Definition:Structure Induced by Permutation
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Definition
Let $\struct {S, \circ}$ be an algebraic structure on a set $S$.
Let $\sigma: S \to S$ be a permutation on $S$.
Let $\circ_\sigma$ be the operation on $S$ induced by $\sigma$:
- $\forall x, y \in S: x \circ_\sigma y := \map \sigma {x \circ y}$
Then $\struct {S, \circ_\sigma}$ is the (algebraic) structure induced by $\sigma$ (on $\circ$).
Also see
- Results about operations induced by permutations can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.9$