User:Dfeuer/Cone Condition Equivalent to Symmetry

Theorem
Let $(G,\circ)$ be a group with identity $e$.

Let $C$ be a cone compatible with $\circ$.

Let $\mathcal R$ be the compatible relation on $G$ induced by $C$.

Then the following are equivalent:
 * $\mathcal R$ is symmetric.
 * $C = C^{-1}$