In the paper titled: "The Elliptic Curve Digital Signature Algorithm" by Don Johnson and Alfred Menezes, there is a statement in section 4.3 "Basic Facts":
$ E(F_q) $ is an abelian group of rank 1 or 2. That is: $ E(F_q) $ is isomorphic to $ Z_{n_1} \times Z_{n_2} $ where $ n_2 $ divides $ n_1 $ for unique positive integers $ n_1 $ and $ n_2 $.
Is there an easy proof of why this is so? My current understanding is that for cryptography we try to find a subgroup of $ E(F_q) $ that has prime order (while avoiding singular ones whose order is $ q $ itself). I thought that the order of $ E(F_q) $ could be a composite number with more than 2 factors. Thus I feel surprised by the statement that $ E(F_q) $ has rank 1 or 2.
I am probably missing a lot of things; perhaps having more than 2 factors has nothing to do with rank. But then why is there such a tight bound on the rank, etc.
Update: The same quoted paragraph adds this further statement which I find hard to see why/how:
Moreover, $ n_2 $ divides $ q - 1 $.