Abelian Group Induces Entropic Structure

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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:

\(\displaystyle \left({a * b}\right) * \left({c * d}\right)\) \(=\) \(\displaystyle \left({a \circ b^{-1} }\right) \circ \left({c \circ d^{-1} }\right)^{-1}\) Definition of $*$
\(\displaystyle \) \(=\) \(\displaystyle \left({a \circ b^{-1} }\right) \circ \left({\left({d^{-1} }\right)^{-1} \circ c^{-1} }\right)\) Inverse of Group Product
\(\displaystyle \) \(=\) \(\displaystyle \left({a \circ b^{-1} }\right) \circ \left({d \circ c^{-1} }\right)\) Inverse of Group Inverse
\(\displaystyle \) \(=\) \(\displaystyle \left({a \circ c^{-1} }\right) \circ \left({d \circ b^{-1} }\right)\) Definition of Abelian Group
\(\displaystyle \) \(=\) \(\displaystyle \left({a \circ c^{-1} }\right) \circ \left({\left({d^{-1} }\right)^{-1} \circ b^{-1} }\right)\) Inverse of Group Inverse
\(\displaystyle \) \(=\) \(\displaystyle \left({a \circ c^{-1} }\right) \circ \left({b \circ d^{-1} }\right)^{-1}\) Inverse of Group Product
\(\displaystyle \) \(=\) \(\displaystyle \left({a * c}\right) * \left({b * d}\right)\) Definition of $*$

$\blacksquare$


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