= Solution
Take $X=G$ and let $g$ act by left multiplication $L_g(x)=gx$. Each $L_g$ is a <permutation>, with inverse $L_{g^{-1}}$. Associativity gives $L_gL_h=L_{gh}$, so $g\mapsto L_g$ is a <group homomorphism> into $\operatorname{Sym}(G)$. If $L_g$ is the identity <permutation>, evaluating it at the identity element gives $g=1$. Thus
$$
\boxed{G\hookrightarrow\operatorname{Sym}(G),\qquad g\longmapsto L_g.}
$$
This is the <Cayley theorem>, realized by the <left regular action>, which is a <faithful group action>.
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