Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 9

The relation $mR=R^{-1}m$ moves every reflection factor to the right of every rotation factor. Canceling pairs $m^2=I$ and reducing rotation powers modulo $n$ shows that every word has one of the formsWithin either family the elements are distinct, since equality would imply equality of two powers of $R$, which has order $n$. No rotation equals a reflection: in the polygon action, rotations preserve planar orientation and reflections reverse it. Thus this complete list has exactly $2n$ elements.

Geometrically, $R$ rotates the polygon by $2\pi /n$ about its centre, and $m$ reflects it in a symmetry axis through the centre. The elements $R^j$ are all rotations; $R^{j}m$ are all reflections. We use the paper's convention that $D_n$ has order $2n$, since another common convention calls this same group $D_{2n}$.

Rotations commute with one another, while reflection conjugation sends $R^j$ to $R^{-j}$. Thus the rotational conjugacy classes are pairs $\{R^j,R^{-j}\}$, except when the two entries coincide. For reflections, direct use of the relations givesIf $n$ is odd, multiplication by two permutes residues modulo $n$, so all reflections are conjugate. The complete class list isTheir sizes are $1$, then $(n-1)/2$ classes of size two, then one class of size $n$, giving $(n+3)/2$ classes and $2n$ elements in total.

If $n$ is even, the half-turn $R^{n/2}$ is central and forms its own singleton. Conjugation changes a reflection exponent by an even amount or negates it, so it preserves parity and reaches every exponent of that parity. The list isThere are $n/2+3$ classes; their sizes are $1,1$, then $n/2-1$ pairs, then two classes of size $n/2$. Geometrically the two reflection classes correspond to vertex-axis and edge-axis reflections, in either order depending on the choice of $m$. These are the conjugacy classes of a finite dihedral group.

Let $n=p\ge 3$ be prime. Any subgroup contained in $⟨R⟩\cong C_p$ 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 $mR{m}^{-1}=R^{-1}$.

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, isThis 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.