= Solution
{c}
The <orbit-stabilizer theorem> identifies the <orbit of a group action> of $s$ with the <set> of <left cosets> of $G_s$. Explicitly, there is a <bijection>
$$
\boxed{G/G_s\longrightarrow G\cdot s,\qquad gG_s\longmapsto g\cdot s.}
$$
Here $G/G_s$ denotes a <set> of <cosets>; it is not asserted to be a <quotient group>. To prove the identification, observe that
$$
gG_s=hG_s\iff h^{-1}g\in G_s
\iff(h^{-1}g)\cdot s=s
\iff g\cdot s=h\cdot s.
$$
This equivalence proves both well-definedness and <injectivity>. Every element of the <orbit of a group action> has the form $g\cdot s$, proving <surjectivity>.
In particular, for a <finite group>, the <cardinalities> satisfy
$$
\boxed{|G\cdot s|=[G:G_s]=\frac{|G|}{|G_s|},\qquad|G|=|G\cdot s|\,|G_s|.}
$$
The last equality follows from <Lagrange's theorem> applied to the <stabilizer subgroup>.
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