A Sylvester–Gallai Type Theorem for Abelian Groups

A finite subset of an Abelian group with respect to addition is called a Sylvester–Gallai set of type if and, for every distinct , there is an element such that , where stands for the zero of the group . We describe all Sylvester–Gallai sets of type . As a consequence, we obtain the following result: if is a finite set of points on an elliptic curve in and (A) if, for every two distinct points , there is a point collinear to and , then either is the Hesse configuration of the elliptic curve or consists of three points lying on the same line; (B) if, for every five distinct points , there is a point such that lie on the same conic, then consists of six points lying on the same conic.