Group of order 15 is abelian How do I prove that a group of order 15 is abelian?
Is there any general strategy to prove that a group of particular order (composite order) is abelian?

There's a unique normal Sylow subgroup of order $5$, because it's index is the smallest prime dividing the group's order.
It follows, after a couple details, that we have a semi-direct product $$G=\Bbb Z_5\rtimes \Bbb Z_3$$, which can only be trivial, because the orders of $\Bbb Z_5^×$ and $\Bbb Z_3$ are relatively prime.
