Transplanting Theorem/Corollary
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Theorem
Let $\struct {S, \circ}$ be an algebraic structure.
Let $f: S \to S$ be an automorphism on $\struct {S, \circ}$.
Then the transplant of $\circ$ under $f$ is $\circ$ itself.
Proof
From the Transplanting Theorem there exists one and only one operation $\circ$ such that $f: \struct {S, \circ} \to \struct {S, \circ}$ is an automorphism.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures