Relation Compatible with Group Operation is Strongly Compatible/Corollary

Theorem

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

Let $\RR$ be a relation compatible with $\circ$.

The following equivalences hold:

$\forall x, y \in G:$

$(1): \quad x \mathrel \RR y \iff e \mathrel \RR y \circ x^{-1}$

$(2): \quad x \mathrel \RR y \iff e \mathrel \RR x^{-1} \circ y$

$(3): \quad x \mathrel \RR y \iff x \circ y^{-1} \mathrel \RR e$

$(4): \quad x \mathrel \RR y \iff y^{-1} \circ x \mathrel \RR e$

Proof

Applying Relation Compatible with Group Operation is Strongly Compatible to $x$, $y$, and $x^{-1}$ we obtain:

$x \mathrel \RR y \iff x \circ x^{-1} \mathrel \RR y \circ x^{-1}$

$x \mathrel \RR y \iff x^{-1} \circ x \mathrel \RR x^{-1} \circ y$

Applying Relation Compatible with Group Operation is Strongly Compatible to $x$, $y$, and $y^{-1}$, on the other hand, yields:

$x \mathrel \RR y \iff x \circ y^{-1} \mathrel \RR y \circ y^{-1}$

$x \mathrel \RR y \iff y^{-1} \circ x \mathrel \RR y^{-1} \circ y$

By the definition of inverse element:

$x \circ x^{-1} = x^{-1} \circ x = y \circ y^{-1} = y^{-1} \circ y = e$

Making these substitutions proves the theorem.

$\blacksquare$