Let l be an odd prime, let F be a finite field of characteristic different from l and let A and B be l-isogenous elliptic curves defined over F. We study how the group structures of A(L) and B(L) vary in finite extensions L/F and prove that if the cardinality of the groups A(F) and B(F) are divisible by l and if A(F) and B(F) are isomorphic, then so are A(L) and B(L) for all finite extensions L of F.

We would like to thank Andrew Sutherland for helpful email discussions and Keith Conrad for pointing us to the proof of Lemma 2. We would also like to thank the anonymous referee for a careful reading of the draft and detailed comments which improved the exposition and content of the paper.