Construct a "multiplication table" for only the rotational symmetries of a pentagon.

The transformations we use must be limited to rotations where our pentagon ultimately ends up in the same area it originally occupied after the rotation.

Given that a pentagon has 5 sides, the smallest such rotation (besides rotating by $0^{\circ}$) is a rotation of $360^{\circ} / 5=72^{\circ}$. One could rotate the pentagon by any multiple of $72^{\circ}$ and occupy its original area, but notice $5 \cdot 72^{\circ} = 360^{\circ}$ is equivalent to no rotation at all.

As such, we really only have 5 distinct transformations: rotating by $0^{\circ}$ (doing nothing), $72^{\circ}$, $144^{\circ}$, $216^{\circ}$, or $288^{\circ}$. Calling these rotations $R_{0}, R_{72}, R_{144}, R_{216}$, and $R_{288},$ we can flesh out the table for their combinations:

  $R_{0}$ $R_{72}$ $R_{144}$ $R_{216}$ $R_{288}$
$R_{0}$ $R_{0}$ $R_{72}$ $R_{144}$ $R_{216}$ $R_{288}$
$R_{72}$ $R_{72}$ $R_{144}$ $R_{216}$ $R_{288}$ $R_{0}$
$R_{144}$ $R_{144}$ $R_{216}$ $R_{288}$ $R_{0}$ $R_{72}$
$R_{216}$ $R_{216}$ $R_{288}$ $R_{0}$ $R_{72}$ $R_{144}$
$R_{288}$ $R_{288}$ $R_{0}$ $R_{72}$ $R_{144}$ $R_{216}$