Permutation Representation defines Group Action

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Theorem

Let $G$ be a group whose identity is $e$.

Let $X$ be a set.

Let $\map \Gamma X$ be the symmetric group of $X$.

Let $\rho: G \to \map \Gamma X$ be a permutation representation, that is, a homomorphism.


The mapping $\phi: G \times X \to X$ associated to $\rho$, defined by:

$\map \phi {g, x} = \map {\map \rho g} x$

is a group action.


Proof

Let $g, h \in G$ and $x \in X$.

We verify that $g * \paren {h * x} = \paren {g h} * x$:

\(\ds g * \paren {h * x}\) \(=\) \(\ds \map {\map \rho g} {\map {\map \rho h} x}\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \map {\paren {\map \rho g \circ \map \rho h} } x\) Definition of Composition of Mappings
\(\ds \) \(=\) \(\ds \map {\map \rho {g h} } x\) $\rho$ is a homomorphism
\(\ds \) \(=\) \(\ds \paren {g h} * x\) Definition of $\phi$


We verify that $e * x = x$.

Let $I_X$ denote the identity mapping on $X$.

\(\ds e*x\) \(=\) \(\ds \map {\map \rho e} x\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \map {I_X} x\) Group Homomorphism Preserves Identity, Set of all Self-Maps under Composition forms Monoid
\(\ds \) \(=\) \(\ds x\) Definition of Identity Mapping

$\blacksquare$


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