Structure Induced by Permutation on Commutative Quasigroup is Commutative Quasigroup

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Theorem

Let $\struct {S, \circ}$ be a quasigroup such that $\circ$ is a commutative operation.

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 such that $\circ_\sigma$ is a commutative operation.


Proof

From definition of a quasigroup, we take it for granted that $\struct {S, \circ}$ is closed under $\circ$.

From Structure Induced by Permutation on Quasigroup is Quasigroup we have that $\struct {S, \circ_\sigma}$ is a quasigroup.

Again, from definition of a quasigroup, we take it for granted that $\struct {S, \circ_\sigma}$ is closed under $\circ_\sigma$.

Then we see that:

\(\ds \forall a, b \in S: \, \) \(\ds a \circ_\sigma b\) \(=\) \(\ds \map \sigma {a \circ b}\) Definition of Operation Induced by Permutation
\(\ds \) \(=\) \(\ds \map \sigma {b \circ a}\) Definition of Commutative Operation
\(\ds \) \(=\) \(\ds b \circ_\sigma a\) Definition of Operation Induced by Permutation

$\blacksquare$


Sources