Definition:Permutation Representation

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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 $\phi_g$.

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