Abelian Group Induces Entropic Structure

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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:

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

$\blacksquare$


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