Group Theory - An Introduction



"When you live in a complex world, you have to simplify it in order to understand it." - Terry Pratchett


In many number systems such as the integers \(\mathbb{Z}\), the reals \(\mathbb{R}\) and the rationals \(\mathbb{Q}\) we can do addition and subtraction. For example, if we have two integers \(a\) and \(b\) we have that \(a+b\) is an integer too. We say that the integers, \(\mathbb{Z}\), are closed under addition. What's more, for every integer \(a\) there's an inverse which we denote as \(-a\) so that \(a+(-a) = 0\), and zero has the special property that \(a+0=a\) for any integer \(a\). This is all pretty simple stuff, so why is it interesting?

Groups are a way of abstracting from this idea, and making more general statements about objects that have these properties. It's precisely because of groups that we can compare things like the integers to objects like a pentagon or a cube. Let me put into perspective how awesome this is. When you were at school, you probably learned about Pythagoras' theorem which states that if you have a right-angled triangle with hypotenuse \(c\) and sides \(a\) and \(b\) then they satisfy the formula \(a^2 + b^2 = c^2\).

Pythagoras' Theorem
Figure 1: Pythagoras' Theorem.

Now, this is probably one of the coolest things in secondary school maths because it's the first introduction to how new maths is discovered. The lengths of the triangle are a geometric feature, and it would be immensely difficult to ascertain what the third length would be given the first two - but that's exactly what Pythagoras' Theorem does; it takes the question out of context and explains a geometric feature by using algebra! In doing this Pythagoras linked two apparently separate branches of maths - algebra and geometry - and showed that we can communicate between these disciplines.

This is exactly what we're doing, or at least trying to do, with groups. Once you notice a few fundamental properties of a structure you can write these properties down and see what features of, say, the rational numbers are caused by the fact that they have unique inverses. To this end, I'll clarify what these fundamental properties are, and we'll call them the axioms of group theory: A group \((G,\ast)\) is a set \(G\) along with a binary operation \(\ast\) which satisfies the following axioms:
[1] Identity: There exists an element of \(G\) called \(e\) such that for any element \(g\) of \(G\) we have that \(e\ast g=g=g\ast e\). In \(\mathbb{Z}\) this element is just zero since for any \(a\), \(a+0=a\).
[2] Inverse: Each element \(g\) of \(G\) has an inverse \(g^{-1}\) (also in \(G\)) such that \(g\ast g^{-1} = e\). For \(\mathbb{Z}\) this would be \(-g\).
[3] Closure If you have two elements \(g,h\) of \(G\) then their product \(g\ast h\) is in \(G\) also, akin to the idea that two integers added together make an integer.
[4] Associativity For elements \(f,g,h\) of \(G\) we have that \(f\ast (g\ast h) = (f\ast g)\ast h\).

These four axioms define a group, and as we've discussed, the integers, rational numbers, and real numbers all form a group (as do the complex numbers, if you've seen them before) under addition as the binary operation. But we can go further; The rotational symmetries of any polygon form a group too! The identity simply consists of not rotating at all, an inverse to a rotation is rotating just as far in the opposite direction, and any two rotations done one after another could be simplified into one rotation - so it even satisfies closure! I've left out associativity, but can you see why rotations are associative (spoiler ahead)?

Pythagoras' Theorem Pythagoras' Theorem Pythagoras' Theorem
Figure 2: Rotational Symmetries (Created with GNuPLOT).

This linking is doing exactly the same as Pythagoras' theorem in that it's linking algebra with geometry, and specifically symmetries! Let's take a look at why the rotational symmetry of a polygon satisfies associativity. Let \(r,s\) and \(k\) be some rotational symmetries of a polygon (for example, a rotation of \(90^\circ\) clockwise of a square). The most intuitive sense in which we can combine rotations \(r\) and \(s\) with a binary operation is by rotating by \(r\) then rotating again by \(s\), giving us the rotation \(r\ast s\), then the question of associativity is this: Is rotating by \(k\) and \(s\) then by \(r\) the same as rotating by \(k\) and then by \(s\) and \(r\)? Well, yes! Thus the rotations of a polygon satisfy the four group axioms!

Perhaps though, it's worth giving examples of things that are not groups. We've looked at addition of integers, so I think the natural next step is to look at multiplication. I think now it's worth stopping reading for a second to look at the group axioms again and see if you can work out why multiplication of integers doesn't form a group - that is, which axiom isn't satisfied?

The problem with multiplication is to do with inverses. While each element of \(\mathbb{Z}\) does have an inverse, the inverse isn't necessarily in \(\mathbb{Z}\), to illustrate this, observe that the inverse of multiplying by two is multiplying by a half, but \(\frac{1}{2}\) isn't an integer! Unfortunately, it doesn't end there though, there's another issue with the fact that multiplying by zero has no concievable inverse at all! Taking these facts into account, we can make \(\mathbb{Q}\) into a group by removing zero and noting that any two fractions multiplied together give another fraction.

Now, we have a few examples of groups in our mind, but what's the point? Where can we go from here? Group theory underlies most of modern mathematics; it's tools spread through number theory, analysis and topology, all the way to solving Rubiks cubes! It's also the first introduction to abstract algebra for most undergraduates, often the questions in group theory courses rely more on proof techniques, rather than, for instance, finding solutions to equations. Each group has a very specific structure, and once we find we can model something using this structure, we are able to talk about it in mathematical terms. For Rubiks cubes this means that we can prove that certain methods will 'solve' a cube without actually having one in front of us[2]. And so, we find that group theory links all kind of topics together by generalising their structure and talking about it using algebra, much like what Pythagoras did with his very famous theorem.

In a later post I will delve deeper into the rich theory of groups to talk about isomorphisms, group actions and the "Orbit-Stabiliser Theorem", hopefully illustrating their power and giving a real feel of what uses they are in fields like chemistry and more abstract uses like topology. Until then, I'd like to give a huge thanks to Maisy for (at some point soon) editing this blog post, and helping me get on my way to starting Awgrym!

Bibliography

[1] Terry Pratchett
[2] "The Mathematics of the Rubiks Cube", Download pdf


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