Dihedral Groups

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Dihedral Symmetry of Order 12
Field: Algebra
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Dihedral Symmetry of Order 12

Each snowflake in the main image has the dihedral symmetry of a natual regular hexagon. The group formed by these symmetries is also called the dihedral group of degree 6. Order refers to the number of elements in the group, and degree refers to the number of the sides or the number of rotations. The order is twice the degree.

Basic Description

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

Dihedral groups arise frequently in art and nature. Many of the decorative designs used on floor coverings, pottery, and buildings have one of the dihedral groups of symmetry. Chrysler’s logo has D_5 as a symmetry group, and that Mercedes-Benz has D_3. The ubiquitous five-pointed star has symmetry group D_5.


There are two different kinds of notation for a dihedral group associated to a polygon with n sides.

In geometry, we usually call it D_n or Dih_n, where n indicates the number of the sides.

In algebra, we call it D_2n, where 2n indicates the number of elements in the group.

On this page, we will use the notation D_n to describe a dihedral group. For D_n, we will call it the dihedral group of order 2n or the group of symmetries of a regular n-gon.

Below is an example of Dihedral symmetry of D_3, D_4, D_5, and D_6.

Example of Dihedral symmetry

A More Mathematical Explanation

Note: understanding of this explanation requires: *Basic Abstract Algebra


The n^\text{th} dihedral group is the symmetry group of the regula [...]


The n^\text{th} dihedral group is the symmetry group of the regular n-sided polygon. The group consists of n reflections, n-1 rotations, and the identity transformation.

Here is an example of D_6. This group contains 12 elements, which are all rotations and reflections. The very first one is the identity transformation.

Image 1

If n is odd each axis of symmetry connects the mid-point of one side to the opposite vertex. If n is even there are \frac{n}{2} axes of symmetry connecting the mid-points of opposite sides and \frac{n}{2} axes of symmetry connecting opposite vertices. In either case, there are n axes of symmetry altogether and 2n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes. In Image 1, through S_0 to S_5 are the axes of symmetries. All the reflections can be described as reflections of the identity through six axes of symmetries.


There are several different way to define a Dihedral Group. We will introduce three of them.

We will use R_0 to represent the identity, R_k, k=1,2,\cdots,n-1, to represent the rotations, and S_k, k=0,1, \cdots, n-1, to represent the reflections.

Complex Plane Presentation

For n \geqslant 3, the dihedral group D_n is defined as the rigid motions of the plane preserving a regular n-gon, with the operation of composition. On complex plane, our model n-gon will be an n-gon centered at the origin, with vertices at the n-th roots of unity. 1 is always an n-th root of unity, but -1 is such a root only if n is even. In general, the roots of unity form a regular polygon with n sides, and each vertex lies on the unit circle.

The n-th roots of unity are roots \omega = e^{\frac{2 \pi i}{n}} of the cyclotomic equation x^n=1.

Since a vector on the complex plane can be described as p=a+bi, a vector with an angle \theta counterclockwise from x-axis can be described as p=r\cos \theta +i r\sin \theta, where r=\sqrt{a^2+b^2} is the magnitude of the vector.

Leonhard Euler's formula says that e^{i \theta}=\cos \theta +i \sin \theta, so any point on the complex plane is p=re^{i \theta}.

For a regular n-gon, the first angle counterclockwise from the x-axis is \frac{2 \pi}{n}, so the primitive root of unity is \omega = e^{\frac{2 \pi i}{n}}.

Letting \omega = \frac{2 \pi i}{n} denote a primitive n^ \text{th} root of unity, and assuming the polygon is centered at the origin, the rotations R_k, k=0, 1, 2, \cdots, n-1(Note: R_0 denotes the identity), are given by

R_k:z \mapsto \omega^k z,\quad z\in\cnums.

Notation \mapsto means a function. For example: input \mapsto output.

For the reflections, S_k, k=0, 1, 2, \cdots, n-1, the functions are given by

S_k: z\mapsto \omega^k \bar{z},\quad z\in\cnums.

Matrix Representation

If we center the regular polygon at the origin, then the elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of D_n as matrices, with composition being matrix multiplication. This is an example of a (2-dimensional) group representation.

For example, the elements of the group D_4 can be represented by the following eight matrices:

\begin{matrix} R_0=\bigl(\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}\bigr), & R_1=\bigl(\begin{smallmatrix}0&-1\\[0.2em]1&0\end{smallmatrix}\bigr), & R_2=\bigl(\begin{smallmatrix}-1&0\\[0.2em]0&-1\end{smallmatrix}\bigr), & R_3=\bigl(\begin{smallmatrix}0&1\\[0.2em]-1&0\end{smallmatrix}\bigr), \\[1em] S_0=\bigl(\begin{smallmatrix}1&0\\[0.2em]0&-1\end{smallmatrix}\bigr), & S_1=\bigl(\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}\bigr), & S_2=\bigl(\begin{smallmatrix}-1&0\\[0.2em]0&1\end{smallmatrix}\bigr), & S_3=\bigl(\begin{smallmatrix}0&-1\\[0.2em]-1&0\end{smallmatrix}\bigr). \end{matrix}

If we represent the columns of each matrix as basis vectors, we can observe directly all the rotations and reflections.

In general, we can write any dihedral group as:

R_k = \begin{bmatrix} \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{bmatrix},

S_k = \begin{bmatrix} \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{bmatrix},

where R_k is a rotation matrix, expressing a counterclockwise rotation through an angle of \frac{2 \pi k}{n}, and S_k is a reflection across a line that makes an angle of \frac{\pi k}{n} with the x-axis.

Group Presentation

A presentation of a group is a description of a set I and a subset R of the free group F(I) generated by I, written as \left \langle (x_i)_{i \in I}|(r)_{r \in R} \right \rangle, where the equation r=1 (the identity element) is often written in place of the element r. A group presentation defines the quotient group of the free group F(I) by the normal subgroup generated by R, which is the group generated by the generators x_i subject to the relations r \in R.

We can use the presentation:

D_n = \left \langle r,s|r^n=1,s^2=1,srs=r^{-1} \right \rangle, or

D_n = \left \langle x,y|x^2=1,y^n=1,(xy)^2=1 \right \rangle to define a group, isomorphic to the dihedral group D_n of finite order 2n, which is the group of symmetries of a regular n-gon.

We will use the second presentation, in which x refers to a reflection, and y refer to a primitive rotation.

For x^2=1, x means an arbitrary mirror image of the n-gon, and 1 means the identity. This equation means that if we reflect the n-gon once, you get x. If reflect the n-gon twice, the result will return to the identity.
For y^n=1, the equation means that y is a rotation, and its nth power y^n equals the identity. That is, if we rotate the n-gon n times, we get back to the identity.
For (xy)^2=1, xy means the mirror image of y. Reflecting the n-gon through the axis of symmetry of xy twice, the result is the identity.

Following the group presentation, we can label all the reflections and rotations in terms of x and y.

  • Identity: 1
  • Rotations: y, y^2, y^3, \cdots, y^{n-1}, and y^n=1, which is the identity.
  • Reflections: x, xy, xy^2, \cdots, xy^{n-1}. There is not xy^n, because y^n=1, and so xy^n=x is the reflection of the identity.


Cayley Table

Image 2 - Cayley Table for D_6

As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry. This operation gives the symmetries of a polygon the algebraic structure of a finite group.

A Cayley table, named after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table.

Image 2 on the right shows the effect of composition in the group D_6 (the symmetries of a hexagon). R_0 denotes the identity; R_1 to R_5 denote counterclockwise rotations by 60, 120, 180, 240,and 300 degrees; and S_0 to S_5 denote reflections across the six diagonals. In general, ab denotes the entry at the intersection of the row with a at the left and the column with b at the top.

In the table, the same or different rotations and reflections work together and result in a new rotation or reflection. For example, look first at the vertical axis to find a element, R_4. Then look at the horizontal axis to get the second element for our composition. We choose S_3. Composing two elements is just the progression of a rotation or a reflection followed by another rotation or a reflection. In this case, our elements are R_4 and S_3. First we rotate the hexagon counterclockwise 240 degrees, and then reflect it along the axis of symmetry of S_3. The result is the same as reflecting the identity transformation through an angle of 60 degrees, which is S_1. See Example 1 below.

Now, look back to Image 2, you will find that the intersection of R_4 in left column and S_3 in top row is S_1. However, you will find the intersection of S_3 in left column and R_4 in top row is R_5. If you like, you can create your own Cayley table for a dihedral group of any order and find the natual rule for it.

Explore the Cayley Table

Perhaps the most important feature of this table is that it has been completely filled in without introducing any new motions.

  • Closure: Algebraically, this says that if A and B are in D_6, then so is AB. This property is called closure, and it is one of the requirements for a mathematical system to be a group.
  • identity: Notice that if A is any element of D_6, then AR_0=R_0 A=A. Thus, combining any element A on either side with R_0 yields A back again. An element R_0 with this property is called an identity, and every group must have one.
  • Inverse: We see that for each element B in D_6, there exists an element A such that AB=BA=R_0. In this case, B is said to be the inverse of A and vise versa. The term inverse is a descriptive one, for if A and B are inverses of each other, then B "un-does" whatever A "does", in the sense that A and B taken together in either order produce R_0, representing no change.
  • Non-Abelian: Another property of D_6 deserves special comment. Obverse that S_1 R_3=R_3 S_1, but S_3 R_4 \neq R_4 S_3. Thus in a group ab may or may not be the same as ba. If it happens that ab=ba for all choices of group elements a and b, we say the group is commutative or --better yet-- Abelian (in honor of the great Norwegian mathematician Niel Abel). Otherwise, we say the group is non-Abelian. All dihedral groups are non-Abelian, except D_1 and D_2.
  • Associativity: For all dihedral groups, it holds true that (ab)c=a(bc) for all a, b, c in the group.
Image 3 - Multiplication Table for D_6

If we want to know what is the composition of any two elements, it is convenient to use a Cayley map, because it tells us the result directly. But when we want to know the gradual change of the compositions, we will need another tool, a multi-colored Muplication Table.

Muptiplication Table

In a Multiplication Table, each color represents one rotation or reflection. In Image 3, pink colors represent rotations, and the deepest pink represents the identity transformation. Green colors represent all the six reflections.

From the changing of color, we can observe that the gradual change of the composition of two elements in D_6. However, we cannot easily determine the exact result of composition by observing directly Image 3.

The abstract group structure is given by:

R_k R_l=R_{k+l}

S_k S_l=R_{k-l}

R_k S_l=S_{k+l}

S_k R_l=S_{k-l}

Uniqueness of the Identity

In a dihedral group D_n, there is only one identity element.

PROOF: Suppose both R_0 and R_0' are identities of D_n. Then,

Eq. 1         aR_0=a for all a in D_n, and
Eq. 2         R_0' a=a for all a in D_n.

The choices of a=R_0' in Eq. 1 and a=R_0 in Eq. 2 yield R_0' R_0=R_0' and R_0' R_0=R_0.

Thus, R_0 and R_0' are both equal to R_0' R_0 and so are equal to each other.


In a dihedral group D_n, the right and left cancellation laws hold; that is, ba=ca implies b=c, and ab=ac implies b=c.

PROOF: Suppose ba=ca. Let a' be an inverse of a.

Then, muliplying on the right by a' yields (ba)a'=(ca)a'.

Associativity yields b(aa')=c(aa').

Then, bR_0=cR_0 and, therefore, b=c as desired.

Similarly, one can prove that ab=ac implies b=c by multiplying by a' on the left.

A consequence of the cancellation property is the fact that in a Cayley table for a dihedral group, each group element occurs exactly once in each row and column. Another consequence of the cancellation property is the uniqueness of inverses.

Uniqueness of Inverses

For each element a in a dihedral group D_n, there is a unique element b in D_n such that ab=ba=R_0.

PROOF: Suppose b and c are both inverses of a.

Then ab=R_0 and ac=R_0, so that ab=ac.

Canceling the a on both sides gives b=c, as desired.

Sock-Shoes property

For dihedral group elements a and b, (ab)^{-1}=b^{-1} a^{-1}.

PROOF: Since (ab)(ab)^{-1}=R_0 and

(ab)(b^{-1} a^{-1})=a(bb^{-1})a^{-1}=aR_0a^{-1}=aa^{-1}=R_0,

we have by the Uniqueness of Inverses theorem that (ab) has only one inverse x such that (ab)x=R_0.

We get x=(ab)^{-1}=(b^{-1} a^{-1}).

3D Rotational Symmetry

Dn consists of n rotations of multiples of \frac{360^\circ}{n} about the origin, and reflections across n lines through the origin, making angles of multiples of \frac{180^\circ}{n} with each other.

If we put a dihedral group in three dimensions, the reflections are also rotations of 180^\circ

Rotation in 3D

The proper symmetry group of a regular polygon embedded in three-dimensional space (if n \geqslant 3). Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group.

Infinite Dihedral Groups

The infinite dihedral group Dih(C_\infty) is denoted by D_\infty. The infinite dihedral group can be described as the group of symmetries of a circle, which has infinite symmetries.

We use the group presentation:

D_\infty = \langle r, s \mid s^2 = 1, srs = r^{-1} \rangle , or

D_\infty = \langle x, y \mid x^2 = y^2 = 1 \rangle to represente the infinite dihedral group.

In the presentation, it says that because there are infinitely many symmetries, we can never rotate back to the identity, and so there are infinitely many rotations and reflections.


Definition: A subgroup is a subset H of group elements of a group G that satisfies the four group requirements. It must therefore contain the identity element. "H is a subgroup of G" is written as H \subseteq G, or sometimes H \leq G.[1]

Now we want to know exactly how many subgroups for D_n, and what they are. Fortunately, mathematician Stephan A. Cavior had already proved this for us in 1975. In the theorem, for any dihedral group in order of 2n, there are \tau(n) + \sigma(n) subgroups in total, including \{1\} and D_{n}. \{1\} is just the identity itself.

Definitions of Terms

  • \tau(n): the number of divisors of n,
e.g. \tau(12)=6.
  • \sigma(n): the sum of divisors of n,
e.g. \sigma(12)=1 + 2 + 3 + 4 + 6 + 12=28.
  • \mathbb{Z}/n \mathbb{Z}: the notation for the cyclic group of order n, can be also written as \mathbb{Z}_n. This is a quotient group presentation.
  • d \mid n: d is a divisor of n.
  • N=\langle b \rangle: group N is a subgroup generated by b. It means N=\{1,b,b^2,b^3,\cdots,b^{n-1}\}.
  • |HN|: the order of HN.
  • [D_{n}:N]=2: the index. [D_{n}:N]=2 means \frac{|D_n|}{|N|}=2.
  • \{ab^{i+km}: \ 0 \leq k < d \}: a set with elements looking like ab^{i+km}.
  • G(i,d): this means a group. i is the index, which labels all the elements; d is the order of the group.

This proof is complicated.

After we know what kind of group can be a subgroup of a dihedral group D_n, the dihedral group of order 2n, we will start to find all subgroups of D_n.

Lemma 1.        The number of subgroups of a cyclic group of order n \geq 1 is \tau(n). [2]

A cyclic group of order n is the group of all the rotations including the identity of the dihedral group of order 2n.

Proof. Let G be a cyclic group of order n. Then G \cong \mathbb{Z}/n \mathbb{Z}. A subgroup of \mathbb{Z}/n \mathbb{Z} is in the form d \mathbb{Z}/n \mathbb{Z} where d \mathbb{Z} \supseteq n \mathbb{Z}. The condition d \mathbb{Z} \supseteq n \mathbb{Z} is obviously equivalent to d \mid n. \

Lemma 2.        Let b be the element of order n in D_{n} and let H be a subgroup of D_{n}. Then either H \subseteq \langle b \rangle or |H \cap \langle b \rangle| =d and |H|=2d for some d \mid n.

Proof. Let N=\langle b \rangle. Clearly N is a normal subgroup of D_{n} because [D_{n}:N]=2. Thus HN is a subgroup of D_{n} and hence the order of dihedral group HN is a divisor of 2n, and we use the notation:

Eq. 1         |HN| \mid 2n. to represent.

On the other hand,

Eq. 2         |HN|=\frac{|H| \cdot |N|}{|H \cap N|}=\frac{n |H|}{|H \cap N|}.
Therefore, by Eq. 1, and Eq. 2. Hence either |H| = |H \cap N| or |H|=2|H \cap N|. If |H|=|H \cap N|, then H = H \cap N and thus H \subseteq N. If |H|=2|H \cap N|, then let |H \cap N|=d and so |H|=2d. Clearly d \mid n because H \cap N is a subgroup of N and |N|=n. \

Lemma 3.        Given d \mid n, let m = n/d. For every 0 \leq i < n let A(i,d) = \{ab^{i+km}: \ 0 \leq k < d \}, where a denotes a reflection and b denotes a primitive rotation. Let B(i,d) = A(i,d) \cup \langle b^m \rangle. Then B(i,d) is a subgroup of D_{n} and |B(i,d)|=2d. We also have |\{B(i,d) : \ 0 \leq i < n \}|=m=n/d.

Proof. If ab^{i+km}=ab^{i+rm}, for some 0 \leq k,r < d, then b^{(k-r)m}=1 and thus d \mid k-r, because |b|=n=md.

Therefore k=r because 0 \leq k,r < d.

So we have proved that |A(i,d)|=d.

Clearly A(i,d) \cap \langle b^m \rangle = \emptyset and |\langle b^m \rangle | = d, because |b|=n.

Thus |B(i,d)|=|A(i,d)| + |\langle b^m \rangle|=2d.

Proving that B(i,d) is a subgroup of D_{n} is very easy.

Just note that every element of A(i,d) is the inverse of itself (because they all have order two) and also note that ab^s = b^{-s}a, for all s, because ab=b^{-1}a.

Finally, the set \{B(i,d) : \ 0 \leq i < n \} has m elements because clearly B(i,d)=B(j,d) if and only if A(i,d)=A(j,d) if and only if i \equiv j \mod m. \

Suppose that H is a subgroup of D_{n}. There are two disjoint cases to consider.

Case 1. H \subseteq \langle b \rangle.

By Lemma 1. the number of these subgroups is \tau(n).

Case 2. H \nsubseteq \langle b \rangle.

In this case, by Lemma 2. we have |H|=2d and |H \cap \langle b \rangle|=d, for some d \mid n.
Let n = md. Since H \cap \langle b \rangle is a subgroup of \langle b \rangle, which is a cyclic group of order n, we have
Eq. 1        H \cap \langle b \rangle = \langle b^m \rangle.
Let A(i,d) and B(i,d) be as were defined in Lemma 3.
Now, since H is not contained in \langle b \rangle, there exists some 0 \leq i < n such that ab^i \in H.
Then, since H is a subgroup, we must have ab^ib^{km} \in H, for all k.
Thus ab^{i + km} \in H and so A(i,d) \subseteq H and therefore, by Eq. 1, we have B(i,d) \subseteq H.
Thus, since |H|=|B(i,d)|=2d, we must have H=B(i,d).
The converse is obvously true, i.e. given d \mid n and 0 \leq i < n, B(i,d) is a subgroup of D_{n}, by Lemma 3., and B(i,d) \nsubseteq \langle b \rangle because it contains A(i,d).
So the subgroups in this case are exactly the ones in the form B(i,d), where 0 \leq i < n and d \mid n.
Thus, by Lemma 3. the number of subgroups in this case is
\sum_{d \mid n} | \{B(i,d) : \ 0 \leq i < n \} | = \sum_{d \mid n} n/d = \sum_{d \mid n} d = \sigma(n).

So, by Case 1. and Case 2. the number of subgroups of D_{n} is \tau(n) + \sigma(n)\ .

Why It's Interesting

In Music

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords.[3]


Dihedral groups as a kind of special symmetric groups are studied in music. In music, we use the operations Transposition and Inversion, which are denoted as T and I, to represente rotations and reflections in dihedral groups.

Musicians usually study D_{12}, because 12 is the length of a normal cycle in music: C C D E E F F G G A B B, and then C again.

Based on this 12 element cycle, D_{12} is important in music theory. Musicians use Transpositions and Inversions (rotations and reflections) of a simple note to create other notes to complete a final composition.

A transposition of a sequence x of pitch classes by n semitones is the sequence T^n(x) in which each of the pitch classes in x has been increased by n semitones.

So for example if

x=3 0 8, where the numbers denote pitches,


T^4(x)=T^4(3 0 8) = 7 4 0.

When doing the operation T^n(x), add n to each digit of x, and use arithmetic modulo 12 (clock arithmetic) when the resulting digit is over 12. For instance, in adding 4 to 8, the result is 12, but 12 \equiv 0 \pmod{12}.\,

Turning to the next operation, inversion I(x) of a sequence x just replaces each pitch class by its negative (in clock arithmetic).

So in the first example above with x = 3 0 8, we have

I(x) = 9 0 4.

To do the operation I(x), we need to do subtraction in clock arithmetic. For instance, if we want to get 12 from 3, we need to add 9. 0 is already 12, so we need to add 0.


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Related Links

Additional Resources


[1] Wikipedia. (n.d.). Dihedral groups. Retrieved from http://en.wikipedia.org/wiki/Dihedral_group

[2] de Cornulier, Yves. (n.d). Group Presentation. From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/GroupPresentation.html

[3] Wikipedia. (n.d.). Cayley table. Retrieved from http://en.wikipedia.org/wiki/Cayley_table

[4] Milson, Robert and Foregger, Thomas. dihedral group. From PlanetMath.org. June, 12. 2007. Retrieved from http://planetmath.org/encyclopedia/DihedralGroup.html

[5] Gallian, Joseph A. Contemporary Abstract Algebra Seventh Edition. Belmont: Brooks/Cole, Cengage Learning. 2010.

[6] Dahlke, Karl. (n.d). Groups, Dihedral and General Linear Groups. Retrieved from http://www.mathreference.com/grp,dih.html

[7] Sharifi, Yaghoub. Subgroups of dihedral groups (1)&(2). Feb, 17, 2011. Retrieved from http://ysharifi.wordpress.com/2011/02/17/subgroups-of-dihedral-groups-1/

[8] [1]Scott, W. R. Group Theory. New York: Dover, 1987.

[9] [2]Hungerford, Thomas W. Graduate Texts in Mathematics - Algebra. New York: Springer, 1974.

[10] [3]Crans, Alissa S., Fiore, Thomas M. and Satyendra, Ramon. Musical Actions of Dihedral Groups. University of South Florida. Nov 3, 2007. Retrieved from http://myweb.lmu.edu/acrans/MusicalActions.PDF

[11] Benson, Dave J. Music: A Mathematical Offering. Cambridge University Press. Nov 2006. Retrieved from http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf

[12] Rowland, Todd and Weisstein, Eric W. (n.d). Root of Unity. From MathWorld--A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/RootofUnity.html

[13] Conrad, Keith. (n.d). DIHEDRAL GROUPS. Retrieved from http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/dihedral.pdf

Future Directions for this Page

  1. More information related to the other groups
  2. Add more about Dihedral Groups in 3D. I only talk about one property in 3D, but there must be some more.
  3. In the subgroups part, it is hard to explain only in words, so I use lots of notation, which is still not very clear. I hope can find a better way to illustrate it.
  4. Add more about applications
  5. Think about non-abelian in matrices which may relate to non-abelian in group theory.

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