"Find the group of symmetries of a chess board", was the original question. As stated is not very rigorous, i intrepreted it as asking to find a (setwise) stabilizer of the group of $\mathbb{R}^2$ isometries acting on $\mathbb{R}^2$ in the expected way. That is, the stabilizer of the set (or a set equivalent in some sense to) $$C :=\{[n, n+1]\times [m, m+1] :(1,1)\leq (n,m)\leq (8,8) \ \ \mathrm{and} \ \ n+m = 1 \ \ \mathrm{(mod\ \ 2)}\}.$$
This is not very pleasant to work with. We can go with cases: it's easy to show that translations don't fix $C$. Due to corners, rotations must at worst be the same of a square; there might be a slick way to show the only plausible reflexions are the same as of a square.
Then by cases we could show non $\pi$ rotation can't work, looking at corners again, thankfully rotation matrix is simple in this case so it's not too hard... We can also show 2 reflexions don't work looking at corners;
It remains to show whether the $\pi$-rotation $r$ work, $rf$ (reflexion followed by 90 degree rotation) and $r^3f$ works (reflexion followed by 270 degree rotation). We can save work since showing 2 of them work implies the other does too. But this is still really tedious..
What are better ways to approach this (kind of) problem, without sacrificing rigor?