Baer's theorem on elation group

Statement
Suppose $$\pi$$ is a fact about::projective plane and $$l$$ is a line on $$\pi$$. Suppose $$P_1, P_2$$ are distinct points on $$l$$ such that $$\Gamma(P_1,l)$$ (the group of fact about::elations with center $$P_1$$ and axis $$l$$) and $$\Gamma(P_2,l)$$ (the group of elations with center $$P_2$$ and axis $$l$$) are both nontrivial groups. Then, the following are true:


 * 1) The group $$\Gamma(l)$$ of all elations with axis $$l$$ is an abelian group.
 * 2) Either $$\Gamma(l)$$ is a fact about::torsion-free group (i.e., none of the non-identity elements have finite order) or $$\Gamma(l)$$ is an fact about::elementary abelian group, i.e., all its non-identity elements have the same prime order.

Related facts

 * Elations with given axis form a group having a partition into subgroups given by elations having elements as center