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 $\forall a, b, c, d \in G: \left({a * b}\right) * \left({c * d}\right) = \left({a * c}\right) * \left({b * d}\right)$.

So: