Let $M$ be a finitely generated module over a PID $R$. Then there are idelas $I_1=(a_1),~I_2=(a_2)~ \cdots, I_n=(a_n)$ of $R$ such that $I_1 \supseteq I_2 \supseteq \cdots \supseteq I_n$ such that $$M \cong R/I_1 \oplus R/I_2 \oplus \cdots \oplus R/I_n, ~~~~~~~~~(1)$$ that is, $M$ can be written as a direct sum of cyclic $R$-modules.
Now, let $E$ be an elliptic curve over $\mathbb{Z}_p$, the ring of $p$-adic integers. Let $E[p^n]$ be the set of $p^n$-torsion points. Then $E[p^n]$ is finitely generated $\mathbb{Z}_p$-module.
What would be the decomposition of $E[p^n]$ similar to that of $(1)$ ?
We note that $E[p] \subseteq E[p^2] \subseteq \cdots E[p^n]$. For any $p^n$-torsion point $P$, there is a $p^{n-1}$-torsion point $Q$ such that $pQ=P$. So I think in this case we have $$I_1=(p) \supseteq I_2=(p^2) \supseteq \cdots I_n=(p^n)$$ so that we have $$E[p^n] \cong \mathbb{Z}_p/p \mathbb{Z}_p \oplus \mathbb{Z}_p/p^2 \mathbb{Z}_p \oplus \cdots \oplus \mathbb{Z}_p/p^n \mathbb{Z}_p.$$
Is it correct ?
For an elliptic curve (with its algebraic group law ) $$E:\{ (x,y) \in \overline{k}, y^2=x^3+ax+b\}\cup O$$ where $a,b\in k$ is a field of characteristic $0$ then $$E[m] \cong \Bbb{Z}/m\Bbb{Z}\times \Bbb{Z}/m\Bbb{Z}$$ Whence $$E[p^n] \cong \Bbb{Z}_p/p^n\Bbb{Z}_p\times \Bbb{Z}_p/p^n\Bbb{Z}_p$$
$G[m] \cong \Bbb{Z}/m\Bbb{Z}\times \Bbb{Z}/m\Bbb{Z}$ is obvious for a complex torus $G=\Bbb{C}/\Lambda$
The Weierstrass functions give some isomorphisms between complex tori and complex elliptic curves
All the torsion points of $E$ will have their coordinates in $\overline{\Bbb{Q}(a,b)}$, that we can identify with a subfield of $\Bbb{C}$ so it suffices to prove the claim for complex elliptic curves.