Inverses of Elements Related by Compatible Relation
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Theorem
Let $\struct {G, \circ}$ be a group.
Let $\RR$ be a relation compatible with $\circ$.
Then:
- $\forall x, y \in G: x \mathrel \RR y \iff y^{-1} \mathrel \RR x^{-1}$
Corollary
- $\forall x, y \in G:$
- $x \mathrel \RR e \iff e \mathrel \RR x^{-1}$
- $e \mathrel \RR x \iff x^{-1} \mathrel \RR e$
Proof
Let $e$ be the group identity of $G$.
By Relation Compatible with Group Operation is Strongly Compatible: Corollary:
- $(1): \quad x \mathrel \RR y \iff e \mathrel \RR y \circ x^{-1}$
By Relation Compatible with Group Operation is Strongly Compatible: Corollary, also:
- $(2): \quad y^{-1} \mathrel \RR x^{-1} \iff e \mathrel \RR \paren {y^{-1} }^{-1} \circ x^{-1}$
- $\paren {y^{-1} }^{-1} = y$
Thus, we can rewrite $(2)$ as:
- $(3): \quad y^{-1} \mathrel \RR x^{-1} \iff e \mathrel \RR y \circ x^{-1}$
Now note that the right hand side of $(3)$ is the same as the right hand side in $(1)$.
We conclude that:
- $x \mathrel \RR y \iff y^{-1} \mathrel \RR x$
$\blacksquare$