Dihedral group ($D_n$)

The dihedral group $D_{2n}$ is generated by a rotation $r$ of order $n$ and a reflection $s$ with $srs=r^{-1}$; it has order $2n$.

Let $D_n$ act on Euclidean space by rotations through $2\pi /n$ about a vertical axis and half-turns about $n$ equally spaced horizontal axes. A round solid torus about the vertical axis, drilled through radially along the $2n$ places where the horizontal axes cross its tube, has a smooth boundary of genus $2n+1$ after equivariant rounding of its rims. All nonidentity rotation axes lie in the complement of this boundary, so the induced group action is a free action of a group. Each bore replaces two closed discs by an annulus, lowering Euler characteristic by two and increasing genus by one. The boundary orientation is preserved; its induced Riemann surface has an unramified quotient space of genus two by the Riemann-Hurwitz formula.

Take $r$ as a counterclockwise quarter-turn and $s$ as reflection across the horizontal axis. The eight symmetries of a square are $e,r,r^2,r^3,s,rs,r^{2}s,r^{3}s$. The last four are reflections across the horizontal axis, the diagonal $y=x$, the vertical axis and the diagonal $y=-x$, respectively. This is a dihedral group of order eight; the subscript convention $D_4$ counts polygon vertices, while some authors use $D_8$ for the same group.

In the order of the conjugacy classes of the symmetry group of a square displayed there, its irreducible complex character table isThe four one-dimensional representations assign independent signs to $r$ and $s$. The remaining row is the character of the ordinary planar representation: a rotation has trace $2\cos \theta$ and a reflection has trace zero. Weighted by class sizes, these rows satisfy character orthogonality. Their squared dimensions sum to eight, so they exhaust the irreducible representations.

The displayed conjugacy classes have sizes one, one, two, two and two. The identity and half-turn are central elements. Reflection sends $r$ to $r^{-1}$, pairing the two quarter-turns. Conjugating a reflection by $r$ sends $r^{k}s$ to $r^{k+2}s$; conjugating by $s$ sends it to $r^{-k}s$. Thus the parity of $k$ is invariant under conjugation, separating the axis reflections from the diagonal reflections, and each displayed pair is one conjugacy class.

Use $D_n$ for the group of order $2n$ with rotation $R$ and reflection $m$. Every element is $R^j$ or $R^{j}m$, $0\le j<n$. The rotational conjugacy classes are the sets $\{R^j,R^{-j}\}$, reduced to singletons for central rotations. For odd $n$, all reflections form one class, giving $(n+3)/2$ classes in total. For even $n$, the identity and $R^{n/2}$ are central singletons, the other rotations form pairs, and the reflections split into even and odd powers of $R$, giving $n/2+3$ classes. Indeed conjugation by $R^s$ changes a reflection exponent by $2s$, while conjugation by $m$ negates it. For odd prime $n$, a normal subgroup containing a reflection must contain all reflections and hence the whole group; the proper normal subgroups are therefore only the identity subgroup and the rotation subgroup.

Let $r$ have order $2m$ with $m$ odd, and let $s^2=1$, $srs=r^{-1}$. The subgroup $H=⟨r^2,s⟩$ is a dihedral group of order $2m$, while $K=⟨r^m⟩$ is a central cyclic group of order two. Since $m$ is odd, $H\cap K=\{1\}$ and $r$ lies in $HK$ by Bézout's identity. Thus the original group of order $4m$ is the direct product of groups $H\times K$. This statement specifies group orders and avoids the two competing conventions for dihedral notation.

Write $D_{2n}=⟨r,s:r^n=s^2=1,srs=r^{-1}⟩$. A subgroup contained in the rotations is a cyclic subgroup $⟨r^d⟩$, with $d\mid n$. If it contains a reflection, every other reflection differs from a chosen $r^{j}s$ by a member of the rotational intersection, so it is $⟨r^d,r^{j}s⟩$, with $j$ specified modulo $d$. These subgroups have orders $n/d$ and $2n/d$ respectively. This gives a complete list, including the trivial and whole subgroups; see Peter Cameron's group-theory exercise solutions, question1.

Write the dihedral group as $D_{2n}=⟨r,s:r^n=s^2=e,srs=r^{-1}⟩$. If $k\mid n$, the cyclic subgroup $⟨r^{n/k}⟩$ has order $k$. For an even divisor $k=2m$ of $2n$, $m\mid n$ and $⟨r^{n/m},s⟩$ has $m$ rotations and $m$ reflections, hence order $k$. An odd divisor of $2n$ always divides $n$. This proves that the necessary divisibility condition from Lagrange's theorem is also sufficient in a dihedral group, although it is not sufficient in arbitrary finite groups.

For the dihedral group of a regular $2n$-gon, $n\ge 2$, use $r^{2n}=s^2=1$ and $srs=r^{-1}$. Its involutions are the central half-turn $r^n$ and all reflections $s{r}^j$. The half-turn has singleton conjugacy class and normal closure $\{1,r^n\}$. Conjugating a reflection changes its exponent by an even integer or negates it, so its conjugacy class is $\{s{r}^{j+2k}:0\le k<n\}$. Products of consecutive reflections generate $r^2$, giving normal closure $⟨r^2,s{r}^j⟩$, of order $2n$ and index two. The two reflection classes are distinguished by exponent parity.