Relation Compatible with Group Operation is Reflexive or Antireflexive

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Theorem

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

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


Then $\RR$ is reflexive or antireflexive.


Proof

Suppose that $\RR$ is not antireflexive.

Then there is some $x \in G$ such that $x \mathrel \RR x$.

Let $y \in G$.

Then by the definition of compatibility:

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

By Group Axiom $\text G 1$: Associativity and Group Axiom $\text G 3$: Existence of Inverse Element:

$y \mathrel \RR y$

Since this holds for arbitrary $y$, $\RR$ is reflexive.

Thus $\mathcal R$ is either reflexive or antireflexive.

$\blacksquare$