Principal homogeneous space (reviewed)

A principal homogeneous space (or torsor) $X$ of an abelian variety $A$ is a variety equipped with an $A$-action that is both free and transitive.

If $X$ has a rational point $O$ then we can view $X$ as an abelian variety with $O$ as its identity element by defining a group operation on $X$ in terms of the $A$-action on $O$.