Dihedral group

Dihedral Group

The Dihedral Group, denoted as D_n, is a mathematical concept that represents the symmetries of a regular polygon with n sides. It is a finite group that consists of rotations and reflections of the polygon.

Definition

The Dihedral Group D_n is defined as the group of symmetries of a regular polygon with n sides. It has 2n elements, which can be categorized into two types: rotations and reflections.

The rotations in D_n are denoted as r_k, where k ranges from 0 to n-1. Each rotation represents a clockwise rotation of the polygon by an angle of 2πk/n. The composition of two rotations r_k and r_l is another rotation r_{(k+l) mod n}.

The reflections in D_n are denoted as s_k, where k ranges from 0 to n-1. Each reflection represents a reflection of the polygon along a line passing through a vertex and the midpoint of the opposite side. The composition of two reflections s_k and s_l is a rotation r_{(k+l) mod n} if k+l is even, and a reflection s_{(k+l) mod n} if k+l is odd.

Properties

The Dihedral Group D_n has several interesting properties:

1. Closure: The composition of any two elements in D_n is also an element of D_n. This property ensures that D_n is a group.

2. Identity Element: The identity element in D_n is the rotation r₀, which represents no rotation.

3. Inverse Element: The inverse of a rotation r_k is the rotation r_(n-k), and the inverse of a reflection s_k is itself.

4. Associativity: The composition of three elements in D_n is associative, meaning that (a * b) * c = a * (b * c) for any elements a, b, and c in D_n.

5. Non-Commutativity: In general, the composition of two elements in D_n is not commutative. That is, a * b is not always equal to b * a.

Examples

Let's consider the Dihedral Group D₃, which represents the symmetries of an equilateral triangle. It has 6 elements: r₀, r₁, r₂, s₀, s₁, and s₂.

The composition of r₁ and r₂ is r₃ (since 1+2=3 mod 3), which represents a full rotation of the triangle. The composition of r₁ and s₂ is s₁ (since 1+2=1 mod 3), which represents a reflection along a line passing through a vertex and the midpoint of the opposite side.

Applications

The Dihedral Group D_n has various applications in different fields of mathematics and science. Some notable applications include:

1. Crystallography: Dihedral groups are used to describe the symmetries of crystals, which have a repeating pattern of atoms.

2. Robotics: Dihedral groups are used in robotics to represent the possible movements and configurations of robotic arms and manipulators.

3. Computer Graphics: Dihedral groups are used in computer graphics to generate symmetrical patterns and animations.

4. Group Theory: The study of Dihedral Groups provides insights into the general theory of groups and their properties.

See Also

• Group Theory
• Symmetry
• Regular Polygon

References

1. Dummit, D. S., & Foote, R. M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons.

2. Rotman, J. J. (1995). An Introduction to the Theory of Groups (4th ed.). Springer.