I'm studying for my final and I came across this homework problem that I had previously done but I don't remember how to do it anymore. It is as follows ($G, H$ are groups):
Suppose that $H \le G$. Prove that if $H \cong \mathbb Z$ or $H \cong \mathbb Z_n$ then $H = ⟨g⟩$ for some $g ∈ G$.
1) So is it basically saying that if $H \cong \mathbb Z_n$ for any $n \in \mathbb N$, then $H$ is a cyclic subgroup of another group?
2) How would I go about proving this? I believe it intuitively because for any $\mathbb Z_n$, we can always for say $H = ⟨1⟩$ for modular addition.
If $\phi: \mathbb Z \to H$ is an isomorphism, then $H = \langle \phi(1) \rangle$.
If $\phi: \mathbb Z_n \to H$ is an isomorphism, then $H = \langle \phi(\bar 1) \rangle$.