Could anyone explain why the abelian group of points on an elliptic curve over a finite field is isomorphic to at most two cyclic groups? Why is it that it cannot be the product of more than two cyclic groups?
I have tried searching for an answer but the best I could find is that it might be because of Lefschetz principle since an elliptic curve is an abelian variety with dimension 2. But I am not very familiar with algebraic geometry so I don't really understand this answer, let alone know if it is correct.
Any help much appreciated.