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:
\(\ds \paren {a * b} * \paren {c * d}\) | \(=\) | \(\ds \paren {a \circ b^{-1} } \circ \paren {c \circ d^{-1} }^{-1}\) | Definition of $*$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \circ b^{-1} } \circ \paren {\paren {d^{-1} }^{-1} \circ c^{-1} }\) | Inverse of Group Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \circ b^{-1} } \circ \paren {d \circ c^{-1} }\) | Inverse of Group Inverse | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \circ c^{-1} } \circ \paren {d \circ b^{-1} }\) | Definition of Abelian Group | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \circ c^{-1} } \circ \paren {\paren {d^{-1} }^{-1} \circ b^{-1} }\) | Inverse of Group Inverse | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \circ c^{-1} } \circ \paren {b \circ d^{-1} }^{-1}\) | Inverse of Group Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a * c} * \paren {b * d}\) | Definition of $*$ |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Exercise $13.12 \ \text{(b)}$