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

Let $\displaystyle G$ be a group of order 15.

By Cauchy, $\displaystyle \exists a \in G$ such that period of $\displaystyle a = 3$.

By Cauchy, $\displaystyle \exists b \in G$ such that period of $\displaystyle b = 5$.

Now this is where I need help.

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

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

Thanks in advance,