Let $p$ denote a fixed point in the plane $P,$
and let $G$ denote the set of all
rotations of the plane about the point $p.$
Each element of $G$ represents an element
of $M(P).$
Agree that two rotations are equal if they
differ by a multiple of $360^\circ.$
Then composition is an operation of $G.$
If $\alpha$ and $\beta$ are rotations about $p,$
then $\beta\circ\alpha$ is the rotation obtained
by first applying $\alpha$ and then $\beta.$
For example, if $\alpha$ denotes clockwise
rotation through $70^\circ,$ and $\beta$
clockwise rotation through $345^\circ,$
then $\beta\circ\alpha$ is clockwise rotation
through $415^\circ,$ or equivalently,
$55^\circ.$
This operation is associative by
For all $a,b\in\R,$ $a\ne0$
let
$\alpha_{a,b}:\R\rightarrow\R$
defined by
$\alpha_{a,b}(x)=ax+b,$
and let $A$ be the set of all such mappings.
Then composition is an operation on $A.$
Notice that $A$ is a subset of $M(\R)$
and that composition is, as always,
associative.
Observe that $\alpha_{a,b}=\alpha_{1,b}\circ\alpha_{a,0}.$
Then, if $a\gt1$ and $b\gt0$
then $\alpha_{a,b}$
corresponds to
$\alpha_{a,0}$
(magnification)
followed by
$\alpha_{1,b}$
(translation).
Alex Notes