I'm faced with the following problem: Let $E/\mathbb{Q}$ be an elliptic curve, that has good reduction at the prime number $l$. Let $p \neq l$ be another prime number and consider the quotient $E(\mathbb{F}_l)/pE(\mathbb{F}_l)$. What can we say about it? If $p$ does not divide $\#E(\mathbb{F}_l)$ then the quotient will be just zero. But otherwise, is there an easy structure of the quotient, like is it cyclic or something? One could just compute the quotient and see what comes out, but is there a theoretical solution?
Thanks guys!
$$E(\mathbb{F}_\ell) \subseteq E(\overline{\mathbb{F}}_\ell) \cong \mathbb{Q}/\mathbb{Z} \times \mathbb{Q}/\mathbb{Z}[1/\ell]$$
So, $E(\mathbb{F}_\ell)$ is always the product of two cyclic groups (at least one with order coprime to $\ell$), and so is its quotient by any modulus.
(note that I count the zero group as cyclic)
This mathoverflow question has more detail. A brief sketch of why this should be true can be seen at the answer to this question.