Why isn't $\mathbb{Z}_2 \times \mathbb{Z}_{30}$ isomorphic to $\mathbb{Z}_{60}$? I know that other groups like $\mathbb{Z}_4 \times \mathbb{Z}_3 \times \mathbb{Z}_5$ are isomorphic to $\mathbb{Z}_{60}$. 

Notice that $\mathbb{Z}_{60}$ is cyclic, while $\mathbb{Z}_2\times\mathbb{Z}_{30}$ is not.
That is for example because $\gcd(30,2)\neq 1$.
