Structure Induced by Permutation on Quasigroup is Quasigroup
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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 $7.9 \ \text {(a)}$