Describe generators and relations for all dihedral groups [tex]$D_{2 n}$[/tex]. (Hint: Let [tex]$x$[/tex] be reflection about a line through the center of a regular n-gon and a vertex, and let [tex]$y$[/tex] be counterclockwise rotation by [tex]$2 \pi / n$[/tex]. The group [tex]$D_{2 n}$[/tex] will be generated by [tex]$x$[/tex] and [tex]$y$[/tex], subject to three relations. To see that these relations really determine [tex]$D_{2 n}$[/tex], use them to show that any product [tex]$x^{i_1} y^{i_2} x^{i_3} y^{i_4} \cdots$[/tex] equals [tex]$x^i y^j$[/tex] for some [tex]$i, j$[/tex] with [tex]$(0 \leq i \leq 1,0 \leq j\ \textless \ n.)$[/tex])
Answer (1)
The dihedral group D 2 n is generated by a reflection x and a rotation y .
The reflection x satisfies the relation x 2 = e .
The rotation y satisfies the relation y n = e .
The interaction between x and y is given by the relation x y x = y − 1 (or equivalently, x y x y = e ).
Any element in D 2 n can be written in the form x i y j where 0 ≤ i ≤ 1 and 0 ≤ j < n .
The generators and relations for D 2 n are ⟨ x , y ∣ x 2 = e , y n = e , x y x = y − 1 ⟩ .
The generators and relations for the dihedral group D 2 n are ⟨ x , y ∣ x 2 = e , y n = e , x y x = y − 1 ⟩ .
Explanation
Understanding the Problem We want to describe the generators and relations for the dihedral group D 2 n . The dihedral group D 2 n is the group of symmetries of a regular n -gon, which includes rotations and reflections. We are given that x is a reflection about a line through the center of the n -gon and a vertex, and y is a counterclockwise rotation by 2 π / n . We need to find the relations that x and y satisfy.
Finding the Order of x Since x is a reflection, applying it twice will return the n -gon to its original orientation. Thus, x 2 = e , where e is the identity element.
Finding the Order of y Since y is a rotation by 2 π / n , applying it n times will result in a full rotation, returning the n -gon to its original orientation. Thus, y n = e .
Finding the Relation Between x and y The reflection x reverses the direction of the rotation y . This means that x y is the same as rotating in the opposite direction and then reflecting. Mathematically, this can be expressed as x y = y − 1 x . Multiplying both sides on the right by x , we get x y x = y − 1 x 2 = y − 1 e = y − 1 . Since y − 1 = y n − 1 , we can also write this as x y x = y n − 1 . Another way to write the relation is x y x y = y − 1 x y = y − 1 y − 1 x 2 = y − 2 . However, we want to show that x y x y = e . Since x y = y − 1 x , then x y x y = y − 1 x y = y − 1 y − 1 x = y − 2 x . This doesn't seem to lead to e . Instead, let's start with x y x = y − 1 . Multiplying both sides on the right by y , we get x y x y = y − 1 y = e .
Generators and Relations The generators for D 2 n are x and y , and the relations are x 2 = e , y n = e , and x y x = y − 1 (or equivalently, x y x y = e ). Any element in D 2 n can be written in the form x i y j where 0 ≤ i ≤ 1 and 0 ≤ j < n . This means that the set { e , y , y 2 , ... , y n − 1 , x , x y , x y 2 , ... , x y n − 1 } contains all elements of D 2 n .
Verification To show that these relations determine D 2 n , we need to show that any product of x and y can be reduced to the form x i y j . We can use the relations to rewrite any occurrence of y x as x y − 1 . For example, consider a product x i 1 y i 2 x i 3 y i 4 ⋯ . If we have y x , we can replace it with x y − 1 . So, we can move all the x 's to the left and all the y 's to the right. Since x 2 = e , we only need to consider x 0 = e and x 1 = x . Since y n = e , we only need to consider y j where 0 ≤ j < n . Thus, any product can be written in the form x i y j where 0 ≤ i ≤ 1 and 0 ≤ j < n .
Examples
Dihedral groups are useful in understanding symmetries in various fields, such as physics, chemistry, and computer graphics. For example, in chemistry, the symmetry of a molecule can be described using a dihedral group. In computer graphics, dihedral groups can be used to generate symmetrical patterns and designs. Understanding the generators and relations of dihedral groups allows us to analyze and manipulate these symmetries more effectively. For instance, knowing that a square has D 8 symmetry helps in designing algorithms that preserve the square's appearance under various transformations like rotations and reflections. The relations x 2 = e , y 4 = e , and x y x = y − 1 (where x is a reflection and y is a 90-degree rotation) completely define how these transformations interact, enabling precise control over symmetrical operations.
