Let $E$ and $E'$ be elliptic curves over a field $k$ An isogeny between $E$ and $E'$ is a finite morphism $f : E\to E'$ of varieties that preserves basepoints.

The two curves are called isogenous if there is an isogeny between them. This is an equivalence relation, symmetry being due to the existence of the dual isogeny. Every isogeny is an algebraic homomorphism and thus induces homomorphisms of the groups of the elliptic curves for $k$ valued points.