an impact by branching out into some new techniques for studying bases, as well as tweaking older ones so that we can make it even easier to understand symmetry and the role it plays in our world.