Structure Induced by Permutation on Commutative Quasigroup is Commutative Quasigroup

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, $\struct {S, \circ}$ is closed under $\circ$.

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

Again, from definition of a quasigroup, $\struct {S, \circ_\sigma}$ is closed under $\circ_\sigma$.

Hence:

\(\ds \forall a, b \in S: \, \)

\(\ds a \circ_\sigma b\)

\(=\)

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

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

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