bigideas

OK, for many readers this is probably the painful part of any of our posts about mathematical subjects but it can’t be avoided: the definitions. The spirit of this mini-series is fairly informal, but we need to get a sense of what a group is before we can get anywhere at all.

As the previous instalment illustrated, groups can have many different manifestations. What they have in common isn’t anything obvious; it’s an abstract structure. And the language mathematicians use to describe such structures is the language of sets. We won’t need any real set theory here, but you might like to glance over our primer on sets before continuing.

Sets With Structure

A group is a set plus some “structure”, and it’s really the structure, not the particular set, that makes a group a group. Unless we’re working with a specific example, we’ll assume we have some set — it doesn’t matter what kinds of objects are in it (in pure set theory those objects are always sets, but working informally we often talk about sets of other objects). The “structure” we’ll put on it will be something called a “binary operation”.

Simply put, a binary operation takes two elements of the set and returns a third one. As an example, think of the set of positive whole numbers N = {1, 2, 3, 4, …} (known as the “natural numbers”), and the binary operation of addition: “+” is an operation that takes any two numbers you give it and gives you back a new number, which in this case is the sum of the two you gave it. We write:

5 + 4 = 9

to mean “if you give 5 and 4 to the + operation, you get 9 back”. “+” is an example of a binary operation. Note that strictly speaking + is a map or function from N² to N, but that’s a detail you can safely ignore.

Properties of Binary Operations

The set N with the operation + can be written <N, +>. We’ll define a group in the standard way shortly in terms of properties that an object like this — a set-with-structure — must have in order to count as a group. We’ll begin by looking at what each of these properties means.

<N, +> exhibits a property called closure, which an operation exhibits if the result of the operation on any two elements of the set is again an element of the set. For any x and y in N, x+y will always be an element of N too. We say that the set N is “closed under addition”.

Not all operations on N are closed. Subtraction, for instance, isn’t; 5-2 is an element of N, but 5-7 isn’t (because -2 isn’t an element of N).

It’s properties like these that define the kind of structure that + is when it’s applied to N. Another one is the existence of an identity element. Consider the set of “whole numbers” W = {0, 1, 2, 3, … }, which is the set of natural numbers plus zero. Take this set and, again, structure it using the addition operation. Now zero behaves differently from any of the other whole numbers, since

0 + x = x

for all whole numbers x. We call zero an “identity element” because of this behaviour. Notice that <N, +> doesn’t have an identity element, but <W, +> does. The existence of an identity element is an algebraic property like closure that’s exhibited by some structured sets and not others.

Now let’s expand our set further to include the negative numbers; this gives us the set known as the integers, Z= {… -3, -2, -1, 0, 1, 2, 3, … }. This set is again closed under addition and has an identity element for that operation, which is zero again. But now we have another feature that N and W both lacked. For every integer x there’s another integer y so that

x + y = 0

If x=5 the y, of course, is -5, since

5 + (-5) = 0

We call y the “inverse” of x if, when the operation combines the two, the result is the identity. In the case of <Z, +> every element has an inverse. Notice that no element of N has an inverse under the addition operation, and in W only zero has an inverse.

Here’s a final, simple example: notice that for any two elements x and y in Z, adding x to y is just the same as adding y to x:

x + y = y + x

If you can reverse the order like this and always get the same result, we say the operation is commutative. Before you dismiss this as an obvious property that operations should have, notice that if our operation were subtraction then you couldn’t just reverse the order:

5 – 3 ≠ 3 – 5

In fact, many important operations, including some we’ll see later, are non-commutative.

Associativity

We need one more property before we can give our formal definition, and it’s a tiny bit less obvious than the ones we’ve just seen. That property tells us what happens when we apply the operation several times in the same expression. In <Z, +>, for example, we can write

(5 + 3) + 2

which means “add five to 3, take the result and add two to it”. Calculating the answer, we work out that 5+3=8, and then that 8+2=10.

But you also know, I hope, that

(5 + 3) + 2 = 5 + (3 + 2)

That is, you can work out 3+2=5 first and add 5 to that to get the same result. In fact, you can always do this; in <Z, +> it’s always true that

(x + y) + z = x + (y + z)

and that means that we can do without the brackets and just write

x + y + z

leaving you to calculate the sum in whichever way seems best. This property is called associativity.

Like commutativity, associativity may look like an obvious property if the only example you have to work with is <Z, +>. To demonstrate that not all structures on sets are associative, consider <Z, ->, which is the integers equipped with the usual subtraction operation. It’s easy to see that

(5 – 3) – 2 = 2 – 2 = 0

but

5 – (3 – 2) = 5 – (-1) = 6

so ordinary subtraction is not an associative operation. Note that subtraction has closure, an identity element (zero again) and inverses. Exercise: What’s the inverse of a non-zero element of <Z, ->?

Formal Definition

Those are all the terms we need, so here’s the formal definition — we’ll use a little circle to represent the operation for now as we don’t want to be tied down to our example of addition:

Definition: A group is a set S equipped with a binary operator º with the following properties:

1. Closure: xºy is an element of S
2. Identity: There is an element of S, denoted e, such that eºx = x and xºe = x for every x in S
3. Inverses: For every x in S, there is another x, denoted x^-1, such that xºx^-1 = e and x^-1ºx = e
4. Associativity: For any x, y and z in S, (x º y) º z = x º (y º z).

Doesn’t sound like much, does it? But the theory of things that obey these rules — group theory — is a deep and beautiful field of pure mathematics with many, many applications.

You may have noticed that we don’t stipulate that a group has to be commutative. In fact some groups are and some aren’t; the ones that are are named after Niels Abel:

Definition: A group is called Abelian if it has the following property:

1. Commutativity: For all choices of x and y in S, x º y = y º x

We’ll have more to say about Abelian groups later, but for now the main definition is the one we’re interested in, and we won’t make any assumptions about whether the group is Abelian or not.

Don’t worry if these definitions aren’t yet 100% clear to you — in the next instalment we’ll take some examples and prove they’re groups, providing some practice with them as well as illustrating how versatile they are.