5.1.2.1 states that every eliptic curve is cyclic.

[pociej]

"As we have seen in the previous section, elliptic curves E(F) have the structure of a commuta- tive group associated to them. It can be shown that this group is finite and cyclic whenever the underlying field F is finite." Which is not always true.
It can be Z/nZxZ/mZ as it is e.g y^{2}=x^{3}-x over F_71 Later in this subchapter we deduct from this of existance of generator.