= Solution
Let $n=p\geq3$ be prime. Any <subgroup> contained in $\langle R\rangle\cong C_p$ is either $\{I\}$ or the whole rotation <subgroup>. Both are normal: the trivial <subgroup> always is, and the rotation <subgroup> has index two, or equivalently is preserved by $mRm^{-1}=R^{-1}$.
If a <normal subgroup> contains any reflection, it contains its entire <conjugacy class>. Part (b) shows that this is every reflection for odd $p$. Products of reflections include a generator of the rotations, since $(Rm)m=R$. Such a <normal subgroup> must therefore be the whole dihedral group.
Thus the complete list of proper <normal subgroups>, including the trivial <subgroup>, is
$$
\boxed{\{I\}\quad\text{and}\quad\langle R\rangle=\{I,R,\ldots,R^{p-1}\}.}
$$
This also agrees with <classification of subgroups of a dihedral group>: the reflection <subgroups> of order two are proper but are not normal when $p$ is odd.
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