Transplanting Theorem

Theorem
Let $\struct {S, \circ}$ be an algebraic structure.

Let $f: S \to T$ be a bijection.

Then there exists one and only one operation $\oplus$ such that $f: \struct {S, \circ} \to \struct {T, \oplus}$ is an isomorphism.

The operation $\oplus$ is defined by:


 * $\forall x, y \in T: x \oplus y = \map f {\map {f^{-1} } x \circ \map {f^{-1} } y}$

The operation $\oplus$ is called the transplant of $\circ$ under $f$.

Existence
To show that $\oplus$ as defined above exists:

Let $u, v \in S$, and let $x = \map f u, y = \map f v$.

Then as $f$ is a bijection, $u = \map {f^{-1} } x, v = \map {f^{-1} } y$.

Thus:

It is seen that $f$ is an isomorphism as required.

Uniqueness
Let $f$ be the isomorphism whose existence has been proven above.

Thus:

So, if $\oplus$ is an operation on $T$ such that $f$ is an isomorphism from $\struct {S, \circ} \to \struct {T, \oplus}$, then $\oplus$ must be defined as by this theorem, and there can be no other such operations.