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

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Statement

Suppose \pi is a projective plane and l is a line in \pi. For every point P \in l (i.e., every point incident to l), define \Gamma(P,l) as the set of all Elation (?)s with center P and axis l. Define:

\Gamma(l) := \bigcup_{P \in l} \Gamma(P,l).

In other words, \Gamma(l) is defined as the set of all elations with axis l.

Then, \Gamma(l) is a group, and the \Gamma(P,l) form a partition of \Gamma(l) (i.e., they are subgroups whose pairwise intersection is trivial and whose union is the whole group).

Related facts

Elations with given axis form abelian group if there exist non-identity elations with that axis and distinct centers on that axis