  As it is well known, one can define an abelian group on the points of an
elliptic curve, using the so called chord-tangent law \cite{dale}, and a chosen
point. However, that very chord-tangent law allows us to define a rather more
obscure algebraic structure, which we call an elliptic group, on the points of
an elliptic curve. In the cases when our curve has a so called flex point
(intersection number with the tangent is $3$), the classical abelian group and
the elliptic group carry the same information. However, if our curve does not
have such a point (which often happens over $\mathbb{Q}$), the abelian group is
not enough to recover the elliptic group.
  The aim of this paper is to study this algebraic structure in more detail,
its connections to abelian groups and at the very end even introduce the notion
of an elliptic ring (a monoid object in the category of elliptic groups).
