Gavin Jared Bala

This is going to be our grand finale. We’re not going to worry too much about rigorously proving that we have all the groups. Instead, we’re going to just use what we’ve learnt to construct as many as we possibly can.

Since in part II we went up to order four, and in part III we went up to order eight, it seems natural to end off by going up to order sixteen.

We’ve seen one infinite class of groups: these are the cyclic groups. They are defined by one generator, that goes back to the identity if you multiply it by itself enough times. That can be written as follows for short: C_n = < a | aⁿ = 1 >.

(That is: a alone generates the group, and aⁿ = 1. This sort of notation, which gives you enough information to reconstruct the entire group table, is called a presentation of a group.)

We’ve also seen how to build up groups using direct products. If we take the direct product of two groups, we essentially are putting them together but not letting them interact with each other. That is, if G and H are groups, then the direct product G x H lets us multiply elements of G by elements of H, but they never interact when they pass each other: gh = hg.

We saw an example earlier, that C₆ = C₃ x C₂:

We can see what’s going on: the a’s and the b’s essentially live in parallel universes. We can express each element using “how much a and how much b is in it”, and rewrite:

And we can see that the a’s and the b’s are multiplying completely separately.

The groups we briefly considered at the end of the last article are, as another example, direct products of a lot of copies of C₂. Each generator squares to 1, and none of them notice each other.

With the help of direct products, the cyclic groups give rise to a whole bunch of easily-defined small groups:

(Some number-theoretical arguments are needed to prove that this is all we get: for example, C₅ x C₃ is really the same group as C₁₅, so we do not show it above.)

However, what we have also had a glance at is the possibility of semidirect products.

Recall that an automorphism is a symmetry of a group: it is some way in which you can rearrange the elements of a group, that preserves the group’s structure.

For example, if we take the Cayley table of C₃:

and swap a and a² throughout:

then we actually have the very same group. Since this is the only nontrivial way to rearrange the elements while making this still work out, we find that the automorphism group of C₃ is a C₂, whose nontrivial element here takes the form of swapping a and a².

Such an automorphism can be thought of as a function. If we call this automorphism φ, then it will leave 1 alone: φ(1) = 1. But we have φ(a) = a² and φ(a²) = a.

We are now in a position to (informally!) define the semidirect product:

Suppose G and H are groups, and there is some way to make a “fuzzy correspondence” between the automorphisms of G and the elements of H. That is, it is possible to select for every element h of H an automorphism φ_h of G, such that the multiplication works out: if h₁h₂ = h₃, then φ_h1φ_h2 = φ_h3. (Meaning that if you perform the symmetry φ_h2 on G and then the symmetry φ_h1, it’s the same as doing the symmetry φ_h3.) This is formally called a homomorphism between H and Aut(G): call it φ. (There may be more than one way to do this.)

Then to define the semidirect product, we allow multiplication between elements of G and elements of H. Except that instead of making them totally separate, we force them to interact by this rule: every time you pass an element of G to the left past an element of H, it is fed through the corresponding homomorphism.

In other words: h * g = φ_h(g) * h.

Whew! That’s complicated. But it’ll become easier to understand what’s going on through a few examples…

Let’s look at our old friend D₆, which we noted last time was a semi-direct product C₃ ⋊ C₂. Our C₃ = {1, a, a²}, and our C₂ = {1, b}:

In this case, the automorphism of C₃ corresponding to b in C₂ is the one that swaps a and a². (The automorphism corresponding to 1 is always the identity, that does nothing.) So φ_b(1) = 1, φ_b(a) = a² and φ_b(a²) = a.

That is why, whenever we pass the element b, this swap happens: ba = a²b and ba² = ab.

It’s simply following the general rule for a semi-direct product, that h * g = φ_h(g) * h.

For a more complicated example, we could look at the case V₄ ⋊ C₃.

How would that work? Well, we know that V₄ has three nontrivial elements, that all square to the identity. And if you multiply any two of them, you get the third one.

That’s completely symmetric, so any rearrangement of them is an automorphism. Therefore, the automorphism group of V₄ is D₆ = S₃.

Now, how can we make the structure of C₃ = {1, x, x²} compatible with that of D₆?

Well, one way to do it is to map a nontrivial element to the one that cycles the letters forward. So, we will declare that φ_x sends 1 to itself; but it sends a to b, b to c, and c to a.

Because of the structural compatibility, we are forced to have that φ_x² does the same thing as performing φ_x twice. That would mean cycling the letters forward twice (or backward once): it sends a to c, c to b, and b to a.

(Note that it’s about structural compatibility, not “being the same group”. Clearly, not every symmetry of V₄ has actually been used!)

Therefore, for example, x * c = φ_x(c) * x = a * x, whereas x² * c = φ_x²(c) * x² = b * x².

The result is a group of order twelve. It is also a group that naturally occurs in geometry, although it’s a more complicated one than any one we’ve seen. It naturally occurs as the rotational symmetry group of the regular tetrahedron, and is called A₄.

Now, we may have noticed that we could just as well have declared that a nontrivial element of C₃ is mapped to the transformation that cycles the letters backward. (We could not have cycled it to a reflection, because a nontrivial element of C₃ has order 3, but a reflection has order 2.) So, in general, there is not just one way to make a semi-direct product.

Indeed, we knew that already, because a direct product is just a semi-direct product when you create “compatibility” by making every element of H correspond to the automorphism that does nothing to G. That recovers h * g = φ_h(g) * h = g * h.

It turns out that in this case the two ways lead to the same thing. We should not expect this in general. A case in point is C₈ ⋊ C₂, which leads to no less than four different groups depending on how exactly you make the C₂ compatible with the symmetries of the C₈.

To see why, let’s have another look at C₈:

If we switch a to a^k, then consistency implies that we must do the same for all its powers. k can’t be even, because then we would get only even powers. However, if k is any odd number (1, 3, 5, or 7), then the structure of the group is retained and the result is a possible automorphism.

Since this operation happens to be self-inverse, this means that there are three possible semi-direct products C₈ ⋊ C₂ (not counting the trivial direct product). They are all built from an order-8 element a and an order-2 element b.

One of them is the dihedral group D₁₆, and gives the symmetry group of the regular octagon: when an a passes a b, we have ba = a⁷b.

Another is the so-called quasidihedral or semidihedral group of order 16, QD₁₆: we have ba = a³b.

The last is the so-called modular group of order 16, M₁₆: we have ba = a⁵b.

This sort of situation, where we have to specify which semi-direct product we mean, becomes very common for higher orders and is the reason we stop at order 16.

Classifying all the semidirect products we can possibly make is quite a difficult task. It involves finding the automorphism group of a group we’re interested in making the product from, which in general is not easy, especially as the group gets large.

Here is, nevertheless, a list of all the small groups we can in principle create. (We also give the names of the groups we haven’t seen yet.)

The groups D_2n turn out to be part of another infinite series of groups: they are the dihedral groups, the symmetries of a regular n-gon. (You might like to try constructing some of those semi-direct products, although at these orders constructing Cayley tables gets a bit tedious.)

This is, in fact, an almost complete list of the groups of order up to 16 (although we are still a very, very long way from being able to think about proving that).

But we may have noticed that we never got Q₈ this way. Indeed, you cannot get Q₈ as a semi-direct product.

It turns out that the problem is quite subtle: the first group in a semi-direct product always has certain properties within the large group – it is a so-called normal subgroup. The quaternion group has lots of normal subgroups (in fact every subgroup is normal); the problem is that they don’t fit together in the right way to make a semi-direct product! The problem is that there is only one order-2 element. Both factors would have to include it, because they have order a power of two; but the two factors making a semi-direct product have to be disjoint. This is a contradiction.

The three missing groups in the above table are: Q₈, Q₈ x C₂, and a kind of enlarged quaternion group that’s usually called Q₁₆.

We’ve thus hit a roadblock in our attempt to build up large groups from small: it turns out that there are lots and lots of ways to do that, and direct and semi-direct products by themselves only scrape the surface.

This is the beginning of a deep problem in group theory!

You may be aware of how whole numbers can be broken down into a chain of prime factors. For example, 168 breaks down as 2³ x 3 x 7.

We can write that as a chain of building up from factors: 1 → 2 → 4 → 28 → 56 → 168

where the factors we multiplied by each time were 2, 2, 7, 2, and 3.

We could of course have done that in a different order, such as: 1 → 3 → 21 → 42 → 84 → 168

where the route we took was 3, 7, 2, 2, and 2.

It turns out that groups also have a similar structure. The Jordan-Hölder theorem says that groups can be built up as a so-called composition series where each factor is a simple group – which is kind of like a prime number, in that it cannot be broken down to produce a normal subgroup. For example: C₁ ⊲ C₃ ⊲ C₆.

The “factors” being “multiplied” here are C₃ and then C₂. However, unlike for numbers, this doesn’t determine what we actually got: we could have “multiplied” C₃ and then C₂ via a semi-direct product instead, to get: C₁ ⊲ C₃ ⊲ D_6.

So it starts to become clear how the finite groups have to be identified. First, somehow the list of finite simple groups, as building blocks akin to the primes, must be obtained. This required a tremendous effort of tens of thousands of pages of work done between 1955 and 2004, and is one of the great monuments of modern mathematics.

Second, somehow it must be characterised exactly how groups can be “multiplied” (i.e. put together) to form bigger groups, a problem known as the extension problem. This is extremely difficult, and little can be said about the situation in general: but many results are known about extensions that have to satisfy certain conditions.

If we are interested in groups of a particular order, then it is a greatly frustrating fact that Lagrange’s theorem has no general converse. Although the order of an element must divide the order of the group, it is not guaranteed that every divisor of the group’s order must be the order of some element. For example, the order-4 group V₄ has no element of order 4, and the order-12 group A₄ has no element of order 6.

The most powerful known result in this direction are called Sylow’s theorems, and they enable one to prove that we truly have all the groups of many small orders. However, when the order is a power of a prime, like 16 or 32, they unfortunately do not tell us much.

Another interesting question is how we should write down those groups. Above, we used the Cayley tables, but for orders higher than eight they get quite cumbersome.

In one example (the cyclic groups), we circumvented this by presenting the group as a kind of puzzle. Instead of giving the whole Cayley table, we presented just the information that a single element generated the group, and that a power of it was 1: C_n = < a | aⁿ = 1 >.This allows us to present a whole lot of groups: for example, V₄ = < a, b | a² = b² = (ab)² = 1 >.

(You might like to think about why this provides enough information to get the whole group table.)

As another example, the one group we never characterised above with order up to 16 was Q₁₆ = < a, b | a⁸ = 1, b² = a⁴, bab^-1 = a^-1 >.

But in general, figuring out what a presentation really is is quite difficult! A very tricky example is this case:
< a, b, c | aba^-1 = b², bcb^-1 = c², cac^-1 = a² >.

This surely looks like a very complicated group, but actually it is just the one-element group!

And in all of this, we have only covered finite groups! The one glance we had at an infinite group is the integers under addition. But there are many others…

One of the simpler ones is the circle group. This is the rotational symmetry group of a circle. How can we rotate a circle to make it look the same? Well, any angle will do, so this symmetry group contains all angles: it’s an infinite group!

But there are others. Consider an infinite floor tiled by one-meter squares. What are the symmetries of this? Well, for example, you can rotate by multiples of a right angle. But you also have some translational symmetries! If you move everything one meter to the right, left, up, or down, everything still looks the same.

A more exotic one is the infinite dihedral group, which is in some sense the symmetry group of an infinite-sided regular polygon – which is just a number line: the rotational symmetries have become translational symmetries, but you also have reflection symmetries that keep any integer or half-integer in place.

And there is a great deal of interesting mathematics behind the structure of infinite groups as well. Here our journey ends: but there are many wonders to see still!

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If you missed any earlier parts of the series, they can be found at the links below.

Part I

Part II

Part III