I know that is possible to define a bijective group homomorphism between $(2)$ and $\mathbb{Z}$ through the function $f(2x) = x$ and similarly for $(3)$ via $f(3x) = x$. But I thought that for two groups to be isomorphic they had to have the same order. If the order is infinity how could you tell the two groups say $(2)$ and $\mathbb{Z}$ are isomorphic to each other?
Could subgroups be isomorphic to the original groups?