Group Action defines Permutation Representation

Theorem
Let $$\Gamma \left({X}\right)$$ be the set of permutations on a set $$X$$.

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

A group action is a homomorphism from $$G$$ to $$\Gamma \left({X}\right)$$.

Proof
Let $$g, h \in G$$.

From the definition of group action, $$\forall \left({g, x}\right) \in G \times X: \phi \left({\left({g, x}\right)}\right) \in X = g \wedge x \in X$$.

Let $$\phi_g: X \to X$$ be the mapping defined as $$\phi_g \left({x}\right) = \phi \left({g, x}\right)$$.

Let $$\phi \left({g, x}\right) = \phi_g \left({x}\right)$$.


 * First we show that $$\phi_g \circ \phi_h \left({x}\right) = \phi_g \phi_h \left({x}\right)$$.

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 * We have $$e \wedge x = x \implies \phi_e \left({x}\right) = x$$.

Comment
Some treatments of this subject take this as the definition of a group action.