Define proper group action on a square

Question

Obviously one can let the dihedral group of order 8 act on a square. How do I define this action mathematically correct? Since the dihedral group leaves the square invariant and just permutes the corners, must I take the set $M=\{(a,b,c,d)\mid 1\leq a,b,c,d\leq4\}$? And then i.e. for the rotation $r\star (1,2,3,4)=(4,1,2,3)$?

Answer 1

A group action is merely a group homomorphism from a group $G$ to a subgroup of $\text{Sym}(X) \cong S_{|X|}$. It appears you want a monomorphism; that is, essentially "another way of describing $G$".

Since $|D_4| = 8$, to obtain such a monomorphism, we require $|X| > 3$. So expressing $D_4$ is a permutation of "$4$ somethings" is going to be optimal. What (symmetric) aspects of a square come in fours?

To understand more fully what I am getting at, note that the square has two diagonals, and that $D_4$ can act on these diagonals (how?). If we see $D_4$ as a subgroup of $S_4$, what are your conclusions regarding $D_4 \cap A_4$?

One of the comments suggest you might see $D_4$ as a subgroup of $\text{GL}_2(\Bbb R)$. To get you started how does:

$\rho = \begin{bmatrix}0&-1\\1&0\end{bmatrix}$

act on the set $X = \{(1,0), (0,1), (-1,0), (0,-1)\}$?