Group Theory!

I think I'm beginning to grasp where group theory is coming from. Have been spending today looking through the notes. Appreciated the fact that...

A Group is a Set and a Binary Operation such that you can arrive at every element in the set with the binary operation, and the result of applying a binary operation to two elements in the set would be another element in the set.

It's like a self-sufficient universe with one law, which is the binary operation. And the elements in the set are the states of the universe, which interact according to the law to give another state in the universe.

Groups by answers.com

Then we moved on to the study of finite groups, refering to groups of finite sets. Like sets with one, two, three, ten, etc... elements.

Specifically, we moved on to the study of functions on a finite set, treating all the functions as a set with the binary operation being the composition law...

And in order for the inverse of a function to exist, it must be a bijection. Such functions are called invertible functions.

So we are dealing with a finite set of functions, acting on a finite set of objects. The set of functions being a group with respect to composition. That's where most of my confusion lies... We're dealing with so many different sets here!

So now you have a function which is a bijection of a set onto itself, basically can be seen as the reordering of the elements of the set, and as Dr Korner puts it, it's like shuffling a deck of cards. Never really understood the meaning until now... Such a bijective function is called a permutation. Cos it is a rearrangement of the elements of a set.

The set of all possible functions on a finite set forms a symmetric group. It is a group because you can arrive at a certain state, i.e. configuration of cards, by many means. The word symmetric is just a name, and hardly has any meaning, at least to me.

Then in this symmetric group , there are ideas like order of a function, and even and odd functions, something which I am just beginning to comprehend. And two permutation functions are said to be disjoint if they act on different elements in the set.

Ok, just a random blabering of my understanding of group theory at the moment. To organise my thoughts. Could serve to give people a glimpse into what I'm doing.