What does congruency mean in $D_4$?

Question

What does congruency mean in $D_4$?

How can I check for example that

For $K = \{k_0, k_2\}$, $$p_x \equiv p_y \pmod K$$

I.e. how to evaluate $(p_x - p_y) \bmod K$, specifically what is $(p_x - p_y)$?

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$k_0$ is a rotation of $0$°.
$k_2$ is a rotation of $180$°.
$p_x$ is the horizontal (x-axis) flip.
$p_y$ is the vertical (y-axis) flip.

Answer 1

$K$ is a subgroup of $D_4$, and $p_x\equiv p_y\pmod K$ means $\bar{p_x}=\bar{p_y}$ in $D_4/K$. Actually $p_y=p_xk_2$.