= Solution
For a <group action> of $G$ on $X$, let $G_x=\{g:gx=x\}$ be the <stabilizer subgroup> of $x$, and $Gx$ its <group orbit>. The <orbit-stabilizer theorem> is the <bijection>
$$
G/G_x\longrightarrow Gx,\qquad gG_x\longmapsto gx.
$$
It is well defined because elements in the same left <coset> differ by an element fixing $x$. Conversely, $gx=hx$ implies $h^{-1}g\in G_x$, so the <cosets> coincide. Surjectivity is the definition of the <group orbit>. For a <finite group>, counting the <cosets> gives
$$
\boxed{|G|=|Gx|\,|G_x|.}
$$
To obtain <Cayley theorem>, let $G$ act on its own underlying set by left multiplication. The map $\lambda_g:x\mapsto gx$ is a <permutation>, with inverse $\lambda_{g^{-1}}$, and $\lambda_g\lambda_h=\lambda_{gh}$. If $\lambda_g$ is the identity <permutation>, evaluating at the identity element gives $g=e$. Thus $g\mapsto\lambda_g$ is an injective <group homomorphism>. When $|G|=n$, labeling this set by $n$ letters identifies its image with a <subgroup> of $S_n$. \b[Every finite group embeds in the symmetric group on its own elements.]
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