Introduction to Groups
Sets
Before reading this page, please read Introduction to Sets. So, hopefully you are familiar with things like this:

Set of clothes: {socks, shoes, pants, ...}

Set of even numbers: {..., 4, 2, 0, 2, 4, ...}

Positive multiples of 3 that are less than 10: {3, 6, 9}


Operations

Now that we have elements of sets it would be nice to do things with them. Specifically, we wish to combine them in some way. This is what an operation is used for.
An operation takes elements of a set, combines them in some way,
and produces another element.
or, more simply:
An operation combines members of a set.

For the clothes above, an operation could be "insert". You can insert the socks into the shoes. You could even insert the shoes into the socks.
And for you artists out there, I can use painting as an example. Let's imagine we have the set of colors {red, green, blue}.
Now we have to define an operation, and one that makes the most sense is mixing. So for example, red mixed with green makes yellow, and red mixed with blue makes purple. 


But saying "red mixed with blue makes purple" is long and annoying. If I have to write a lot, I'm going to want to shorten that up. So I'm going to let "mixed with" be symbolized by + and "makes" be symbolized by =.
So "red mixed with blue makes purple" becomes "red + blue = purple". 
Binary Operations
So far we have been a little bit too general. So we are going to be just a little bit more specific. A binary operation is just like an operation, except that it takes 2 elements, no more, no less, and combines them into one.
You already know a few binary operators, even though you may not know that you know them:
 5 + 3 = 8
 4 × 3 = 12
 10  4 = 6
 5 / 5 = 1
These all take two integers and combine them in different ways to get one number. Notice the last example, 5 / 5 = 1. It still takes two elements, even if they are the exact same elements.
Now above it looks like there are 4 operations. You will learn in a minute that there are really only two!
Well Defined

One thing about operators is that they must be well defined. But reverse that. They must be defined well.
Think about applying those two words, "defined well" to the English language. If a word is defined well, you know exactly what I mean when I say it.

 The word "angry" is defined pretty well, as you know exactly what I mean when I say it.
 But if I said the word, "date", would you know that I was talking about a piece of fruit, rather than the calendar date?
Now let's apply this to math. If I give you two numbers and a well defined operations, you should be able to tell me exactly what the result will be.
For example, there is only one answer to 5 + 3. That is because the operator is well defined.
But there are some things that look like operators which aren't well defined.
For example, square roots. When we write x^{2} = 25, or rather x = ± √(25), there are two answers to this question.
If you tell me the answer is 5, I could just say, "Nope, the answer is 5. You're wrong." Because 5×5 = 25 and (5)×(5) = 25.
With well defined operators, there is only one possible answer.
Now as a final note with operations, many times we will use * to denote an operation. We don't mean multiplication, although we certainly can use it for that. But normally, we just mean "some operation". When we do mean multiplication, it will be clear.
Introduction to Groups
Now that we understand sets and operators, you know the basic building blocks that make up groups. Simply put:
A group is a set combined with an operation
So for example, the set of integers with addition.
But it is a bit more complicated than that. We can't say much if we just know there is a set and an operator. What more could we describe? We need more information about the set and the operator. This is why groups have restrictions placed on them. That is, they have more properties.
Formal Definition of a Group
A group is a set G, combined with an operation *, such that:
 The group contains an identity
 The group contains inverses
 The operation is associative
 The group is closed under the operation.
Let's look at those one at a time:

1. The group contains an identity. If we use the operation on any element and the identity, we will get that element back.
For the integers and addition, the identity is "0". Because 5+0 = 5 and 0+5 = 5

In other words it leaves other elements unchanged when combined with them.
There is only one identity element for every group
The symbol for the identity element is e, or sometimes 0. But you need to start seeing 0 as a symbol rather than a number. 0 is just the symbol for the identity, just in the same way e is. It's defined that way. In fact, many times mathematicians prefer to use 0 rather than e because it is much more natural.
Formal Statement:
There exists an e in the set G, such that a * e = a and e * a = a, for all elements a in G 

2. The group contains inverses. If we have an element of the group, theres another element of the group such that when we use the operator on both of them, we get e, the identity.
For the integers and addition, the inverse of 5 is 5. (because 5 + 5 = 0)

In just the same way, for negative integers, the inverses are positives. 5 + 5 = 0, so the inverse of 5 is 5. In fact, if a is the inverse of b, then it must be that b is the inverse of a.
Inverses are unique. You can't name any other number x, such that 5 + x = 0 besides 5.
Make a note that while there exists only one identity for every single element in the group, each element in the group has a different inverse.
The notation that we use for inverses is a^{1}. So in the above example, a^{1} = b. In the same way, if we are talking about integers and addition, 5^{1} = 5.
Formal Statement:
For all a in G, there exists b in G, such that a * b = e and b * a = e. 

3. Associative. You should have learned about associative way back in basic algebra. All it means is that the order in which we do operations doesn't matter.
a * (b * c) = (a * b) * c

Notice that we still went a...b...c. All that changes was the parentheses. We'll get back to this later ...
Formal Statement:
For all a, b, and c in G, a * (b * c) = (a * b) * c 

4. Closed under the operation. Imagine you are closed inside a huge box. When you are on the inside, you can't get to the outside. In that same way, once you have two elements inside the group, no matter what the elements are, using the operation on them will not get you outside the group

If we have two elements in the group, a and b, it must be the case that a*b is also in the group. This is what we mean by closed. It's called closed because from inside the group, we can't get outside of it.
And as with the earlier properties, the same is true with the integers and addition. If x and y are integers, x + y = z, it must be that z is an integer as well.
Formal Statement:
For all elements a, b in G, a*b is in G 
So, if you have a set and an operation, and you can satisfy every one of those conditions, then you have a Group.
Only Two Operations

Way back near the top, I showed you the four different operators that we use with the numbers we are used to:
+  × /
But in reality, there are only two.


When we subtract numbers, we say "a minus b" because it's short. But what we really mean is "a plus the additive inverse of b".
The minus sign really just means add the additive inverse. But you would want to be crazy saying that over and over again, so we just say "minus".

Can you take a guess at what division is? In the same way, it just means "multiply by the multiplicative inverse". 

So there is really only addition and multiplication!

Example Time!
Whew! Confused? You probably are. This is where examples come in.
Example 1: Addition and {0}
Well this is an odd example. But let's try out the three steps. Let's find the identity element. Well, that should be to hard. If we add 0 to anything else in the group, we would hope to get 0. Since the only other thing in the group is 0, and 0 + 0 = 0, we have found the identity.
Now we need to find inverses. Well, again, we only have one element. So what's the inverse of 0? We want 0 + 0¹ = 0. Well, 0 + 0 = 0, so 0¹ = 0. And 0 is in the group, so 0¹ is also in the group. Since we've tried all the elements, all one of them, we're done.
Associative? a + (b + c) = (b + c) + a? Well, since there is only one element, a = b = c. So 0 + (0 + 0) = (0 + 0) + 0? Of course.
Finally, is it closed? If we take any element a, and any element b, will a + b be in the group? Well, since there is only one element, 0, then a = 0 and b = 0. Is 0 + 0 in the group? You bet it is. So it's closed.
So {0} is a group with repect to addition.
Example 2: Multiplication and {1, 1}
Back to the four steps. First, is there an identity? Well, this is going to be easy, there are only three possibilties. Either:
 1 is the identity,
 1 is the identity,
 or there isn't one.
1*1 = 1 and 1*1 = 1. So it looks like 1 is the identity. Should have expected that.
Now we need to find inverses. If we have a in the group, then we need to be able to find an a¹ such that a * a¹ = 1 (or rather, e). So let's start off with 1.
1 * 1 = 1, so we know that if a = 1, a¹ = 1 as well. Now 1 * 1 = 1. So if a = 1, then a¹ = 1 as well! Since we have found an inverse for every element, we know the group is closed with respect to inverses.
Is it associative? a * (b * c) = (a * b) * c. Well, since we have only 2 numbers, we can try every possibility. And if you really want to, you can. But it should be pretty obvious that it is.
Finally, is it closed? Is 1*1 in the group? Yep. How about 1 * 1? Yep. And 1 * 1? Yep. And finally, 1 * 1? Of course. So it is closed under the operation.
And we're done! {1, 1} is a group under multiplication.
Example 3: Integers and addition
Consider the integers. Can you name the identity element of integers when it comes to addition? We want to find a + e = e + a = a. OK, you know already. 0 is the identity. That is because a + 0 = 0 + a = a, for any integer a.
Sticking with the integers, let's say we have a number a. Can we find it's inverse? That is, does there exist an a¹ such that a + a¹ = a¹ + a = e? For example, 5 + 5¹ = 0? What is 5¹? 5 would be the answer. To a + a = e, for the integers.
If I add two integers together, will the result be an integer? Yes. So it is closed.
Finally, does a + (b + c) = (a + b) + c? It does! And guess what, we just showed that the integers are a group with respect to addition.
Example 4: Integers and multiplication
Let's go through the three steps again. First, we need to find the identity. So we want a * e = e * a = a. 5 * e = 5. What would e be? 1, of course.
Now we need to find out if integers under multiplication have inverses. So if we take a number a, can we find a¹ such that a * a¹ = e? Let's try 5 again. 5 * 5¹ = 1. So what is 5¹? It's 1/5.
But that isn't in the integers! Ahhhh! The integers don't contain multiplicative inverses, so they can't be a group with respect to multiplication.
So we have shown that using one operation, the integers are a group, and under another, they aren't.
Why Groups?
So why do we care about these groups? Well, that's a hard question to answer. Not because there isn't a good one, but because the applications of groups are very advanced.
For example, they are used on your credit cards to make sure the numbers scanned are correct.
They are used by space probes so that if data is misread, it can be corrected. They are even used to tell if polynomials have solutions we can find.
Here is one good reason:
Solving Equations
As it turns out, the special properties of Groups have everything to do with solving equations. If we had a*x = b, where a and b were in a group G, the properties of a group tell us that there is one solution for x, and that this solution is also in G.
a * x = b 
a^{1} * a * x = a^{1} * b 
(a^{1} * a) * x = a^{1} * b 
(e) * x = a^{1} * b 
x = a^{1} * b 
Since it must be that both a^{1} and b are in G, a^{1} * b must be in G as well.
Also, since we know the operator * must be well defined, this must be a unique solution. Otherwise, the operator wouldn't be defined very well, now would it?
Special Types of Groups: Abelian

Before I go on to talk about Abelian, let me point out that it is pronounced abelian. Not abelian. I made that mistake when I was first reading about groups, and I still have yet to break the habit. 
If a * e = a, doesn't that mean that e * a = a?
And similarly, if a * b = e, doesn't that mean that b * a = e?
Well, as a matter of fact, it does. But we are careful here because in general, it is not true that
a * b = b * a. However, when it is true that a * b = b * a for all a and b in the group, then we call that group an abelian group.
That fact is true for integers, and this is why we call the integers with addition an abelian group.
