Proof that symmetries of a square form a group under composition I am curious if there is less wonky/wordy way to prove that the eight symmetries of the square form a group under composition than by checking each property for each element. I can prove this in a long and not-so-pretty way. But I would really love to see a better proof. Thanks in advance, ladies and gents.

More generally, it is true that the set $S$ of symmetries of _any_ geometrical object form a group under the operation, "followed by". For it is clear that "$a$ followed by ($b$ followed by $c$)" is the same as "($a$ followed by $b$) followed by $c$"; it is clear that "do nothing" is in $S$, and that for all $a$ in $S$, $a$ followed by "do nothing" and "do nothing" followed by $a$ are both the same as $a$; and it is clear that any symmetry that you can perform, you can undo, so there are inverses.