Dihedral Groups and Symmetry

In this lesson you’ll learn what the dihedral group $D_n$ is, how to compute in it using generators and relations, why reflections don’t commute with rotations, and how the same ideas describe symmetry in three dimensions.

A symmetry of a figure is a motion that leaves it looking unchanged. Symmetries always form a group: doing nothing is the identity, any symmetry can be reversed, and one symmetry after another is again a symmetry.

For a regular $n$-gon there are exactly $2n$ of them, and they make up the dihedral group $D_n$:

• $n$ rotations, by multiples of $360°/n$.
• $n$ reflections, across lines through the centre.

$$|D_n| = 2n$$

A warning on notation: some books write $D_{2n}$ for this group, emphasizing its order rather than the polygon’s side count. This site uses $D_n$ for the $n$-gon, so $|D_4| = 8$.

For a square, the reflection axes come in two flavours: two through opposite edge midpoints, and two through opposite corners. For odd $n$ every axis passes through one vertex and the opposite edge’s midpoint, so all $n$ reflections look alike. That asymmetry between even and odd $n$ shows up repeatedly.

Every element of $D_n$ is built from two moves: one rotation $r$ and one reflection $s$. Every element can be written as

$$r^k \quad \text{or} \quad r^k s, \qquad k = 0, 1, \dots, n-1$$

That’s $n + n = 2n$ elements, as promised. Three relations govern all computation:

$$r^n = e, \qquad s^2 = e, \qquad srs = r^{-1}$$

The last one is the important one, usually written $sr = r^{-1}s$. It says a reflection turns a rotation into its opposite, which is why $D_n$ is non-abelian for $n \ge 3$.

Physically this is obvious once you try it. Flip a square, then rotate it clockwise; that’s the same as rotating counter-clockwise first and then flipping. Reflecting reverses your sense of “clockwise.”

Writing a group as generators plus relations is called a presentation:

$$D_n = \langle r, s \mid r^n = s^2 = e,\ srs = r^{-1} \rangle$$

This is a complete rulebook. Any product of $r$‘s and $s$‘s can be pushed into the standard form $r^k$ or $r^k s$ by repeatedly applying the relations, so you never need the multiplication table.

The move that matters: whenever you see $sr$, rewrite it as $r^{-1}s$. That pushes every $s$ to the right end. For example in $D_5$:

$$(r^2 s)(r^3 s) = r^2 (s r^3) s = r^2 (r^{-3} s) s = r^{-1} s^2 = r^{-1} = r^4$$

The step $sr^3 = r^{-3}s$ follows from applying $sr = r^{-1}s$ three times.

Two useful facts fall out. Every element of the form $r^k s$ has order 2, because

$$(r^k s)(r^k s) = r^k (s r^k) s = r^k r^{-k} s^2 = e$$

So all $n$ reflections are self-inverse, which matches the intuition that flipping twice does nothing. And the rotations form a cyclic subgroup $\langle r \rangle \cong \mathbb{Z}_n$ of index 2.

The same reasoning works on solids, where the groups get bigger and more interesting.

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For the cube, count the rotations by choosing which face points up (6 ways) and then how it is spun about the vertical axis (4 ways): $6 \times 4 = 24$ rotations. Remarkably this group is isomorphic to $S_4$, because each rotation permutes the cube’s four long diagonals, and every permutation of those four diagonals is achievable.

Including reflections gives 48 symmetries, the full symmetry group of the cube. The distinction between rotations only (proper symmetries) and rotations plus reflections matters in chemistry, where a molecule and its mirror image can behave very differently.

All rotations of 3D space, not just those preserving a particular solid, form the infinite non-abelian group $SO(3)$. Every symmetry group of a solid is a finite subgroup of it, and there is a complete classification of those subgroups: the cyclic groups, the dihedral groups, and exactly three exceptional ones from the tetrahedron, cube/octahedron, and dodecahedron/icosahedron.

Example 1: List the elements of $D_3$.

Solution. $|D_3| = 6$. Rotations: $e$, $r$ (120°), $r^2$ (240°). Reflections: $s$, $rs$, $r^2s$, one through each vertex.

$$D_3 = \{e,\ r,\ r^2,\ s,\ rs,\ r^2s\}$$

$D_3 \cong S_3$: the symmetries of a triangle are exactly the permutations of its three vertices, and both groups have order 6. This is the only $n$ for which $D_n$ and $S_n$ coincide, since $|D_n| = 2n$ and $|S_n| = n!$ agree only at $n = 3$.

Example 2: Simplify $s r^2 s r$ in $D_4$.

Solution. Push each $s$ rightwards using $sr = r^{-1}s$.

$$s r^2 s r = (s r^2) s r = (r^{-2} s) s r = r^{-2} s^2 r = r^{-2} r = r^{-1} = r^3$$

The answer is $r^3$, a 270° rotation. Sanity check: the expression contains two reflections, and two reflections always compose to a rotation, so a rotation is the right kind of answer.

Example 3: Find the order of every element of $D_4$.

Solution.

• $e$: order 1.
• $r$: order 4, since $r^4 = e$ and no smaller power is $e$.
• $r^2$: order 2, since $(r^2)^2 = r^4 = e$.
• $r^3$: order 4.
• $s$, $rs$, $r^2s$, $r^3s$: order 2 each, by the general fact that $(r^ks)^2 = e$.

So $D_4$ has one element of order 1, five of order 2, and two of order 4. Note every order divides 8, as Lagrange requires, but there is no element of order 8, so $D_4$ is not cyclic.

Compare $\mathbb{Z}_8$, which also has order 8 but has an element of order 8. Both have order 8; they are different groups. So do $\mathbb{Z}_4 \times \mathbb{Z}_2$, $\mathbb{Z}_2^3$, and the quaternion group $Q_8$ - five groups of order 8 in total.

Example 4: Is $D_4$ abelian?

Solution. No. Take $r$ and $s$:

$$rs \quad \text{versus} \quad sr = r^{-1}s = r^3 s$$

These differ because $r \ne r^3$ in $D_4$. Non-abelian.

$D_1$ has order 2 and $D_2$ has order 4, and both are abelian. From $n = 3$ upward, $r \ne r^{-1}$ and the group is non-abelian.

Example 5: The center of $D_4$.

Solution. The center is the elements commuting with everything.

$r$ fails, since $rs \ne sr$. Any reflection $r^ks$ fails, since it doesn’t commute with $r$ by the same relation.

Test $r^2$: it must commute with $s$. Using the relation twice, $sr^2 = r^{-2}s = r^2s$ in $D_4$ because $r^{-2} = r^2$ when $r^4 = e$. So $r^2$ does commute with $s$, and it commutes with all rotations automatically.

$$Z(D_4) = \{e, r^2\}$$

The 180° rotation is central because it is its own inverse, so the reflection has nothing to reverse. For odd $n$, $Z(D_n) = \{e\}$, since there is no half-turn.

Example 6: Subgroups of $D_4$.

Solution. By Lagrange, orders can only be 1, 2, 4, 8.

• Order 1: $\{e\}$.
• Order 2: $\{e, r^2\}$, and $\{e, x\}$ for each of the four reflections. Five of them.
• Order 4: $\langle r \rangle = \{e,r,r^2,r^3\}$; $\{e, r^2, s, r^2s\}$; $\{e, r^2, rs, r^3s\}$. Three, and the last two are Klein four-groups.
• Order 8: $D_4$ itself.

Ten subgroups in all. Contrast $\mathbb{Z}_8$, which has exactly four. Cyclic groups have one subgroup per divisor; other groups can have many more, and that difference is a good measure of how much structure a group has.

Crystallography and materials. A crystal’s rotational symmetry must be compatible with a repeating lattice, and that constraint permits only 2-, 3-, 4- and 6-fold rotation axes. Five-fold symmetry is impossible in a periodic crystal, which is why the 1984 discovery of quasicrystals with 5-fold diffraction patterns was so startling and eventually earned a Nobel Prize.

Chemistry. Molecular point groups are dihedral or related groups, and they determine which vibrational modes absorb infrared light. Benzene has $D_{6h}$ symmetry, and its spectrum is a direct readout of that group’s structure. Whether a molecule is chiral, meaning distinguishable from its mirror image, is exactly the question of whether its symmetry group contains any reflection, which has direct pharmacological consequences.

Computer graphics and robotics. Composing rotations is group multiplication in $SO(3)$, and because it’s non-abelian, applying rotations in a different order gives a different result. Unit quaternions provide a double cover of $SO(3)$ and avoid the gimbal-lock degeneracies of Euler angles, which is why nearly every game engine and spacecraft attitude controller uses them.

Architecture and design. Rose windows, tiling patterns and logos are built from dihedral symmetry, and the group tells you exactly how many distinct motifs you need to draw: one, repeated by the group action. Islamic geometric art systematically explores wallpaper groups, of which there are exactly 17.

Physics. Conservation laws come from continuous symmetry groups by Noether’s theorem, while discrete symmetries govern selection rules. The observed violation of parity symmetry in the weak interaction was a genuine shock precisely because reflection had been assumed to be a symmetry of nature.