Baer's theorem on elation group

The statement of this article contains an assertion of the form that for a certain kind of group, either every element has finite order (i.e., the group is a Periodic group (?)) or every non-identity element has infinite order (i.e., the group is a Torsion-free group (?) or aperiodic group). The actual statement in this case may be considerably stronger.
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Statement

Suppose $\pi$ is a 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 Elation (?)s 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 Torsion-free group (?) (i.e., none of the non-identity elements have finite order) or $\Gamma(l)$ is an 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