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 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