# Very Elementary Group Theory Problem

1. Sep 14, 2010

### end_game

Hi. I'm new to Group Theory and wanted to see if I had the right train of thought for this problem.

1. The problem statement, all variables and given/known data

Let S be a set with an associative law of composition and with an identity element. Prove that the subset of S consisting of invertible elements is a group.

2. Relevant equations

A group is a set G together with a law of composition that is associative and has an identity element, and such that every element of G has an inverse.

3. The attempt at a solution

I'll denote my subset of S by G.

I know G has the identity element of S as the identity element is invertible.

The associativity for the law of composition is inherited and the existence of inverses we get for free. All that I'm left with is verifying that G has closure. That is for a,b in G, ab is in G.

It's sufficient to show ab is invertible.

But, $$(ab)(b^{-1}a^{-1})=e$$

where e is the identity element. So $$ab$$ is invertible and thus is an element of G and G is a group.

Last edited: Sep 14, 2010
2. Sep 14, 2010

### end_game

Not sure why my tex code isn't working...

3. Sep 14, 2010

### Staff: Mentor

Use [ tex] and [ /tex] (without the spaces). Don't use [ latex] and [/ latex].

4. Sep 14, 2010

### Hurkyl

Staff Emeritus
I think [itex] works too. The problem is the wrong slash.

5. Sep 14, 2010

### end_game

Thanks. I think I got it now (the tex code that is).

Last edited: Sep 14, 2010