Definition: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 $X$.
A permutation representation of $G$ is a group homomorphism from $G$ to $\struct {\map \Gamma X, \circ}$.
Associated to Group Action
Let $\phi: G \times X \to X$ be a group action.
Define for $g \in G$ the mapping $\phi_g : X \to X$ by:
- $\map {\phi_g} x = \map \phi {g, x}$
The permutation representation of $G$ associated to the group action is the group homomorphism $G \to \struct {\map \Gamma X, \circ}$ which sends $g$ to $\phi_g$.
Also see
Sources
- 2008: I. Martin Isaacs: Finite Group Theory ... (next) Chapter $1$: Sylow Theory: $\S 1A$