# Group Theory

This post started out as a response to a question in the comments of my last post on groupoids. Answering those questions, and thinking more about the answers while sitting on the train during my commute, I realized that I left out some important things that were clear to me from thinking about this stuff as I did the research to write the article, but which I never made clear in my explanations. I'll try to remedy that with this post.
So - in the last post I explained a bit about the categorical viewpoint on why
we should care about groupoids. Every groupoid contains groups. The groups…

In my introduction to groupoids, I mentioned that if you have a groupoid, you can find
groups within it. Given a groupoid in categorical form, if you take any object in the
groupoid, and collect up the paths through morphisms from that object back to itself, then
that collection will form a group. Today, I'm going to explore a bit more of the relationship
between groupoids and groups.
Before I get into it, I'd like to do two things. First, a mea culpa: this stuff is out on the edge of what I really understand. My category-theory-foo isn't great, and I'm definitely
on thin ice here. I think…

Today's entry is short, but sweet. I wanted to write something longer, but I'm very busy at work, so this is what you get. I think it's worth posting despite its brevity.
When we look at groups, one of the problems that we can notice is that there are things
that seem to be symmetric, but which don't work as groups. What that means is that despite the
claim that group theory defines symmetry, that's not really entirely true. My favorite example of this is the fifteen puzzle.
The fifteen puzzle is a four-by-four grid filled with 15 tiles, numbered from 1 to 15, and one empty space. You can…

So far, I've spent some time talking about groups and what they mean. I've also given a
brief look at the structures that can be built by adding properties and operations to groups -
specifically rings and fields.
Now, I'm going to start over, looking at things using category theory. Today, I'll start
with a very quick refresher on category theory, and then I'll give you a category theoretic
presentation of group theory. I did a whole series of articles about category theory right after I moved GM/BM to ScienceBlogs; if you want to read more about category theory than this brief…

When we start looking at fields, there are a collection
of properties that are interesting. The simplest one - and
the one which explains the property of the nimbers that
makes them so strange - is called thecharacteristic of the field. (In fact, the
characteristic isn't just defined for fields - it's defined
for rings as well.)
Given a field F, where 0F is the additive
identity, and 1F is the multiplicative identity,
the characteristic of the field is 0 if and only if no
sequence of adding 1F to itself will ever result
in 0F; otherwise, the characteristic is the
number of 1Fs you need to…

When I learned abstract algebra, we very nearly skipped over rings. Basically, we
spent a ton of time talking about groups; then we talked about rings pretty much as a
stepping stone to fields. Since then, I've learned more about rings, in the context of
category theory. I'm going to follow the order in which I learned things, and move on
to fields. From fields, I'll jump back a bit into some category theory, and look at
the category theoretic views of the structures of abstract algebra.
My reasoning is that I find that you need to acquire some understanding of what
the basic objects and…

If you're looking at groups, you're looking at an abstraction of the idea of numbers, to try to reduce it to minimal properties. As I've already explained, a group is a set of values with one operation, and which satisfies several simple properties. From that simple structure comes the
basic mathematical concept of symmetry.
Once you understand some of the basics of groups and symmetry, you can move in two directions. You can ask "What happens if I add something?"; or you can ask "What happens if I remove something?".
You can either add operations - which can lead you to a two-operation…

After that nasty diversion into economics and politics, we now return to your
regularly scheduled math blogging. And what a relief! In celebration, today I'll give
you something short, sweet, and beautiful: quotient groups. To me, this is a shining
example of the beauty of abstract algebra. We've abstracted away from numbers to these
crazy group things, and one reward is that we can see what division really means. It's
more than just a simple bit of arithmetic: division is a way of describing a fundamental
structural relationship that pervades mathematics.
So what is division all about?…

In my last post on group theory, I screwed up a bit in presenting an example. The example was using a pentagram as an illustration of something called a permutation group. Of course, in
my attempt to simplify it so that I wouldn't need to spend a lot of time explaining it, I messed up. Today I'll remedy that, by explaining what permutation groups - and their more important cousins, the symmetry groups are, and then using that to describe what a group action is, and how the group-theory definition of symmetry can be applied to things that aren't groups.
As I alluded to in the last post,…

In the last post, I talked about what symmetry means. A symmetry is an immunity to some kind of transformation. But I left the idea of transformation informal and intuitive. In this post, I'm going
to move towards formalizing it.
The group theoretic notion of immunity to transformation is defined in terms of group isomorphisms. A group isomorphism is a structure preserving mapping between two different groups. If you know category theory, it's defined very easily: a group isomorphism is an iso arrow in the category of groups. Of course, that's a bit of a hand-wave, because I haven't…

As I said in the last post, in group theory, you strip things down to a simple collection of values and one operation, with four required properties. The result is a simple structure, which completely captures the concept of symmetry. But mathematically, what is symmetry? And how can something as simple and abstract as a group have anything to do with it?
Let's look at a simple, familiar example: integers and addition. What does symmetry mean in terms of the set of integers and the addition operation?
Suppose I were to invent a strange way of writing integers. You know nothing about how I…