Write out Cayley tables for groups formed by the symmetries of a rectangle
and for ($Z_4$, +). How many elements are in each group? Are the groups the
same? Why or why not?
$$(Z_4, +)\ 0\ 1\ 2\ 3
\\
\ \ \ \ \ \ \ \ 0|\ 0\ 1\ 2\ 3
\\
\ \ \ \ \ \ \ \ 1|\ 1\ 2\ 3\ 0
\\
\ \ \ \ \ \ \ \ 2|\ 2\ 3\ 0\ 1
\\
\ \ \ \ \ \ \ \ 3|\ 3\ 0\ 1\ 2
$$
But I don't understand how I would write out a Cayley table for the symmetries of a rectangle? Would I label each corner a number?
Best Answer
Here's one way to do it: let $e$ denote the identity symmetry, let $a$ correspond to a reflection, and let $b$ denote the rotation by $180^\circ$ about the center. Note that the symmetries of the rectangle are then $\{e,a,b,ab\}$.
Alternatively, use the formulation given here.