bigideas

“Group theory” is the name of a rich and deep field of mathematics. Group theorists study objects called “groups” in the way zoologists study animals. Groups turn up in virtually all fields of mathematics and in many other disciplines, and in this mini-series we’ll introduce the basic ideas involved.

We’ll define exactly what a group is soon, but before then it would be useful to get a feel for what, in general, group theory is about, and that’s the study of reversible transformations. But that probably doesn’t tell you much, so let’s start with some examples.

Think of the hour on a clock-face — 1 o’clock, 2 o’clock and so on. We can add or subtract a number of hours from the current time and get a new time — 4 o’clock plus 2 hours is 6 o’clock. 11 o’clock plus four hours is 3 o’clock, and so on. We’re transforming one time into another, and since adding x hours and then subtracting x hours again leaves you back where you started this transformation is reversible. This situation can be modelled as a group.

Using a 24-hour clock, like the one at Greenwich Observatory (left), leaves the situation similar but different; now 11 o’clock plus four hours is 15 o’clock, but 23 o’clock plus four hours is 3 o’clock again. The similarities and differences between 12-hour and 24-hour time are nicely clarified when you think of the two time systems as two different, but related, groups.

Now think of a standard deck of playing cards. There happen to be a bit more than

80,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000

(52 factorial) possible orderings of those cards. When you pick up the pack and shuffle it, you change the ordering from the original one to a new one. If you knew what the original ordering was then you could carefully “shuffle” the cards back to that state, so shuffling is another reversible transformation. Shuffling cards, like telling the time, can be illuminated by group theory.

A classic example of a group arises in the study of symmetries, and many writers will tell you that group theory is the study of symmetry. In a sense it is, but in another sense they’re redefining the word “symmetry” to mean “something that has a group structure”. A “symmetry” of a shape can be defined as a physical transformation (such as rotation or reflection) that leaves that shape looking the same as it did before.

The Alhambra palace, for instance, is well-known for its ornate decorations that exhibit many different kinds of symmetry, as Islamic art often does in general. Such symmetries are often considered to be key features of aesthetic beauty. As we will see, they can be described using the theory of groups just as the previous examples can, and this fact leads some to consider group theory to exhibit great mathematical beauty on account of the abstract “symmetries” it embodies. The study of three-dimensional symmetries has been useful in molecular chemistry.

Other physical transformations can also be described using group theory. The homeomorphisms you meet in topology are an example since, by definition, every homeomorphism has an inverse that “undoes it”. These aren’t obviously “symmetries”, but nor are the examples involving clocks and cards. Groups appear in other contexts too: they’re important in quantum theory, cryptography and algebraic topology, to name just a few.

We haven’t said what a group is yet, or what kinds of things we can do with it, but I hope this gives you a flavour of the wide range of things group theory can be used to talk about. In the next instalment we’ll get down to business and define our terms.