How To Think Like An Algebraist

I look at a Group Theory problem set to 18 year olds to demonstrate concepts of algebraic structure

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If you were an enthusiastic and talented high school mathematics student in the UK in 1988, you would have studied elementary Group Theory as part of your Advanced Level (A-level) curriculum. Sadly this is not the case any more. Several years ago, Group Theory was removed from the A-level curriculum and now is only available to those pursuing undergraduate degrees in mathematics related subjects.

This is sad, because Group Theory is a great introduction to how to think like an algebraist. This way of thinking allows you to deal with discrete and abstract objects in an organized and systematic way, a skill which is essential for the study of advanced mathematics, and which I have found extremely useful in life in general. Thinking in this way takes some getting used to, and those that have done a little bit of it at school will likely adapt to it quicker if they proceed to study mathematics at degree level.

I came across this problem in a 1988 Cambridge entrance paper and I think it is a great illustration of my point. If you can think about this problem in an appropriately ordered and logical way, the solution is neat, tidy and very elegant indeed. But without any training in how to think like an algebraist, you would struggle to attack this problem.

Before I present the problem, let me introduce you to some of the small amount of group theory that high school students used to learn back in the olden days 😊

What is a Group?

A Group is a way to abstract and generalize some common structures which we see in mathematics. As a simple example, let’s take the set of integers and lets look at the operation of addition on this set. There are some properties of these integers and this operation which we take for granted but which are essential to how they work and to their usefulness. Here are the properties:

1. The integers are closed under addition, meaning that adding any two integers will produce another integer.
2. The integers are associative under addition, meaning that if you have three integers a, b and c, then a + (b + c) =