Abelian Group Induces Entropic Structure

Theorem
Let $$\left({G, \circ}\right)$$ be an abelian group.

Let the operation $$*$$ be defined on $$G$$ such that $$\forall x, y \in G: x * y = x \circ y^{-1}$$.

Then $$\left({G, *}\right)$$ is an entropic structure.

Proof
We need to prove that $$\forall a, b, c, d \in G: \left({a * b}\right) * \left({c * d}\right) = \left({a * c}\right) * \left({b * d}\right)$$.

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