Silverman, Exercise 3.12: Here, $E$ is an elliptic curve and $E[m]$ is $m-$ torsion subgroup of it and $\operatorname{Aut}(E)$ is the group of invertible isogenies from $E$ to $E$.
Let $m \geq 2$ be an integer, prime to char (K) if char (K) $>$0. We have to show that the natural map $$ \operatorname{Aut}(E) \to \operatorname{Aut}(E[m])$$ is injective except for $m =2$.
Now, I've got a hint on how to go about showing this from a previously asked SE question here: But what I don't understand is, what could be this natural map the book refers to?
I think if $\phi \in \operatorname{Aut}(E)$ be an invertible isogeny, then I just have to define a map from $E$ to $E[m])$ in order to have a commutative diagram from where I can guess what $\phi$ maps to..is this line of thinking correct?
I would appreciate any kind of helpful suggestions. Thanks.
It should just be restriction, i.e. $\phi\mapsto\phi|_{E[m]}$. To show this is actually an automorphism of $E[m]$, you just need to show that $\phi$ actually maps $E[m]$ to itself, i.e. you need to show that $\phi(E[m])\subseteq E[m]$. But if $P\in E[m]$ then $mP=O$, and we find $m\,\phi(P)=\phi(mP)=\phi(O)=O$.