Structure Induced by Permutation on Quasigroup is Quasigroup

Theorem

Let $\struct {S, \circ}$ be a quasigroup.

Let $\sigma: S \to S$ be a permutation on $S$.

Let $\struct {S, \circ_\sigma}$ be the structure induced by $\sigma$ on $\circ$:

$\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$

Then $\struct {S, \circ_\sigma}$ is also a quasigroup.

Proof

By definition of quasigroup:

$\forall a, b \in S: \exists ! x \in S: x \circ a = b$

$\forall a, b \in S: \exists ! y \in S: a \circ y = b$

Let $a, b \in S$.

As $\sigma$ is a permutation, it is by definition both surjective and injective.

We have that:

$\exists ! x: x \circ a = b$

Thus:

\(\ds \exists ! x \in S: \, \)

\(\ds \map \sigma {x \circ a}\)

\(=\)

\(\ds b\)

\(\ds \leadsto \ \ \)

\(\ds x \circ_\sigma a\)

\(=\)

\(\ds b\)

Definition of Operation Induced by Permutation

Similarly, we have that:

$\exists ! x: a \circ x = b$

Thus:

\(\ds \exists ! x \in S: \, \)

\(\ds \map \sigma {a \circ x}\)

\(=\)

\(\ds b\)

\(\ds \leadsto \ \ \)

\(\ds a \circ_\sigma x\)

\(=\)

\(\ds b\)

Definition of Operation Induced by Permutation

The result follows.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.9 \ \text {(a)}$