René Schoof's 1995 paper contains the following statement about an elliptic curve $E$ (at the bottom of page 233):
[...], we use the subgroup $E[l]$ of $l$-torsion points of $E(\overline{\mathbb F_p})$: $$E[l] = \{\,P\in E(\overline{\mathbb F_p}) \,:\, l\cdot P=0 \,\} \text.$$ The group $E[l]$ is isomorphic to $\mathbb Z/l\mathbb Z\times\mathbb Z/l\mathbb Z$.
Why is this the case? This seems somewhat arbitrary to me, why not $\mathbb Z/l\mathbb Z$ or any other group whose exponent is $l$? How would you go about proving this (just hints are welcome as well)?