Prove that a group of order 15 is abelian, and in fact cyclic.

Let be a group of order 15.

By Cauchy, such that period of .

By Cauchy, such that period of .

Now this is where I need help.

I think I want to say that: Since is a group, . Then somehow show that the element has period 15 thus making it a cyclic group that generates .

Am I going about this wrong? If not, what is my next step?

Thanks in advance,