# Properties of Relation Compatible with Group Operation/CRG1

## Theorem

Let $\left({G, \circ}\right)$ be a group.

Let $x, y, z \in G$.

Let $\mathcal R$ be a relation compatible with $\circ$.

Then the following equivalences hold:

$x \mathrel {\mathcal R} y \iff x \circ z \mathrel {\mathcal R} y \circ z$
$x \mathrel {\mathcal R} y \iff z \circ x \mathrel {\mathcal R} z \circ y$

## Proof

By Relation Compatible with Group Operation is Strongly Compatible, $\mathcal R$ is strongly compatible with $\circ$.

Thus by the definition of strong compatibility, the theorem holds.

$\blacksquare$