Introduction

In this article we’re going to talk about groups. A solid understanding of what a group is, what it can represent, and what it enables you to do is fundamental to the study of more advanced mathematics that uses groups to describe their components. In fact, some of the higher tiers of mathematics does away with numbers altogether and work is done with groups of numbers rather than particular representatives like the number 1, for example; it’s like saying that cars (as a group) have four wheels instead of saying a Toyota has four wheels, and a Ford has four wheels, and a BMW has four wheels, etc… We just say the group of all cars have four wheels and that makes analysis of cars much easier to deal with because we don’t have to explain ourselves for each type.

As we will see, a group actually a couple of fundamental mathematical objects lumped together. Rarely are fundamental objects studied on their own, they’re almost always much more interesting and useful when studied together, a group is no exception.

A group is made up of a set X together with a binary operation \cdot that provides the instructions for combining together the elements from the set X. For example, a set on its own, say {1,2,3} does not tell you how to combine 1 and 2 together. The answer could be 3 , as in addition, but it could just as well be 2, as in multiplication, or even \frac{1}{2}. A group is specific as to which operation is being used. Thus, \mathbb{Z} is not a group (it is a set – the set of all integers) but (\mathbb{Z},+) is a group and it tells us to combine elements by adding them. However, (\{1,2,3\},+) is not a group because it fails the closure axiom, i.e. 2 + 3 = 5 \notin \{1,2,3\}. As we’ll see the law of combining elements must satisfy Closure, Associativity, iNverse, and Identity (note the use of the mnemonic “CANI“).

The Binary Operation

We start our foray in to groups with the familiar set of integers \dots,-2,-1,0,+1,+2,\dots , we denote them by the symbol \mathbb{Z} , and it contains all the numbers from negative infinity to positive infinity. Alone, the set of integers is not enough to form a group, it is merely a set. To turn the integers in to a group we need a) three rules of membership in to the group – which will be explained shortly, and b) a way to combine two objects into one, or in other words, we need a binary operator and it must satisfy CANI. Binary operators are like little machines that eat two things and spit out one thing. They are not overly complicated devices and without much thought it should be apparent that the mere act of addition is itself a binary operator; so is multiplication and a host of other elementary operations that you’ve probably come across. When you perform addition you take two numbers, say 1 and 2 and produce one number, the number 1+2 , or simply 3 (note that the “symbol” 1+2 represents the same element as the “symbol” 3 – a trick that will be used often in maths). It should be noted that the number three (as being one plus two) is also an integer; we would have problems if it weren’t because closure would be violated. In fact, most non-infinite sets of integers fail closure under addition because you can just keep adding positive integers together to get larger and larger positive integers.

More precisely, a binary operation, f on a set G is a map which sends the Cartesian product of two elements, G\times G (this is not multiplication, it is an ordered pair!), to G ; or in symbols:f\,:\,G\times G \rightarrow G . So in our example the Cartesian product of 1 and 2 is just the ordered pair (1,2) , and this pair gets mapped to the single element 3 \in \mathbb{Z} . Furthermore, since the result of performing the operation on a pair of objects from a set is again an object in the same set we say that the operation is a closed binary operation.

You will see many examples of binary notation written with different symbols. The binary operation that looks like this; a+b is almost always the addition of two numbers in the usual sense. When you perform the addition binary operator multiple times: a+a+\underbrace{\cdots}_{n\,\mbox{times}}+a , you can derive another common binary operation called multiplication, and it’s written like this an ; the symbol is often omitted due to potential confusion with the Cartesian product. Other examples of binary operators include the composition operator, written as a \circ b ; the exponential operator: a^b f; the dot product: a\cdot b , the list goes on…

The Membership Rules

It turns out that the binary operator that is addition, whose symbol will be denoted by + , is a good candidate to put with the integers, \mathbb{Z} , to form a group. To see why this is the case let’s start by picking three representatives from the set of integers, say 1 , -2 , and 3 . What happens when we add the first two together and then the third:

((-1)+2)+3 = 4

We get the number 4 which is valid because it is integer as well, i.e. 4 \in \mathbb{Z}. Now what happens if we add the second and third together and then the first?

-1+(2+3) = 4

Again, we get 4 . This is important because the integers with addition is associative, in other words it doesn’t matter in which order we add integers together we get the same integer.

The next question we must ask of our integers and addition is this: Is there an identity? An identity is some object of a set that does absolutely nothing when you combine it with some other object using the binary operator. Can you think of an integer that is like that? An integer which does absolutely nothing to all other integers when you add it to them? How about 0 ? Adding zero to any integer does nothing so we say that the integer 0 is the identity of the integers under addition.

The third and final question we must ask is this: For every integer, does there exist another integer that when you add them together you get the identity (which in this case is zero)? Since this is a question regarding every integer we must pick a general integer using algebra notation, and call it a \in \mathbb{Z} . Then, is there another element a' , also in the integers such that a+a' = 0 ? In other words how do we get zero by adding a and a' together? Well, since the integers include all the negative numbers why not just pick -a for our a' ? Then we really have a+(-a) = 0 and this is a true statement for every integer. We call “-a ” the inverse integer under addition.

To summarise, we have four conditions to impose on our set and binary operation:

  • Is the set Closed under the binary operation?
  • After picking any three objects from our set, we require the combination of them with the binary operation to be Associative.
  • For every object in our set we can always combine it with some second object using the binary operation and get the identity. That second element is called the iNverse of the first object.
  • There exists a unique object from our set that, along with the binary operation, does absolutely nothing – and this object is called the Identity.

Our First Group

When we declare that we have a group we write its components in parentheses like this: (\mathbb{Z},+) . This indicates that a group is always defined by two things: 1) a set of objects, and 2) a way of combining two of the objects to a single object that is also in the set.

We can be even more general and abstract than this. We can choose any set and call it G , and any binary operation and call it \star and then if associativity is closed under the operation, there exists an identity, and if for every element of G there exists an inverse, then (G,\star ) is called a group.