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...
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Posted on February 7, 2008 10:03 PM • 1 Comments •
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...
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Posted on February 5, 2008 12:17 PM • 23 Comments •
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...
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Posted on January 24, 2008 4:12 PM • 16 Comments •
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,...
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Posted on January 20, 2008 7:18 PM • 16 Comments •
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...
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Posted on January 16, 2008 3:53 PM • 11 Comments •
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...
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Posted on January 11, 2008 1:40 PM • 20 Comments •
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...
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Posted on December 28, 2007 8:10 PM • 14 Comments •
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...
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Posted on December 27, 2007 2:32 PM • 41 Comments •
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...
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Posted on December 13, 2007 5:28 PM • 15 Comments •
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...
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Posted on December 10, 2007 12:12 PM • 9 Comments •
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...
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Posted on December 4, 2007 11:33 AM • 20 Comments •
