Properties of Relation Compatible with Group Operation

Theorem

Let $\struct {G, \circ}$ be a group with identity element $e$.

Let $\RR$ be a relation on $G$ which is compatible with $\circ$.

The following properties hold:

Relation Compatible with Group Operation is Strongly Compatible

$\RR$ is strongly compatible with $\circ$:

$\forall x, y, z \in G:$
$x \mathrel \RR y \iff x \circ z \mathrel \RR y \circ z$
$x \mathrel \RR y \iff z \circ x \mathrel \RR z \circ y$

Corollary

$\forall x, y \in G:$
$(1): \quad x \mathrel \RR y \iff e \mathrel \RR y \circ x^{-1}$
$(2): \quad x \mathrel \RR y \iff e \mathrel \RR x^{-1} \circ y$
$(3): \quad x \mathrel \RR y \iff x \circ y^{-1} \mathrel \RR e$
$(4): \quad x \mathrel \RR y \iff y^{-1} \circ x \mathrel \RR e$

Inverses of Elements Related by Compatible Relation

$\forall x, y \in G: x \mathrel \RR y \iff y^{-1} \mathrel \RR x^{-1}$

Corollary

$\forall x, y \in G:$
$x \mathrel \RR e \iff e \mathrel \RR x^{-1}$
$e \mathrel \RR x \iff x^{-1} \mathrel \RR e$