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:

\(\displaystyle \left({a * b}\right) * \left({c * d}\right)\)

\(=\)

\(\displaystyle \left({a \circ b^{-1} }\right) \circ \left({c \circ d^{-1} }\right)^{-1}\)

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 $*$

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Exercise $13.12 \ \text{(b)}$