With $E$ as an elliptic curve defined over a field $K$ and $K^{al}$ it's algebraic closure, $E[m]$ it's $m$-torsion subgroup.
Silverman's Arithmetic of Elliptic Curves says that we have a representation
$$ G_{K^{al}/K} \to Aut(E[m])$$.
Now, I am only familiar with representation theory of finite groups and here, I was wondering how we can consider $E[m]$ as a vector space?
$E[m]$ is group isomorphic to the direct product $\mathbb{Z}/ m \mathbb{Z} × \mathbb{Z} / m \mathbb{Z}$, and it can also be considered as $\mathbb{Z} / m \mathbb{Z}$-module, and maybe now we'll use the module definition of a group representation (which I don't remember, would be great if someone could provide it as I can't seem to find it online) to show that the above.
Can somebody confirm or deny my above proof idea?
Thanks in advance.
EDIT: the book says that $Aut(E[m])$ is isomorphic to $GL_2(\mathbb{Z} / m \mathbb{Z})$ and it's from here that I got the hint that maybe we'll have to think of $E[m]$ as a $\mathbb{Z} / m \mathbb{Z}$- module (module because it can't be a vector space) and hence maybe use module definition of a group representation and not the one I learnt for vector spaces in 'Representation theory of finite groups' course.