In Washington's "Elliptic Curves: Number Theory and Cryptography",
the proof that $E[n] \simeq Z_n$ x $Z_n$ is concluded by referring to the structure theorem of finite abelian groups ($E[n] \simeq Z_{n_1} $ x $...$ x $Z_{n_k}$ with $n_i | n_{i+1}$).
If we have a prime $l$ with $l | n_i$ for all i, why can we conclude that "$E[l] \subseteq E[n]$ has order $l^2$"? The proof continues with "Multiplication by n annihilates $E[n] \simeq Z_{n_1}$ x $Z_{n_2}$, so we must have $n_2 | n$". Why is that the case?
The conclusion step that $n_1 = n_2$ is clear but I still can't see why the condition that the characteristic $p$ of the field does not divide $n$ is necessary.