Transplanting Theorem/Corollary

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
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.