= Solution
{c}
The <stabilizer subgroup> of $H$ for this <conjugation> <group action> is its <normalizer>
$$
\operatorname{Stab}(H)=N_G(H)=\{g\in G:gHg^{-1}=H\}.
$$
If it is all of $G$, the equality $gHg^{-1}=H$ holds for every $g\in G$, which is precisely the definition of a <normal subgroup>.
If instead $\operatorname{Stab}(H)=H$, the <orbit-stabilizer theorem> says that the <group orbit> of $H$ has size $[G:H]$. Every conjugate $gHg^{-1}$ is isomorphic to $H$, with <group isomorphism> $h\mapsto ghg^{-1}$, so this <group orbit> is a <subset> of $\Omega$. Therefore
$$
\boxed{|\Omega|\geq[G:H].}
$$
The <group action> need not be transitive on all of $\Omega$; this one <group orbit> supplies the lower bound.
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