I'm trying to understand why how you can determine whether two groups of the form $\mathbb{Z}_n$ are isomorphic to each other. More specifically why is $\mathbb{Z}_2 \oplus \mathbb{Z}_3 \cong \mathbb{Z}_6$ but $\mathbb{Z}_3 \oplus \mathbb{Z}_5 \ncong \mathbb{Z}_{15}$?
Is there some method that I'm missing of determining that two groups are isomorphic?