Elation of a projective plane

Statement
A collineation of a projective plane is termed an elation if it has a center(i.e., a point such that every line through that point is preserved by the collineation) and an axis (i.e., a line such that every point on the line is preserved by the collineation) containing the center.

Facts

 * Elations with given axis form a group having a partition into subgroups given by elations having elements as center
 * Elations with given axis form abelian group if there exist two non-identity elations with that axis and distinct centers on that axis