Properties of Relation Compatible with Group Operation
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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
- $\forall x, y \in G:$
- $x \mathrel \RR e \iff e \mathrel \RR x^{-1}$
- $e \mathrel \RR x \iff x^{-1} \mathrel \RR e$