bigideas

Now we have the formal definition of a group, we’ll look at some first examples and reveal their group structure. This will offer good practice in using the definition, so ideally you should read each example and try to prove for yourself that it has the required properties before moving on.We’ll insert a ¶ symbol at suitable points.

Ordinary Arithmetic

You probably remember doing arithmetic in school; you know, adding up, multiplication and so on. Perhaps it was a bit like this:

Let’s begin with some very ordinary groups that you’re certainly familiar with. In the previous instalment we met <Z, +>, and looked at its properties. This is a group. ¶ We showed it was closed, it has an identity element (zero) and inverses (for every x, the inverse of x is -x, remembering that -(-2) is 2) and it’s associative. We also met <Z, -> and observed that it’s not a group ¶; although it’s closed, has an identity and has inverses for all elements, it’s not associative.

Now consider the set <Z, ×> of integers under multiplication. This isn’t a group ¶ because although it is closed an has an identity element (1, this time), not all elements have inverses. That is, there’s no x in Z that makes this equation true:

4 × x = 1

Put another way, 4 has no inverse in Z.

Well, we all know the “proper” solution to this equation — x is a quarter, since four times a quarter is one (there are four quarters in a “whole”). But a quarter isn’t an integer. Let’s change our underlying set to Q^*, the set of all non-zero positive and negative fractions. Now, <Q^*, ×> is certainly a group ¶. Every element of Q^* can be written as

a/b

where a and b are non-zero integers. The inverse of a/b is just b/a, since

a/b × b/a = ab/ba = 1

which is just what we need. You can check associativity for yourself. Notice that <Q^*, ÷> isn’t a group ¶, because as with subtraction we fall at the hurdle of associativity. Q is the set of fractions including zero, and notice that <Q, ×> isn’t a group ¶ because noting multiplied by zero is 1; zero has no inverse (remember, zero isn’t the identity element if your operation is multiplication!). You can probably see that not only are <Z, +> and <Q^*, ×> groups, they’re also Abelian. <Q, +> is an Abelian group too (you need the zero this time — why?).

Next consider the set nZ of integer multiples of some whole number n — so 3Z, for instance, looks like this: {… -12, -9, -6, -3, 0, 3, 6, 9, 12, … }. It should be clear to you that <nZ, +> is a group for any n in W. If n is zero we get the set 0Z = {0}, which isn’t very interesting. In fact we give this group a special name:

Definition: The trivial group is the group whose underlying set consists of only the identity element, {e}.

We’ll have more to say about the trivial group and <nZ, +> in a later instalment.

Modular Arithmetic

So the everyday arithmetic operations of addition and multiplication both have the same structure, inasmuch as they’re groups. We’ll now create some interesting new arithmetic operations that are similar to these and have many applications. They also give rise to some new groups.

We first make a definition, since it’s an extremely important one and what follows gives us the chance to use it:

Definition: The order or a group is the size of its underlying set.

Without getting too funky, it should be clear enough to you that <Z, +> and <Q, ×> are both of infinite order. In this section we’ll construct some groups of finite order (or, in the parlance, “finite groups”).

Let’s go back to clocks; specifically, adding hours to the time. There are 24 hours in a day, so let’s work with the set

Z₂₄ = {0, 1, 2, 3, 4, 5, 6, ,7, 8, 9, 10, 11, 12 ,13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}

The set Z_n means “the first n integers, including zero”, and it has order n. These can represent the hour part of the display on your computer’s clock, but they also represent a certain number of hours after midnight. We can imagine adding them together, either in a straightforward way:

12 + 3 = 15

or with “overlapping”, where we go past “midnight”:

12 + 15 = 3

Why? Because 12 + 15=27, which is 24 + 3. This operation — with which you’re already familiar — is called “addition modulo 24″, and the rule is quite simple:

Add the two numbers together, then subtract 24 from the result until you get a number that’s less than 24.

So 13 modulo 24 is 13, 26 modulo 24 is 2 because 26 – 24 leaves 2. You subtract 24, of course, because adding 24 hours just leaves you back where you started in terms of clock time; you’ve just advanced another day. We’ll write “addition modulo 24″ as “+₂₄“.

Is <Z₂₄, +₂₄> a group? ¶ We must have closure, since the result of subtracting 24 repeatedly must lead to a whole number between zero and 23 inclusive, which is exactly what Z₂₄ contains. We have an identity element, zero, since just as in ordinary arithmetic adding zero to anything leaves it unchanged. We have inverses, although they’re a bit harder to spot:

x^-1 = 24 – x

Why? Because then

x + x^-1 = x + 24 – x = 24 = 0 modulo 24

Remember: keep subtracting 24 until you have a number smaller than (not equal to!) 24. Associativity follows from the associativity of <Z, +>. We haven’t used the specific fact that we’re working modulo 24 here — it could be any number. So in fact we have a stronger result: <Z_n, +_n> is a group for any n > 0.

The family of groups formed by modular arithmetic are known as the cyclic groups for reasons that will hopefully become apparent, and will be of great importance to us. Musically-inclined readers may like to know that if you number the pitch classes C, C# and so on from 0 to 11 you get the group <Z₁₂, +₁₂>, where addition can be thought of as transposition. This group is the foundation of pitch class set theory, which is a rich and rather beautiful field of musicology.

So far we’ve looked at examples that are to do with addition and multiplication — kids’ stuff, in other words, although the structures we’ve created will be of importance to us later. I also mentioned symmetry among our initial examples, and even that group theory might be considered the study of symmetries. It must be pretty unclear how that can be true given the kinds of groups we’ve been looking at; we’ll try to fix that in the next instalment.