I have question regarding exercise 3.12 of J. Silverman "The arithmetic of Elliptic curves". It states the following:
Let $m \geq 2$ be an integer, prime to $\text{char}(K) > 0$. Prove that the natural map $\text{Aut}(E) \longrightarrow \text{Aut}(E[m])$ is injective except for $m=2$, where the kernel is $[\pm1]$.
I want to prove this without too much prior knowledge of the structure of $\text{Aut}(E)$ (to be more precise, without the use of theorem III.10.1 in Silverman).
($E[m] =\{ P \in E : [m]P=O\}$).
Anyone got any ideas?