Need help proving a group is abelian

1. Oct 13, 2011

vince72386

I have a midterm tomorrow morning and I am completely lost on how to finish the problem, I was told a question tomorrow will mirror this one so any help is appreciated.

Question:

Prove any group of order 9 is abelian.

Let G be a group such that |G|=9

One of these elements has to be the identity.

The remaining 8 will consist of 4 elements and their respective inverses.

Where do I go from here?

2. Oct 13, 2011

ArcanaNoir

perhaps making a cayley table will help.

3. Oct 13, 2011

vince72386

How do I go about creating a cayley table?

4. Oct 13, 2011

ArcanaNoir

You put all the elements, presumable a, b, c, d, e, f, g, h, i in a table, across and down (like a multiplication table) and then fill in a*a=? a*b=?

But for nine elements this may not be the best way to approach this problem.

5. Oct 13, 2011

vince72386

What is another way of approaching it without constructing the tables?

6. Oct 13, 2011

Dick

If |G|=9, then if G has an element of order 9, then it's a cyclic group and it's abelian. Problem solved. If not then all nonidentity elements of G must have order 3, right? Start from there.