A Theorem of Sylvester–Gallai Type for Abelian Groups

A finite subset $X$ of an Abelian group $A$ with respect to addition is called a Sylvester–Gallai set of type $m$ if $|X|\ge m$ and, for every distinct $x_1,\dots,x_{m-1} \in X$, there is an element $x_m \in X \setminus \{x_1,\dots,x_{m-1}\}$ such that
$$ x_1+\dots+x_m=o_A, $$
where $o_A$ stands for the zero of the group $A$. We describe all Sylvester–Gallai sets of type $m$. As a consequence, we obtain the following result: if $Y$is a finite set of points on an elliptic curve in $\mathbb P^2(\mathbb C)$ and
(A) if, for every two distinct points $x_1,x_2 \in Y$, there is a point $x_3 \in Y \setminus \{x_1,x_2\}$ collinear to $x_1$ and $x_2$, then either $Y$ is a Hesse configuration of an elliptic curve or $Y$ consists of three points lying on the same line;
(B) if, for every five distinct points $x_1,\dots,x_5 \in Y$, there is a point $x_6 \in Y \setminus \{x_1,\dots,x_{5}\}$ such that $x_1,\dots,x_6$ lie on the same conic, then $Y$ consists of six points lying on the same conic.