Group of symmetries of a regular polygon

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The following content provides a detailed exposition on the structure and characteristics of the dihedral group. For those who might find themselves adrift in the vast ocean of abstract algebra, this serves as a rather direct, if somewhat unenthusiastic, guide.

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Group theory

Basic notions

• Subgroup
• Normal subgroup
• Group action
• Quotient group
• (Semi-)direct product
• Direct sum
• Free product
• Wreath product

Group homomorphisms

• kernel
• image

Types

• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable

Lists

• Glossary of group theory
• List of group theory topics

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Finite groups

• Cyclic group Z n
• Symmetric group S n
• Alternating group A n
• Dihedral group D n
• Quaternion group Q

Theorems

• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p -group
• Elementary abelian group
• Frobenius group
• Schur multiplier

Classification of finite simple groups

• cyclic
• alternating
• Lie type
• sporadic

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Discrete groups and Lattices

• Integers (${\displaystyle \mathbb {Z} }$)
• Free group

Modular groups

• PSL(2, ${\displaystyle \mathbb {Z} }$)
• SL(2, ${\displaystyle \mathbb {Z} }$)
• Arithmetic group
• Lattice
• Hyperbolic group

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Topological and Lie groups

• Solenoid
• Circle
• General linear GL( n )
• Special linear SL( n )
• Orthogonal O( n )
• Euclidean E( n )
• Special orthogonal SO( n )
• Unitary U( n )
• Special unitary SU( n )
• Symplectic Sp( n )
• G 2
• F 4
• E 6
• E 7
• E 8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop

Infinite dimensional Lie group

• O(∞)
• SU(∞)
• Sp(∞)

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Algebraic groups

• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve

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In mathematics, specifically within the realm of group theory, a dihedral group isn't some esoteric construct for the initiated. It is, quite simply, the group that describes all the symmetries of a regular polygon — an object whose sides and angles are all equal. This includes not only the obvious rotations but also the more subtle reflections. These groups, despite their apparent simplicity, are rather foundational. They represent some of the most straightforward examples of finite groups and, as such, are surprisingly prevalent in various fields, from pure group theory and geometry to the more applied domain of chemistry. One might even say they're inescapable, like a bad habit.

Now, because nothing in mathematics can ever be entirely straightforward, the notation used to denote these dihedral groups inconveniently varies between geometry and abstract algebra. In geometry, one typically encounters D n or Dih n when referring to the symmetries of an n -gon, which, to be clear, is a group of order 2 n . However, in the more abstract corners of algebra, the very same group is often denoted D 2 n . This article, in a concession to geometric clarity, will adhere to the geometric convention of D n . Let's not make things more complicated than they already are.

Definition

The term "dihedral" itself is derived from the Greek elements "di-" (meaning "two") and "-hedron" (from hédra, meaning "face of a geometrical solid"). So, literally, it refers to something with "two faces." In the context of polygons, this implies the two distinct surfaces of the shape, or more accurately, the two sides from which one might view its symmetries.

Elements

Consider a regular polygon with $n$ sides. One might assume its symmetries are straightforward, but they are more numerous than a casual glance suggests. Such a polygon possesses precisely $2n$ distinct symmetries, assuming, of course, that $n \geq 3$. These comprise $n$ rotational symmetries and an equal number of $n$ reflection symmetries. These specific rotations and reflections are the very elements that constitute the dihedral group denoted as ${\displaystyle \mathrm {D} _{n}}$.

The nature of these reflection symmetries slightly shifts depending on the parity of $n$. If $n$ is an odd number, each axis of symmetry elegantly bisects one side and simultaneously passes through the opposite vertex. It's a rather neat arrangement. If, however, $n$ is even, the axes of symmetry diverge into two distinct types: there are $n/2$ axes that connect the midpoints of opposing sides, and another $n/2$ axes that pass directly through opposite vertices. Regardless of $n$'s parity, the total count remains $n$ axes of symmetry, always leading to $2n$ elements within the full symmetry group. A rather crucial property to note is that performing one reflection about an axis of symmetry, and then immediately following it with another reflection about a different axis, invariably results in a rotation. The angle of this resultant rotation will always be exactly twice the angle separating the two initial reflection axes.

To illustrate this concept, one might consider the familiar stop sign, which, mathematically speaking, is a regular octagon. The image provided demonstrates the profound effect of the sixteen elements that make up ${\displaystyle \mathrm {D} _{8}}$ on such an object. The first row meticulously displays the outcome of the eight distinct rotations, while the second row showcases the consequence of the eight reflections. Each operation begins with the stop sign oriented exactly as it appears in the top-left corner, emphasizing the transformative power of these group elements.

Group structure

As is the case with any well-behaved geometric object, the composition of any two symmetries of a regular polygon will, without fail, yield yet another symmetry of that very same object. This inherent closure, when combined with the operation of composing symmetries to produce new ones, bestows upon the symmetries of a polygon the elegant algebraic structure of a finite group. It's almost as if they were designed for it.

Crucially, when considering the composition of symmetries, it's important to understand that the lines of reflection, typically labeled S 0 , S 1 , and S 2 in illustrative examples, are fixed in space. They do not shift or move as the symmetry operation (be it a rotation or a reflection) is applied to the polygon itself. This distinction becomes paramount when one attempts to perform successive compositions of symmetries, ensuring that the underlying reference frame remains consistent.

The subsequent Cayley table serves as an explicit demonstration of the effect of composition within the dihedral group of order 6, more commonly known as ${\displaystyle \mathrm {D} _{3}}$ — which, for the uninitiated, represents the symmetries of an equilateral triangle. Here, the element ${\displaystyle \mathrm {r} _{0}}$ represents the identity, meaning no change at all. ${\displaystyle \mathrm {r} _{1}}$ and ${\displaystyle \mathrm {r} _{2}}$ denote precise counterclockwise rotations of 120° and 240°, respectively. The elements ${\displaystyle \mathrm {s} _{0}}$, ${\displaystyle \mathrm {s} _{1}}$, and ${\displaystyle \mathrm {s} _{2}}$ correspond to reflections across the three distinct lines of symmetry, as depicted in the accompanying visual aid.

	r0	r1	r2	s0	s1	s2
r0	r0	r1	r2	s0	s1	s2
r1	r1	r2	r0	s1	s2	s0
r2	r2	r0	r1	s2	s0	s1
s0	s0	s2	s1	r0	r2	r1
s1	s1	s0	s2	r1	r0	r2
s2	s2	s1	s0	r2	r1	r0

For example, observe the entry ${\displaystyle \mathrm {s} _{2}\mathrm {s} _{1}=\mathrm {r} _{1}}$. This is not merely an arbitrary assignment but a concrete result: performing the reflection ${\displaystyle \mathrm {s} _{1}}$ first, and then applying the reflection ${\displaystyle \mathrm {s} _{2}}$, is mathematically equivalent to a single rotation of 120°. A crucial, if sometimes overlooked, convention is that the order of elements in these composition operations proceeds from right to left. This reflects the standard mathematical convention where an element is understood to act upon the expression immediately to its right. Furthermore, it quickly becomes apparent that this composition operation is decidedly not commutative. That is, the order in which you apply these symmetries fundamentally matters, a property that distinguishes non-abelian groups.

More generally, the dihedral group ${\displaystyle \mathrm {D} _{n}}$ is composed of elements typically designated as ${\displaystyle r_{0},\dots ,r_{n-1}}$ (representing rotations) and ${\displaystyle s_{0},\dots ,s_{n-1}}$ (representing reflections). Their composition is governed by a set of well-defined formulae:

${\displaystyle {\begin{aligned}\mathrm {r} _{i}\,\mathrm {r} _{j}=\mathrm {r} _{i+j},&\qquad \mathrm {r} _{i}\,\mathrm {s} _{j}=\mathrm {s} _{i+j},\\\mathrm {s} _{i}\,\mathrm {r} _{j}=\mathrm {s} _{i-j},&\qquad \mathrm {s} _{i}\,\mathrm {s} _{j}=\mathrm {r} _{i-j}.\end{aligned}}}$

It is imperative to note that in all these operations, the addition and subtraction of the subscripts ($i$ and $j$) are performed using modular arithmetic with a modulus of $n$. This ensures that the resulting subscript always falls within the valid range of 0 to $n-1$, effectively wrapping around the polygon's sides.

Matrix representation

For those who appreciate precision and a more concrete visualization, dihedral groups offer a rather elegant matrix representation. By centering the regular polygon precisely at the origin of a Cartesian coordinate system, the elements of the dihedral group can be understood as linear transformations of the plane. This allows each element of ${\displaystyle \mathrm {D} _{n}}$ to be represented by a matrix, where the composition of symmetries translates directly into matrix multiplication. This is a quintessential example of a two-dimensional group representation, providing a powerful tool for analyzing the group's actions.

Consider, for instance, the elements of the dihedral group of order 8, denoted as ${\displaystyle \mathrm {D} _{4}}$ — which, as you might recall, is the group symmetry of a square. These eight distinct elements can be represented by the following eight matrices, offering a clear algebraic mapping of their geometric actions:

${\displaystyle {\begin{matrix}\mathrm {r} _{0}=\left({\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}}\right),&\mathrm {r} _{1}=\left({\begin{smallmatrix}0&-1\\[0.2em]1&0\end{smallmatrix}}\right),&\mathrm {r} _{2}=\left({\begin{smallmatrix}-1&0\\[0.2em]0&-1\end{smallmatrix}}\right),&\mathrm {r} _{3}=\left({\begin{smallmatrix}0&1\\[0.2em]-1&0\end{smallmatrix}}\right),\\[1em]\mathrm {s} _{0}=\left({\begin{smallmatrix}1&0\\[0.2em]0&-1\end{smallmatrix}}\right),&\mathrm {s} _{1}=\left({\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}}\right),&\mathrm {s} _{2}=\left({\begin{smallmatrix}-1&0\\[0.2em]0&1\end{smallmatrix}}\right),&\mathrm {s} _{3}=\left({\begin{smallmatrix}0&-1\\[0.2em]-1&0\end{smallmatrix}}\right).\end{matrix}}}$

In this specific arrangement, these matrices effectively represent the symmetries of a square that is both axis-aligned and centered at the origin. They operate on the plane through multiplication with column vectors of coordinates, typically expressed as ${\textstyle {\bigl (}{\begin{smallmatrix}x\\y\end{smallmatrix}}{\bigr )}}$. The matrix ${\displaystyle \mathrm {r} _{0}}$ is the identity element, leaving the square unchanged. The matrices ${\displaystyle \mathrm {s} _{0}}$ and ${\displaystyle \mathrm {s} _{2}}$ correspond to reflections across the horizontal and vertical axes, respectively. Meanwhile, ${\displaystyle \mathrm {s} _{1}}$ and ${\displaystyle \mathrm {s} _{3}}$ represent reflections across the main diagonals of the square. The remaining elements, ${\displaystyle \mathrm {r} _{1}}$, ${\displaystyle \mathrm {r} _{2}}$, and ${\displaystyle \mathrm {r} _{3}}$, are the distinct rotations around the center, moving the square without flipping it.

r0	r1	r2	r3
The square's initial position	Rotation by 270°	Rotation by 180°	Rotation by 90° anticlockwise

s0	s1	s2	s3
Horizontal reflection	Vertical reflection	Diagonal NW–SE reflection	Diagonal NE–SW reflection

In a more general formulation, the matrices corresponding to the elements of the dihedral group ${\displaystyle \mathrm {D} _{n}}$ take on the following elegant, trigonometric form:

${\displaystyle {\begin{aligned}\mathrm {r} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&-\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&\cos {\frac {2\pi k}{n}}\end{pmatrix}}\ \ {\text{and}}\\[5pt]\mathrm {s} _{k}&={\begin{pmatrix}\cos {\frac {2\pi k}{n}}&\sin {\frac {2\pi k}{n}}\\\sin {\frac {2\pi k}{n}}&-\cos {\frac {2\pi k}{n}}\end{pmatrix}}.\end{aligned}}}$

Here, the element ${\displaystyle \mathrm {r} _{k}}$ is a classic rotation matrix, meticulously expressing a counterclockwise rotation by an angle of ${\displaystyle 2\pi k/n}$ radians. Conversely, the element ${\displaystyle \mathrm {s} _{k}}$ represents a reflection across a line that forms an angle of ${\displaystyle \pi k/n}$ radians with the positive $x$-axis. These generalized forms are crucial for understanding how dihedral groups scale and behave for any regular $n$-gon.

Other definitions

For those who appreciate a more abstract, structural perspective, ${\displaystyle \mathrm {D} _{n}}$ can be precisely defined as the semidirect product of the cyclic group ${\displaystyle \mathrm {C} _{2}=\{1,s\}}$ acting upon the cyclic group ${\displaystyle \mathrm {C} _{n}}$. This action is achieved through the automorphism ${\displaystyle \varphi _{s}(r)=r^{-1}}$, which essentially means that the reflection element $s$ inverts the rotational element $r$.

Consequently, this structure leads to the following standard presentation for ${\displaystyle \mathrm {D} _{n}}$:

${\displaystyle {\begin{aligned}\mathrm {D} _{n}&=\left\langle r,s\mid \operatorname {ord} (r)=n,\operatorname {ord} (s)=2,srs^{-1}=r^{-1}\right\rangle \\&=\left\langle r,s\mid \operatorname {ord} (r)=n,\operatorname {ord} (s)=2,srs=r^{-1}\right\rangle \\&=\left\langle r,s\mid r^{n}=s^{2}=(sr)^{2}=1\right\rangle .\end{aligned}}}$

Here, $r$ represents a generator for the rotational symmetries (an element of order $n$), and $s$ represents a generator for a reflection symmetry (an element of order 2). The relation ${\displaystyle srs^{-1}=r^{-1}}$ (or equivalently, ${\displaystyle srs=r^{-1}}$ since $s^{-1}=s$) captures the interaction between these two types of symmetries: a reflection followed by a rotation followed by another reflection is equivalent to the inverse of the original rotation. The final line, ${\displaystyle r^{n}=s^{2}=(sr)^{2}=1}$, provides a concise set of defining relations, stating that $n$ rotations return to identity, two reflections return to identity, and a rotation followed by a reflection, then by another rotation, also results in identity.

Utilizing the relation ${\displaystyle s^{2}=1}$, which implies $s = s^{-1}$, we can also derive the relationship ${\displaystyle r=s\cdot sr}$. This implies that ${\displaystyle \mathrm {D} _{n}}$ can also be generated by two elements $s$ and $t:=sr$. This substitution then reveals an alternative, equally valid presentation for ${\displaystyle \mathrm {D} _{n}}$:

${\displaystyle \mathrm {D} _{n}=\left\langle s,t\mid s^{2}=1,t^{2}=1,(st)^{n}=1\right\rangle .}$

This particular form is rather significant, as it highlights that ${\displaystyle \mathrm {D} _{n}}$ inherently belongs to the class of Coxeter groups. These groups are defined by generators and relations where all relations are of the form $(g_i g_j)^{k_{ij}}=1$, which makes them a fundamental object of study in geometric group theory.

Small dihedral groups

When discussing dihedral groups, it's customary to start with $n \geq 3$ for polygons. However, the definitions can be extended to smaller values, though they become somewhat "degenerate" and exhibit unique characteristics.

Cycle graphs
D 1 = Z 2	D 2 = Z 2 2 = K 4	D 3	D 4	D 5
D 6 = D 3 × Z 2	D 7	D 8	D 9	D 10 = D 5 × Z 2

D 3 = S 3

D 4

An example illustrating subgroups derived from a hexagonal dihedral symmetry might be useful here, but the diagram is omitted from the original.

Let's examine the "small" cases, which are, frankly, a bit peculiar:

• D 1 is isomorphic to Z 2, the cyclic group of order 2. This represents the symmetries of a "1-gon" – essentially a line segment, which has one rotation (the identity) and one reflection. Not terribly exciting, but mathematically consistent.
• D 2 is isomorphic to K 4, the Klein four-group. This corresponds to the symmetries of a "2-gon" or a line segment, with two rotations (0° and 180°) and two reflections. It's the smallest non-cyclic group, and quite distinct from its higher-order siblings.

These two small dihedral groups are exceptional, standing apart from the others in a few key ways:

• D 1 and D 2 are the only abelian dihedral groups. For any $n \geq 3$, the dihedral group ${\displaystyle \mathrm {D} _{n}}$ is decidedly non-abelian, meaning the order of operations matters. This is a fundamental distinction.
• For $n \geq 3$, ${\displaystyle \mathrm {D} _{n}}$ is a subgroup of the symmetric group S n. However, for $n=1$ or $n=2$, this relationship breaks down. The order of D n (which is $2n$) is greater than $n!$ for these values ($2 \times 1 = 2 > 1! = 1$ for $n=1$, and $2 \times 2 = 4 > 2! = 2$ for $n=2$). Therefore, D n is simply too large to be a subgroup of S n in these specific instances.
• The inner automorphism group of D 2 is trivial, meaning it contains only the identity automorphism. For all other even values of $n$, this inner automorphism group is isomorphic to D n / Z 2. This subtle point reveals how D 2 behaves differently in terms of its internal structure.

The visual representations known as cycle graphs for dihedral groups present a rather consistent pattern. They are characterized by an $n$-element cycle, which embodies the rotational symmetries, and $n$ distinct 2-element cycles, which represent the individual reflection symmetries. In these graphs, the darkened vertex invariably denotes the identity element of the group, with all other vertices corresponding to the remaining elements. Each cycle is formed by the successive powers of any element connected directly to the identity element, offering a clear diagrammatic representation of the group's structure.

The dihedral group as symmetry group in 2D and rotation group in 3D

The abstract group D n, in its most intuitive and commonly visualized form, manifests as the group of Euclidean plane isometries that maintain a fixed origin. These groups are not merely theoretical constructs; they form one of the two fundamental series of discrete point groups in two dimensions. Specifically, D n encompasses $n$ distinct rotations, each a multiple of $360^\circ/n$ around the central origin. Additionally, it includes $n$ distinct reflections across lines that all pass through the origin, with these lines making angles that are multiples of $180^\circ/n$ with respect to each other. This collection of operations precisely defines the symmetry group of a regular polygon with $n$ sides. While this is most commonly considered for $n \geq 3$, the concept extends, albeit with some degeneracy, to the cases where $n=1$ (a plane with a single point offset from the "center" of a "1-gon") and $n=2$ (a "2-gon" or line segment).

Formally, D n is elegantly generated by a rotation $r$ of order $n$ and a reflection $s$ of order 2. These two generators are linked by a crucial relation:

${\displaystyle \mathrm {srs} =\mathrm {r} ^{-1}\,}$

Geometrically, this relation is rather profound and intuitive: when you observe a rotation through a mirror (a reflection), what you perceive is an inverse rotation. The direction of rotation is reversed.

In the language of complex numbers, this translates to multiplication by ${\displaystyle e^{2\pi i \over n}}$ for the rotation and complex conjugation for the reflection. This offers an alternative, yet equally valid, perspective on the group's operations.

When expressed in matrix form, we can define the fundamental rotation and reflection as:

${\displaystyle \mathrm {r} _{1}={\begin{bmatrix}\cos {2\pi \over n}&-\sin {2\pi \over n}\\[4pt]\sin {2\pi \over n}&\cos {2\pi \over n}\end{bmatrix}}\qquad \mathrm {s} _{0}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}}$

From these base elements, we can then define all other rotations as ${\displaystyle \mathrm {r} _{j}=\mathrm {r} _{1}^{j}}$ and all other reflections as ${\displaystyle \mathrm {s} _{j}=\mathrm {r} _{j}\,\mathrm {s} _{0}}$ for ${\displaystyle j\in \{1,\ldots ,n-1\}}$. With these definitions in place, the product rules for D n can be concisely written as:

${\displaystyle {\begin{aligned}\mathrm {r} _{j}\,\mathrm {r} _{k}&=\mathrm {r} _{(j+k){\text{ mod }}n}\\\mathrm {r} _{j}\,\mathrm {s} _{k}&=\mathrm {s} _{(j+k){\text{ mod }}n}\\\mathrm {s} _{j}\,\mathrm {r} _{k}&=\mathrm {s} _{(j-k){\text{ mod }}n}\\\mathrm {s} _{j}\,\mathrm {s} _{k}&=\mathrm {r} _{(j-k){\text{ mod }}n}\end{aligned}}}$

These rules, as noted previously, are simply a more formal expression of the composition table, ensuring that all operations are performed using modular arithmetic modulo $n$. (For further context, one might compare these operations to the broader topic of coordinate rotations and reflections.)

For the specific case of the dihedral group D 2, it is generated by a rotation $r$ of 180 degrees and a reflection $s$ across the $x$-axis. The elements of D 2 can then be enumerated as $\{e, r, s, rs\}$, where $e$ is the identity or null transformation, and $rs$ represents a reflection across the $y$-axis.

The four elements of D 2 (x-axis is vertical here)

As previously mentioned, D 2 is isomorphic to the Klein four-group, a small but significant abelian group.

It is for $n > 2$ that the operations of rotation and reflection generally cease to commute, rendering D n a non-abelian group. Take D 4 as a prime example: a 90-degree rotation followed by a reflection yields a distinctly different result than a reflection followed by that same 90-degree rotation. The order of operations, in these cases, is not merely a suggestion; it's a fundamental determinant of the outcome.

D 4 is nonabelian (x-axis is vertical here).

Thus, beyond their rather obvious utility in analyzing symmetry problems within the plane, these groups serve a crucial pedagogical role. They are among the simplest, most accessible examples of non-abelian groups, and as such, they frequently appear as convenient counterexamples to theorems that, unfortunately for some, are exclusively restricted to abelian groups. They quickly disabuse one of the notion that all groups behave as nicely as the cyclic group or the integers under addition.

The $2n$ distinct elements of D n can be systematically listed as $e, r, r^2, \ldots, r^{n-1}$ for the rotations, and $s, rs, r^2s, \ldots, r^{n-1}s$ for the axis-reflections. It's worth noting that all these reflection elements inherently have an order of 2, meaning applying them twice returns the object to its original state. A simple rule of thumb for composition is this: the product of two rotations or two reflections will always result in a rotation. Conversely, the product of a rotation and a reflection will invariably yield another reflection.

Up to this point, our discussion of D n has primarily positioned it as a subgroup of O(2), which is the group encompassing all rotations (about the origin) and reflections (across axes through the origin) of the plane. However, the notation D n also extends to denote a subgroup of SO(3) that also possesses the abstract group type D n. This refers to the proper symmetry group of a regular polygon when it is considered as an object embedded in three-dimensional space (assuming $n \geq 3$). Such a geometric figure can be conceptualized as a degenerate regular solid, one where its faces are counted twice. This peculiar characteristic is precisely why it is also termed a dihedron (derived from Greek, meaning "solid with two faces"), which, in turn, provides the etymological root for the "dihedral group." This naming convention follows the pattern of other well-known symmetry groups, such as the tetrahedral, octahedral, and icosahedral group, which respectively refer to the proper symmetry groups of a regular tetrahedron, octahedron, and icosahedron.

Examples of 2D dihedral symmetry

Sometimes, a visual cue helps the concepts solidify, even if the patterns are rather obvious.

• 2D D 16 symmetry – Imperial Seal of Japan, representing eightfold chrysanthemum with sixteen petals.

• 2D D 6 symmetry – The Red Star of David

• 2D D 12 symmetry — The Naval Jack of the Republic of China (White Sun)

• 2D D 24 symmetry – Ashoka Chakra, as depicted on the National flag of the Republic of India.

Properties

The universe, it seems, has a preference for parity, and the properties of the dihedral groups D n with $n \geq 3$ are no exception. Their characteristics often hinge on whether $n$ is an even or an odd number. For instance, the center of D n — that collection of elements that commute with every other element in the group — consists solely of the identity element if $n$ is odd. A rather sparse center, one might say. However, if $n$ is even, the center expands to include two elements: the identity itself and the element ${\displaystyle r^{n/2}}$. When viewed as a subgroup of O(2), this ${\displaystyle r^{n/2}}$ corresponds to inversion, a rotation by 180 degrees. Since scalar multiplication by $-1$ inherently commutes with any linear transformation, it's perfectly clear why this element finds its way into the center of the group.

In the context of 2D isometries, the presence of this additional element in the center for even $n$ means effectively adding inversion to the group. This operation generates new rotations and mirrors that are precisely positioned between the existing ones, creating a richer set of symmetries.

More generally, if $m$ is a divisor of $n$, then D n will contain precisely $n/m$ subgroups that are themselves of type D m. Additionally, it will possess one subgroup isomorphic to the cyclic group ${\displaystyle \mathbb {Z} _{m}}$. Consequently, the total number of subgroups within D n (for $n \geq 1$) can be calculated as ${\displaystyle d(n) + \sigma(n)}$, where $d(n)$ represents the number of positive divisors of $n$, and ${\displaystyle \sigma(n)}$ is the sum of those positive divisors of $n$. For a more concrete illustration, one can consult a list of small groups for cases where $n \leq 8$.

The dihedral group of order 8 (D 4 ) holds a particular distinction: it is the smallest example of a group that defies classification as a T-group. A T-group is one where all subnormal subgroups are normal. In D 4, while its two Klein four-group subgroups are themselves normal within D 4 , they each contain order-2 subgroups (generated by a [reflection](/Reflection_(mathematics), or "flip")) that are normal within the Klein four-group but are not normal within the larger D 4 . This intricate structure makes D 4 a useful counterexample in advanced group theory.

Conjugacy classes of reflections

The behavior of reflections within dihedral groups also exhibits a dependence on the parity of $n$. All reflections are conjugate to one another if and only if $n$ is odd. This means that for an odd $n$, any reflection can be transformed into any other reflection by conjugation with some element of the group. However, if $n$ is even, the reflections neatly partition into two distinct conjugacy classes.

Geometrically, this distinction is quite vivid when considering the isometries of a regular $n$-gon: for an odd $n$, the group's rotations are sufficient to move any mirror axis to any other mirror axis, as every axis passes through both a vertex and the midpoint of an opposite side. But for an even $n$, the situation is different. Only half of the mirror axes can be reached from a given mirror by rotations within the group. The even polygon explicitly displays two sets of axes: those that pass through pairs of opposite vertices, and those that pass through the midpoints of pairs of opposite sides. Each of these sets forms its own conjugacy class.

From an algebraic perspective, this phenomenon for odd $n$ is a direct consequence of the conjugate Sylow theorem. When $n$ is odd, each reflection, combined with the identity, forms a subgroup of order 2. This subgroup is a Sylow 2-subgroup, since 2 (which is $2^1$) is the highest power of 2 that divides the group's order, $2n$ (where $n$ is odd, so $2n = 2 \times (\text{odd number})$). Conversely, when $n$ is even, these order-2 subgroups generated by reflections are not Sylow subgroups, because 4 (a higher power of 2) divides the group's order (since $n$ is even, $2n$ is divisible by 4).

For $n$ even, a fascinating consequence arises: there exists an outer automorphism that explicitly interchanges these two types of reflections. More precisely, it's a class of outer automorphisms, all of which are conjugate to each other by an inner automorphism. This outer automorphism can be conceptualized as a rotation by ${\displaystyle \pi/n}$ (half the minimal rotation angle), an operation that is not part of the original dihedral group but reveals its deeper structural connections.

Automorphism group

The automorphism group of D n is remarkably isomorphic to the holomorph of ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$. This means it is isomorphic to Hol(${\displaystyle \mathbb {Z} /n\mathbb {Z} }$) = $\{ax + b \mid (a,n)=1\}$, and its order is given by $n\phi(n)$, where $\phi$ is Euler's totient function. The totient function, for those needing a refresher, counts the number of positive integers less than or equal to $n$ that are coprime to $n$.

Understanding the structure of the automorphism group can be approached by considering the actions on the group's generators: a reflection and an elementary rotation (specifically, a rotation by $k(2\pi/n)$, where $k$ is coprime to $n$). Whether these automorphisms are inner or outer depends critically on the parity of $n$.

• For $n$ odd, the dihedral group is centerless, meaning only the identity element commutes with all other elements. Consequently, every non-identity element defines a non-trivial inner automorphism.
• For $n$ even, however (with the exception of $n=2$), the rotation by 180° (which corresponds to a reflection through the origin) constitutes the non-trivial element within the group's center.

Based on this, we can further refine the understanding of the inner automorphism group:

• When $n$ is odd, the inner automorphism group has an order of $2n$.
• When $n$ is even (and not equal to 2), the inner automorphism group has an order of $n$.

Revisiting the conjugacy classes of reflections:

• For $n$ odd, all reflections are conjugate to one another, forming a single class.
• For $n$ even, as discussed, they split into two distinct classes (those passing through two vertices and those passing through two faces). These classes are related by an outer automorphism, which can be visualized as a rotation by $\pi/n$ (half the minimal rotation angle).

The rotations within D n form a normal subgroup. Conjugation by a reflection has the effect of changing the sign (or direction) of the rotation, while otherwise leaving its magnitude unchanged. Therefore, any automorphisms that multiply the angles of rotation by some factor $k$ (where $k$ is coprime to $n$) are considered outer automorphisms, unless $k = \pm 1$.

Examples of automorphism groups

Let's consider a few concrete examples, as they tend to clarify matters more effectively than abstract prose:

• D 9: This group possesses 18 inner automorphisms. When viewed as a 2D isometry group, D 9 is characterized by mirrors positioned at 20° intervals. These 18 inner automorphisms correspond precisely to rotations of these mirrors by multiples of 20°, along with the reflections. As an isometry group, these are, in fact, all the automorphisms. However, as an abstract group, D 9 also exhibits an additional 36 outer automorphisms. For instance, an operation that multiplies all angles of rotation by 2 would be an outer automorphism.
• D 10: This group has 10 inner automorphisms. In its manifestation as a 2D isometry group, D 10 features mirrors spaced at 18° intervals. The 10 inner automorphisms correspond to rotations of these mirrors by multiples of 36° and the reflections themselves. As an isometry group, there are 10 additional automorphisms that are conjugates by isometries outside the group, effectively rotating the mirrors by 18° relative to the inner automorphisms. As an abstract group, beyond these 10 inner and 10 outer automorphisms (as isometries), there are yet another 20 outer automorphisms. An example would be multiplying rotations by 3.

To put these numbers in perspective, one might compare the values of Euler's totient function, $\phi(n)$, for $n=9$ and $n=10$. For $n=9$, $\phi(9)=6$. For $n=10$, $\phi(10)=4$. These values explain the observed multiplication factors in the number of automorphisms when compared to the two automorphisms that simply preserve or reverse the order of rotations as isometries.

It is a rather specific, almost peculiar, fact that the only values of $n$ for which Euler's totient function $\phi(n) = 2$ are $n=3, 4,$ and $6$. Consequently, this implies that there are only three dihedral groups that are isomorphic to their own automorphism groups: these are D 3 (which has an order of 6), D 4 (with an order of 8), and D 6 (with an order of 12). A rather neat little anomaly in the grand scheme of things.

Inner automorphism group

The inner automorphism group of D n is, as one might expect, also dependent on the parity of $n$. It is isomorphic to:

• D n itself, if $n$ is odd. This means that every automorphism generated by conjugation is itself an element of D n.
• D n / Z 2 if $n$ is even. For the special case of $n=2$, D 2 / Z 2 is trivial, meaning it is the identity group (of order 1), which aligns with D 2 being abelian and having a trivial inner automorphism group.

Generalizations

The concept of dihedral groups, while seemingly specific to regular polygons, provides a fertile ground for several important generalizations that extend its utility and theoretical reach:

• The infinite dihedral group stands as a direct extension, representing an infinite group that mirrors the algebraic structure of its finite counterparts. It can be elegantly conceptualized as the group of symmetries of the integers (${\displaystyle \mathbb {Z} }$), where reflections and translations take the place of finite rotations.
• The orthogonal group O(2), which is the complete symmetry group of the circle, shares a profound kinship with the dihedral groups. It effectively represents the limiting case of D n as $n$ approaches infinity, encompassing continuous rotations and reflections.
• The broader family of generalized dihedral groups serves as an overarching category that subsumes both the finite and infinite examples mentioned above, along with a multitude of other groups that share similar structural properties.
• The quasidihedral groups, sometimes also referred to as modular groups, constitute another distinct family of finite groups. These groups possess properties that are remarkably similar to, yet subtly distinct from, the traditional dihedral groups, making them an interesting area of comparative study in group theory.

See also

For those who find themselves wanting to delve deeper into related concepts, the following articles provide further avenues for exploration:

• Coordinate rotations and reflections
• Cycle index of the dihedral group
• Dicyclic group
• Dihedral symmetry groups in 3D
• Dihedral symmetry in three dimensions