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 |
$\blacksquare$
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions