Need help proving a group is abelian

[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.


Answer:

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?

 

[ArcanaNoir]

perhaps making a cayley table will help.

 

[vince72386]

How do I go about creating a cayley table?

 

[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.

 

[vince72386]

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

 

[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.