Abelian Group Induces Entropic Structure

Theorem
Let $\struct {G, \circ}$ 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 $\struct {G, *}$ is an entropic structure.

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

So:

Also presented as
This is usually presented in the form:

Let the operation $-$ be defined on $\struct {G, +}$ such that:
 * $\forall x, y \in G: x - y = x + \paren {-y}$

Then $\struct {G, -}$ is an entropic structure.