I'm an undergrad student fairly keen on algebra. Over the different algebra courses I've taken, I've often encountered the so-called $p$-Prüfer group on exercises but somehow never got around to them. Now I'm trying to take care of that, but there are some statements I've seen about this group which I don't know how to prove (maybe because I lack some more background in group theory, especially in the study of infinite abelian groups?)
Definition A $p$-group is a $p$-Prüfer group if it is isomorphic to $C_{p^\infty}=\{e^{\frac{2k\pi i}{p^n}}:k\in \mathbb{Z}, n\in\mathbb{Z}^+\} \subset (\mathbb{C}^\times, \cdot)$
What I'm having trouble to prove is:
The following are $p$-Prüfer groups:
1) An infinite $p$-group whose subgroups are totally ordered by inclusion,
2) An infinite $p$-group such that every finite subset generates a cyclic group,
3) An infinite abelian $p$-group such that $G$ is isomorphic to every proper quotient,
4) An infinite abelian $p$-group such that every subgroup is finite
Just for the record, what I (think I) could prove was that the following are $p$-Prüfer groups:
5) An injective envelope of $C_{p^n}$, for any $n\geq 1$,
6) A Sylow $p$-subgroup of $\frac{\mathbb{Q}}{\mathbb{Z}}$,
7) The direct limit of $0\subset C_p \subset C_{p^2}\subset ...$
Here $C_{p^n}$ denotes a cyclic group of order $p^n$.
Any other characterizations of the $p$-Prüfer group are welcome.