Definition:Permutation Representation/Group Action
< Definition:Permutation Representation(Redirected from Definition:Permutation Representation Associated to Group Action)
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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 $\phi: G \times X \to X$ 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
- Group Action determines Bijection, which shows that $\phi_g \in \struct {\map \Gamma X, \circ}$
- Group Action defines Permutation Representation, which shows that this defines a homomorphism
Sources
- 2003: David S. Dummit and Richard M. Foote: Abstract Algebra (3rd ed.) Chapter $1$: Introduction to Groups: $\S 1.7$: Group Actions