by Anna Mustata
Part 0 – Disclaimer
This article is a rather experimental attempt at explaining how I think about mathematics in an aesthetic and intuitive way, and hopefully help the reader see the purpose of studying certain things in mathematics and the reason behind what is done. Essentially, I want to portray mathematics in a way that is appealing and that shows more than the final results and technical proofs. Please keep in mind that none of the explanations I give in this article are fully-fledged proofs, just intuitive motivations for things being a certain way. Also, quotation marks denote terms which are NOT the official or technical term for what I am describing, but are a good intuitive way of thinking about it.
Part I – Introduction
Everything happens in a context. Sometimes, in mathematics, this context is obvious: Euclidean geometry happens on a flat, infinite plane. Hyperbolic geometry happens in the interior of a circle.
In algebra, this is harder to see, but it can be visualized. Real numbers can be added on a number line. Integers can be added similarly if you restrict the points on this line that can be used. Complex numbers can be added in a plane with two axes, real and imaginary. But what happens when we try to get more abstract? Is there a space where we can add, say, polynomials in the variable x? At this point – as tends to happen when things get abstract – our simple geometric intuition seems to fail us. What we can’t show in pictures, we can describe mathematically through a set of rules. Such landscapes where mathematics can happen are known as groups.
As with most areas in mathematics, diving headfirst into group theory without a guide will quickly leave you tangled in minutiae and wondering why you bothered approaching it in the first place. Therefore, let us outline a few key philosophical ideas to keep an eye out for in order to ensure that we are moving forwards in our attempt to study groups.
The first and most straightforward is to remember to think of groups as a landscape. Even when we switch from literal, geometric landscapes to a list of rules, we can still think of the metaphorical “shape” of a group as something that determines how we can “move” through it when we do mathematics in that context.
Another thing that is true throughout mathematics is that everything is about information. Specifically – and this is likely to be best appreciated by computer scientists – it is about making sure that the amount of information needed to describe a useful idea is not greater than the amount of information that can be generated from it; otherwise, our idea is useless. Combining this with the concept of landscape leads us to look for symmetry; that is, for patterns that allow us to examine a small area of this landscape and deduce something about the shape of the rest of it.
Finally, in order to progress, we need to recognize how we might get stuck, and how to become unstuck. Generally, there are two extremes where progress tends to peter out: too much abstraction, and too little. Too much abstraction eliminates all details we might care to know and leaves us with a meaningless, hollow outline. Not enough leaves us repeating the same process indefinitely, never catching a glimpse of the larger picture that might allow us to improve our measure of “information in – information out”. For the best effect, we must combine the two perspectives so that they fuel each other, zooming in and out much as one might when looking at a fractal: first, finding a detail, then using that detail to extrapolate a larger outline, then once more looking closely in order to fill in that outline.
Keeping those things in mind, let’s dive in!
Hopefully at this point you have a fairly good intuitive idea of what we mean by a group; however, this is mathematics, so let’s get a rigorous definition in place. In order to have a mathematical “landscape”, you need objects, and you need to be able to do things with them. Therefore, a group will be defined over a set S and an operation * (which can be anything that combines two objects to give a third). We don’t want to be able to walk off the edge of this mathematical landscape, so applying * to two elements in S must produce another element in S. There are three more axioms that need to be satisfied for (S, *) to be a group.
1) * must be associative: that is, brackets don’t matter, or a*(b*c) = (a*b)*c. This is saying that any ordered sequence of elements in S uniquely define a “path” to another element in S.
2) We also want our landscape to have a “starting point” (this would be 0 on the number line, for example). We will call this the identity element (usually written as e or Id). Moving by the identity element should not affect your path, so for any x in S, x*e = x = e*x.
3) In order to actually make computations and do things, we need to be able to retrace our steps; so, every element x in S must have an inverse y such that x*y = e.
A few algebraic manipulations will show that every element has exactly one inverse and that if x*y = e then y*x = e. (Note: it is not always true that x*y = y*x).