Introduction to Groups
Sets
Before reading this page, please read Introduction to Sets, so you are familiar with things like this:
- Set of clothes: {hat, shirt, jacket, 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 is 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.
And for you digital artists out there, I can use color as an example. Let's imagine our set is all triples of numbers (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'll now be 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
- 4 − 4 = 0
These all take two numbers and combine them in different ways to get one number. Notice the last example, 4 - 4 = 0. It still takes two elements, even if they are the exact same elements.
Also note: we often don't use "division" as a binary operation on a set, because it may not stay inside the set (closure can fail) and division by 0 isn't allowed. For example, in the integers, 1 ÷ 2 isn't an integer.
Now above it looks like there are 3 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, in other words defined well.
Think about those two words "defined well" in English. 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 say the word "date", is it a piece of fruit, or a calendar date?
Now let's apply this idea to operations. If I give you two numbers and a well-defined operation, you should be able to tell me exactly what the result is.
Example: there's only one answer to 5 + 3. That's because the operator is well defined.
But there are some things that look like operators which aren't well defined.
Example: x2 = 25
If I ask "what's x?", you could answer 5 or −5. Both are correct because
- 5×5 = 25, and
- (−5)×(−5) = 25
If you tell me the answer is 5, I could just say, "Nope, the answer is −5. You're wrong."
If we pick the principal square root, √25 = 5, then the operation is well-defined.
With well defined operators, there's only one possible answer.
Finally, when we use * to denote an operation, we don't always mean multiplication, it just means "some operation." If we mean multiplication specifically, we'll say so.
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's 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's, they have more properties.
Formal Definition of a Group
A group is a set G, combined with an operation *, such that:
- The group is closed under the operation
- The operation is associative
- The group contains an identity element
- The group contains inverses for every element
Let's look at those one at a time:
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 won't get you outside the group.
Read more about closure.
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.
For example, look at the integers and addition. If x and y are integers, and x + y = z, it must be that z is an integer as well.
For all elements a, b in G, a*b is in G
Associative. You should have learned about associative laws way back in basic algebra. All it means is that the order in which we do certain operations doesn't matter.
a * (b * c) = (a * b) * c
Notice that we still went a...b...c. All that changed was the parentheses. We'll get back to this later ...
For all a, b, and c in G, a * (b * c) = (a * b) * c
The group contains an identity. If we use the operation on any element and the identity, we'll 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's only one identity element for the entire group.
The symbol for the identity element is usually e, or sometimes 0.
There exists an e in the set G, such that a * e = a and e * a = a, for all elements a in G
The group contains inverses. If we have an element of the group, there's 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 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's only one shared identity element for the whole group, each individual element has its own unique inverse.
The generic notation that we use for an inverse is a-1. So in the above example, a-1 = b. (Note: When talking specifically about integers and addition, we usually write this inverse as −5 instead of 5-1 to avoid confusion with fractions).
For all a in G, there exists b in G, such that a * b = e and b * a = e.
So, if we have a set and an operation, and we can satisfy every one of those conditions, then we have a Group.
Only Two Operations
Way back near the top, we looked at four familiar operators:
In basic number arithmetic, these can be simplified!
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 it is 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 basic arithmetic really just comes down to addition and multiplication!
While number arithmetic simplifies to addition and multiplication, group operations can be anything!
We can have groups based on rotating a square or shuffling cards or reflecting shapes or combining colors or ... anything really.
An operation is just any rule for combining two elements to get a single result.
Examples!
Whew! Confused? You probably are. This is where examples come in.
Example: 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 shouldn't be too hard. If we add 0 to anything else in the group, we 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 + 0-1 = 0. Well, 0 + 0 = 0, so 0-1 = 0. And 0 is in the group, so 0-1 is also in the group. Since we've tried all the elements, all one of them, we're done.
Associative? We need to check: a + (b + c) = (a + b) + c. Since there's only one element, a = b = c = 0, so 0 + (0 + 0) = (0 + 0) + 0. True.
Finally, is it closed? If we take any element a, and any element b, will a + b be in the group? Well, since there's 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 respect to addition.
Example: 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 possibilities. 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 a-1 such that a * a-1 = 1 (or rather, e). So let's start off with 1.
1 * 1 = 1, so we know that if a = 1, a-1 = 1 as well. Now -1 * -1 = 1. So if a = -1, then a-1 = -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: 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's 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's, does there exist an a-1 such that a + a-1 = a-1 + a = e? For example, 5 + 5-1 = 0? What's 5-1? −5 is 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: 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's e? 1, of course.
Now we need to discover if integers under multiplication have inverses. So if we take a number a, can we find a-1 such that a * a-1 = e? Let's try 5 again. 5 * 5-1 = 1. So what's 5-1? 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's one good reason:
Solving Equations
In a group, we can solve equations like a * x = b because every element has an inverse.
If we multiply both sides on the left by a-1, associativity lets us regroup the terms:
a-1 * (a * x) = a-1 * b
(a-1 * a) * x = a-1 * b
e * x = a-1 * b
x = a-1 * b
The solution is unique because we can "cancel" using inverses: if a * x = a * y, multiplying both sides by a-1 shows that x = y.
Special Types of Groups: Abelian
Before I go on to talk about Abelian, let me point out that it is pronounced a-be-lian. Not abe-lian. I made that mistake when I was first reading about groups, and I still have yet to break the habit.
It is named after the mathematician Niels Henrik Abel.
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 isn't true that
a * b = b * a. But 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.