Properties of Relation Compatible with Group Operation/CRG3

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Theorem

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

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

Let $x, y \in G$.


Then:

$x \mathrel{\mathcal R} y \iff y^{-1} \mathrel{\mathcal R} x^{-1}$


Proof

Let $e$ be the group identity of $G$.

By User:Dfeuer/CRG2 $(1)$:

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

By User:Dfeuer/CRG2 $(2)$, also:

$(2): \quad y^{-1} \mathrel{\mathcal R} x^{-1} \iff e \mathrel{\mathcal R} \left({y^{-1}}\right)^{-1} \circ x^{-1}$

By Inverse of Group Inverse $\left({y^{-1}}\right)^{-1} = y$.

Thus, we can rewrite $(2)$ as:

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


Now note that the right hand side of $(3)$ is the same as the right hand side in $(1)$.

We conclude that:

$x \mathrel{\mathcal R} y \iff y^{-1} \mathrel{\mathcal R} x$

$\blacksquare$