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.
Archive for the 'Group Theory' category
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 that I've worked things out enough to get this right, but I'm
not sure. So category-savvy commenters, please let me know if you see any major problems, and I'll do my best to fix them quickly; other folks, be warned that I might have blown some of the details.
Second, I'd like to point you at Wikipedia's page on groupoids as a
reference. That article is quite good. I often look at the articles in Wikipedia and
MathWorld when I'm writing posts, and while wikipedia's articles are rarely bad, they're also
often not particularly good. That is, they cover the material, but often in a
somewhat disorganized, hard-to-follow fashion. In the case of groupoids, I think Wikipedia's
article is the best general explanation of groupoids that I've seen - better than most
textbooks, and better than any other web-source that I've found. So if you're interested in
finding out more than I'm going to write about here, that's a good starting point.
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 make a move in the puzzle by sliding a tile adjacent to the empty
space into the empty. In the puzzle, you scramble up the tiles, and then try to move them back so that they're in numerical order. The puzzle, in its initial configuration, is shown to the right.
If you look at the 15 puzzle in terms of configurations - that is, assignments of the pieces to different positions in the grid - so that each member of the group describes a single tile-move in a configuration, you can see some very clear symmetries. For example, the moves that are possible when the empty is in any corner are equivalent to the moves that are possible when the empty is in any other corner. The possible moves when the space is in any given position are the same except for the labeling of the tiles around them. There's definitely a kind of symmetry there. There are also loops - sequences of moves which end in exactly the same state as the one in which they began. Those are clearly symmetries.
But it's not a group. In a group, the group operation most be total - given any pair of values x and y in the group, it must be possible to combine x and y via x+y. But with the 15 puzzle, there moves that can't be combined with other moves. If x = "move the '3' tile from square 2 to square 6", and y = "move the '7' tile from square 10 to square 11", then there's no meaningful value for "x+y"; the two moves can't be combined.
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 introduction, you can look at the category theory archives.
Like set theory, category theory is another one of those attempts to form a fundamental
abstraction with which you can build essentially any mathematical abstraction. But where sets
treat the idea of grouping things together as the fundamental abstraction, category
theory makes the idea of mappings between things as the fundamental abstraction.
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 the
characteristic 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 add together to get
That sounds confusing - but it really isn't. It's just
hard to write in natural language. A couple of examples will
make it clear.
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 morphisms mean, before the category theoretic view makes any
sense. Once you understand the basic concepts of abstract algebra, category theory
becomes very useful for understanding how things fit together. Many complicated things become clear in terms of category theoretic descriptions and structures - but first, you need to understand what the elements of those structures mean.
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 structure called a ring; or you can add properties - in which the simplest step leads you to something called an abelian group. When it comes to removing, you can remove properties, which leads you to a simpler structure called a groupoid. Eventually, I'm going to follow both the upward and the downward paths. For now, we'll start with the upward path, since it's easier.
Building up from groups, we can progress to rings. A group captures one simple property of
a set of number-like objects. A ring brings us closer to capturing the structure of the system
of numbers. The way that it does this is by adding a second operation. A group has one operation
with symmetric properties; a ring adds a second symmetric operation, with a well-defined relationship
between the two operations.
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.
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.
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.