Group Theory

I know I said I was going to talk about induction today, and I apologize for those of you that we're psyched to learn about it, but I'm going to talk about something else. I promise that I will explain induction because it is very important, but I'm going to wait until I actually need it for something to talk about it.

Instead, I'm going to start talking about group theory. In my last post I talked a lot about group theory and group theory is in my subject reference and a few of my suggested readings are about group theory, so today you're going to learn what group theory is.

Group theory is, unsurprisingly, the study of groups, so I should first define a group.

Definition: Group
Let G be any set and let ∙ be a binary operation defined on the elements of G. Then G is a group if it satisfies the following four axioms:

Closure

For every elements, a and b, in G, a∙b is also in G.

Associativity

For every elements, a, b, and c, in G, (a∙b)∙c = a∙(b∙c).

Identity

There exists some element, e, in G such that for every element, a, in G, a∙e = a = e∙a.

Inverses

For every element, a, in G there exists some element, b, in G such that a∙b = e = b∙a. We call b the inverse of a and I will typically denote it by a^-1.

So, that probably requires some explanation. To make a group, we need to start with a set, G. The set can be a set of literally anything - numbers, functions, symmetries, polygons, paths, or anything else you can think of. Then you need an "operation." The operation can also be anything - it just has to be some way to combine two of the elements in the set. If the elements of the set are numbers, maybe the operation is multiplication or addition. If the elements are functions, maybe the operation is composition. An operation just has to take two of the elements of the set and combine them in some way. Then, once you have a set and an operation, the operation has to follow these four rules:

Closure

Any time you combine two elements of the set, you have to get an element of the set. We call this closure because it means that no matter what you do, you cannot use the operation to leave the set. And its a pretty logical requirement - if you've got a set of numbers, when you combine them you want to get a number back, not a function or a polygon.

Associativity

Associativity just means that if we have a list of elements of the set, a, b, and c, it doesn't matter if we do b∙c first and then a∙(b∙c) or if we do (a∙b) first and then (a∙b)∙c. Associativity is a boring condition and we never use it to do anything particularly cool, but it is necessary sometimes, and it is kind of an obvious thing. However, something very important must be said about what associativity does not mean. It does not mean that a∙b = b∙a. If you have a list of elements that you're going to combine, the order in which you do the operations does not matter but the order that the elements are in is still important. Sometimes it is true that a∙b = b∙a for each a and b in G, but this condition is called commutivity and is not related to associativity.

Identity

The identity condition states that there is some element - we'll call it e - in G such that combining e with anything in G doesn't actually do anything. Just like how anything times one is itself and anything plus zero is itself, the identity element of the group does not change whatever you combine it with.

Inverses

The existence of inverses is a pretty important condition. It says that for each element of the set, you can get back to the identity. As examples, with addition the inverse of 5 is -5 and with multiplication the inverse of 12 is 1/12. Notice, too, that its important that if a^-1 is the inverse of a, then a is the inverse of a^-1 because a∙a^-1 = e = a^-1∙a.

You should notice that this definition is pretty simple. It only has 4 total conditions and the first two are natural and rather intuitive. This is the power of group theory - we will see that there are a lot of interesting facts that can be developed about groups and yet there are a lot of things that satisfy the definition of being a group, thus giving a lot of very specific results about a wide range of mathematical ideas.

I expect there to be some questions about this because it is pretty confusing if you've never seen groups before, so ask away. And in my next post I plan on giving some concrete examples of groups.

References

Text Reference: Gallain Chapter 2

Wikipedia: Group

Wikipedia: Group Theory