Isomorphisms in finite abelian groups 1

Question

True of false? If G and H are two groups with the same order and both are abelian, then they are isomorphic.

Answer 1

Am implication of this is that every finite abelian group is isomorphic to some cyclic group, which is not true.

Answer 2

$\textbf{HINT-}$ Consider $\mathbb{Z_4}$ and $\mathbb{Z_2 \times Z_2}$. So FALSE.

Answer 3

Any finite abelian group is isomorphic to direct product of prime power orders. So I can apply this for these two groups we get they are not isomorphic. Because if G is cyclic and H is not.