I'd be interested in a short proof of and a reference (first source, not a textbook as e.g. one of Silverman's monographs) for the following result:
Let $q$ be a prime power and $C/F_q$ be an elliptic curve. Then there are integers $m,n \geq 1$ with $$ C(F_q) \cong \mathbb{Z}_m \times \mathbb{Z}_{mn},~~~ gcd(m,q) =1. $$