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

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 fact about::elations 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