Baer's theorem on elation group

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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.

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