# 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 is a Projective plane (?) and is a line on . Suppose are distinct points on such that (the group of Elation (?)s with center and axis ) and (the group of elations with center and axis ) are both nontrivial groups. Then, the following are true:

- The group of all elations with axis is an abelian group.
- Either is a Torsion-free group (?) (i.e., none of the non-identity elements have finite order) or is an Elementary abelian group (?), i.e., all its non-identity elements have the same prime order.