Definition:Group Action/Permutation Representation
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Definition
Let $G$ be a group.
Let $X$ be a set.
Let $\struct {\map \Gamma X, \circ}$ be the symmetric group on $X$.
Let $\rho: G \to \struct {\map \Gamma X, \circ}$ be a permutation representation.
The group action of $G$ associated to the permutation representation $\rho$ is the group action $\phi: G \times X \to X$ defined by:
- $\map \phi {g, x} = \map {\rho_g} x$
where $\rho_g : X \to X$ is the permutation representation associated to $\rho$ for $g \in G$ by $\map {\rho_g} x = \map \phi {g, x}$.
Also see
- Permutation Representation defines Group Action, where it is shown that $\phi$ is indeed a group action
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): action: 1. (of a group on a non-empty set $S$)