Dihedral Groups
Dihedral Symmetry of Order 12 |
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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.
Contents
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 as a symmetry group, and that Mercedes-Benz has . The ubiquitous five-pointed star has symmetry group .
Notation
There are two different kinds of notation for a dihedral group associated to a polygon with sides.
In geometry, we usually call it or , where indicates the number of the sides.
In algebra, we call it , where indicates the number of elements in the group.
On this page, we will use the notation to describe a dihedral group. For , we will call it the dihedral group of order or the group of symmetries of a regular -gon.
Below is an example of Dihedral symmetry of and .
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Basic Abstract Algebra
Elements
The dihedral group is the symmetry group of the regula [...]Elements
The dihedral group is the symmetry group of the regular -sided polygon. The group consists of reflections, rotations, and the identity transformation .
Here is an example of . This group contains 12 elements, which are all rotations and reflections. The very first one is the identity transformation.
If is odd each axis of symmetry connects the mid-point of one side to the opposite vertex. If is even there are axes of symmetry connecting the mid-points of opposite sides and axes of symmetry connecting opposite vertices. In either case, there are n axes of symmetry altogether and 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 to are the axes of symmetries. All the reflections can be described as reflections of the identity through six axes of symmetries.
Definition
There are several different way to define a Dihedral Group. We will introduce three of them.
We will use to represent the identity, , to represent the rotations, and , to represent the reflections.
Complex Plane Presentation
For , the dihedral group is defined as the rigid motions of the plane preserving a regular -gon, with the operation of composition. On complex plane, our model -gon will be an -gon centered at the origin, with vertices at the n-th roots of unity . is always an -th root of unity, but is such a root only if is even. In general, the roots of unity form a regular polygon with sides, and each vertex lies on the unit circle.
The -th roots of unity are roots of the cyclotomic equation .
Since a vector on the complex plane can be described as , a vector with an angle counterclockwise from x-axis can be described as , where is the magnitude of the vector.
Leonhard Euler's formula says that , so any point on the complex plane is .
For a regular -gon, the first angle counterclockwise from the x-axis is , so the primitive root of unity is .
Letting denote a primitive root of unity, and assuming the polygon is centered at the origin, the rotations , (Note: denotes the identity), are given by
Notation means a function. For example: .
For the reflections, , , the functions are given by
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 as matrices, with composition being matrix multiplication. This is an example of a (2-dimensional) group representation.
For example, the elements of the group can be represented by the following eight matrices:
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:
,
,
where is a rotation matrix, expressing a counterclockwise rotation through an angle of , and is a reflection across a line that makes an angle of with the x-axis.
Group Presentation
A presentation of a group is a description of a set and a subset of the free group generated by , written as , where the equation (the identity element) is often written in place of the element . A group presentation defines the quotient group of the free group by the normal subgroup generated by , which is the group generated by the generators subject to the relations .
We can use the presentation:
, or
to define a group, isomorphic to the dihedral group of finite order , which is the group of symmetries of a regular -gon.
We will use the second presentation, in which refers to a reflection, and refer to a primitive rotation.
- For , means an arbitrary mirror image of the -gon, and means the identity. This equation means that if we reflect the -gon once, you get . If reflect the -gon twice, the result will return to the identity.
- For , the equation means that is a rotation, and its th power equals the identity. That is, if we rotate the -gon times, we get back to the identity.
- For , means the mirror image of . Reflecting the -gon through the axis of symmetry of twice, the result is the identity.
Following the group presentation, we can label all the reflections and rotations in terms of and .
- Identity:
- Rotations: , and , which is the identity.
- Reflections: . There is not , because , and so is the reflection of the identity.
Properties
Cayley Table
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 (the symmetries of a hexagon). denotes the identity; to denote counterclockwise rotations by 60, 120, 180, 240,and 300 degrees; and to denote reflections across the six diagonals. In general, denotes the entry at the intersection of the row with at the left and the column with 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, . Then look at the horizontal axis to get the second element for our composition. We choose . 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 and . First we rotate the hexagon counterclockwise 240 degrees, and then reflect it along the axis of symmetry of . The result is the same as reflecting the identity transformation through an angle of 60 degrees, which is . See Example 1 below.
Now, look back to Image 2, you will find that the intersection of in left column and in top row is . However, you will find the intersection of in left column and in top row is . 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 and are in , then so is . This property is called closure, and it is one of the requirements for a mathematical system to be a group.
- identity: Notice that if is any element of , then . Thus, combining any element on either side with yields back again. An element with this property is called an identity, and every group must have one.
- Inverse: We see that for each element in , there exists an element such that . In this case, is said to be the inverse of and vise versa. The term inverse is a descriptive one, for if and are inverses of each other, then "un-does" whatever "does", in the sense that and taken together in either order produce , representing no change.
- Non-Abelian: Another property of deserves special comment. Obverse that , but . Thus in a group may or may not be the same as . If it happens that for all choices of group elements and , 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 and .
- Associativity: For all dihedral groups, it holds true that for all a, b, c in the group.
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 . However, we cannot easily determine the exact result of composition by observing directly Image 3.
The abstract group structure is given by:
Uniqueness of the Identity
In a dihedral group , there is only one identity element.
PROOF: Suppose both and are identities of . Then,
- for all in , and
- for all in .
The choices of in Eq. 1 and in Eq. 2 yield and .
Thus, and are both equal to and so are equal to each other.
Cancellation
In a dihedral group , the right and left cancellation laws hold; that is, implies , and implies .
PROOF: Suppose . Let be an inverse of .
Then, muliplying on the right by yields .
Associativity yields .
Then, and, therefore, as desired.
Similarly, one can prove that implies by multiplying by 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 in a dihedral group , there is a unique element in such that .
PROOF: Suppose and are both inverses of .
Then and , so that .
Canceling the on both sides gives , as desired.
Sock-Shoes property
For dihedral group elements and , .
PROOF: Since and
,
we have by the Uniqueness of Inverses theorem that has only one inverse such that .
We get .
3D Rotational Symmetry
consists of rotations of multiples of about the origin, and reflections across lines through the origin, making angles of multiples of with each other.
If we put a dihedral group in three dimensions, the reflections are also rotations of
The proper symmetry group of a regular polygon embedded in three-dimensional space (if ). 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 is denoted by . The infinite dihedral group can be described as the group of symmetries of a circle, which has infinite symmetries.
We use the group presentation:
, or
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.
Subgroups
Definition: A subgroup is a subset of group elements of a group that satisfies the four group requirements. It must therefore contain the identity element. " is a subgroup of " is written as , or sometimes .^{[1]}
Now we want to know exactly how many subgroups for , 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 , there are subgroups in total, including and . is just the identity itself.
Definitions of Terms
- : the number of divisors of ,
- e.g. .
- : the sum of divisors of ,
- e.g. .
- : the notation for the cyclic group of order , can be also written as . This is a quotient group presentation.
- : is a divisor of .
- : group is a subgroup generated by . It means .
- : the order of .
- : the index. means .
- : a set with elements looking like .
- : this means a group. is the index, which labels all the elements; 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 , the dihedral group of order , we will start to find all subgroups of .
- The number of subgroups of a cyclic group of order is . ^{[2]}
A cyclic group of order is the group of all the rotations including the identity of the dihedral group of order .
- Let be the element of order in and let be a subgroup of . Then either or and for some .
Proof. Let . Clearly is a normal subgroup of because . Thus is a subgroup of and hence the order of dihedral group is a divisor of , and we use the notation:
- to represent.
On the other hand,
- .
- Given , let . For every let , where denotes a reflection and denotes a primitive rotation. Let . Then is a subgroup of and . We also have
Proof. If , for some , then and thus , because .
Therefore because .
So we have proved that .
Clearly and , because .
Thus .
Proving that is a subgroup of is very easy.
Just note that every element of is the inverse of itself (because they all have order two) and also note that , for all , because .
Finally, the set has elements because clearly if and only if if and only if
Suppose that is a subgroup of . There are two disjoint cases to consider.
Case 1. .
- By Lemma 1. the number of these subgroups is .
Case 2. .
- In this case, by Lemma 2. we have and , for some .
- Let . Since is a subgroup of , which is a cyclic group of order , we have
- Let and be as were defined in Lemma 3.
- Now, since is not contained in , there exists some such that .
- Then, since is a subgroup, we must have , for all .
- Thus and so and therefore, by Eq. 1, we have .
- Thus, since , we must have .
- The converse is obvously true, i.e. given and , is a subgroup of , by Lemma 3., and because it contains .
- So the subgroups in this case are exactly the ones in the form , where and .
- Thus, by Lemma 3. the number of subgroups in this case is
So, by Case 1. and Case 2. the number of subgroups of is .
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 and , to represente rotations and reflections in dihedral groups.
Musicians usually study , 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, 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 of pitch classes by semitones is the sequence in which each of the pitch classes in has been increased by semitones.
So for example if
- , where the numbers denote pitches,
then
- .
When doing the operation , add to each digit of , 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
Turning to the next operation, inversion of a sequence just replaces each pitch class by its negative (in clock arithmetic).
So in the first example above with , we have
- .
To do the operation , 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.
Teaching Materials
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Related Links
Additional Resources
- An applet to explore dihedral groups: http://www.mathlearning.net/learningtools/Flash/Dihedral/dihedralExplorer.html
References
[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
- More information related to the other groups
- Add more about Dihedral Groups in 3D. I only talk about one property in 3D, but there must be some more.
- 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.
- Add more about applications
- Think about non-abelian in matrices which may relate to non-abelian in group theory.
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