Explain an Orbit: Simple Definition & Example

[hottytoddy]

I'm trying to study a proof for a quiz right now, and even though I have the answer, I don't get it because I don't understand what an orbit is. I got no definition in class, and can't find anything in the book. I looked on wikipedia, but that was completely over my head.

Can someone please explain an orbit as simply as possible, and perhaps with an example?

Thank you!

 

[maze]

http://img98.imageshack.us/img98/6195/orbit2oj8.gif

hmm looks like the image code is broken

 

[ravioli]

This sentence from Wikipedia basically sums up an what an orbit is as briefly as possible...

The orbit of a point x in X is the set of elements of X to which x can be moved by the elements of G.

So a few questions for you. What's a G-set? What's an action? Where is all of this coming from?

I'm curious, what book are you using for your class? I've used Introduction to Abstract Algebra by Keith Nicholson, and it provides a pretty decent discussion on this material. Also, what proof are you looking at right now?

 

[hottytoddy]

Somehow, everyone in my class gets this, but me. I don't know how I missed it, but I've never learned this stuff... I'm past the proof, but now I have another problem and just don't understand what it means.

The book we're using is the third edition of Abstract Algebra, I.N. Herstein. The only references in the book, according to the index, are problems.

So a few questions for you. What's a G-set? What's an action? Where is all of this coming from?

The wiki sentence: Is it saying that if X is {1,2,3,4,5} then, for some G, O(x) for x=1 could be {2,4}? I don't know what a G-set is, and I can only assume an action refers to a function?

 

[ravioli]

Well the orbit of x will include x itself, but I think you're starting to get the idea. Basically the orbit is all of the possible values that you can achieve when operating the element x with all of the elements in G, where G is a group. The action is the operation that's used, so yeah basically a function.

Another resource that you can look at is Wolfram MathWorld. They usually provide defintions, with a few extra information about the topic.