Property of the $p^n$-Selmer group Consider the $p^n$-Selmer group of an elliptic curve $E/\mathbb{Q}$, $\operatorname{Sel}_{p^n}(E)$. Must we always have $\operatorname{Sel}_{p^n}(E) \cong (\mathbb{Z}/p^n\mathbb{Z})^{s}$ for some $s$? It seems that all the examples I've seen with $p^n = 2, 3$ this seems to be true, but why?

Let $p=2$ and $n=2$, and consider $\text{Sel}_4(E/\mathbb{Q})$, the Selmer group associated to the isogeny $[4]:E\to E$. By definition, the Selmer group and the Tate-Shafarevich group fit in an exact sequence:
$$0\to E(\mathbb{Q})/4E(\mathbb{Q})\to \text{Sel}_4(E/\mathbb{Q})\to \text{Sha}(E/\mathbb{Q})[4]\to 0.$$
Suppose that $E(\mathbb{Q})\cong \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}$. Then, 
$$E(\mathbb{Q})/4E(\mathbb{Q})\cong \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/4\mathbb{Z}.$$
Moreover, suppose that $\text{Sha}(E/\mathbb{Q})[4]$ is trivial. Then, 
$$\text{Sel}_4(E/\mathbb{Q})\cong E(\mathbb{Q})/4E(\mathbb{Q})\cong \mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/4\mathbb{Z}.$$
Of course, showing that $\text{Sha}(E/\mathbb{Q})[4]$ is trivial is hard, but if you are willing to assume the Birch and Swinnerton-Dyer conjecture, then you may look for examples where the analytic order of Sha is $1$ (or, more generally, odd analytic order). You can find a list of $100$ examples that satisfy the above properties here. For instance,
$$E: y^2+xy=x^3-x$$
is one such example.
