dihedral group

In group theory, the dihedral groups are certain groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations. Specifically, the finite dihedral group D_2n has 2n elements and is generated by a rotation r with order n and a reflection f with order 2; the two elements satisfy

r f = f r ^-1.

If n ≥ 3, then D_2n is the symmetry group of a regular polygon with n sides. (Note that some authors use the notation D_n instead of our D_2n.) The simplest dihedral group is D₄, which is generated by the rotation r of 180 degrees, and the reflection f across the y-axis. The elements of D₄ can then be represented as {e, r, f, rf}, where e is the identity or null transformation and rf is the reflection across the x-axis. D₄ is isomorphic to the Klein four-group. If the order of D_2n is greater than 4, the operations of rotation and reflection in general do not commute and D_2n is not abelian; for example, in D₈, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees: Thus, beyond their obvious application to problems of symmetry in the plane, these groups are among the simplest examples of non-abelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups. The 2n elements of D_2n can be written as e, r, r²,...,r^n-1, f, fr, fr²,...,fr^n-1. The first n listed elements are rotations and the remaining n elements are axis-reflections (all of which have order 2). The product of two rotations or two reflections is a rotation; the product of a rotation and a reflection is a reflection. Some equivalent definitions of D_2n are:

The automorphism group of the graph consisting only of a cycle with n vertices (if n ≥ 3).
The group with presentation

({r,f}; {rⁿ, f ², (rf)²}), or
({a,b}; {a ², b ², (ab)ⁿ}) (and indeed the only finite groups that can be generated by two elements of order 2 are the dihedral groups and the cyclic groups)

The semidirect product of cyclic groups C_n and C₂, with C₂ acting on C_n by inversion (thus, D_2n always has a normal subgroup isomorphic to C_n)

If we consider D_2n (n ≥ 3) as the symmetry group of a regular n-gon and number the polygon's vertices, we see that D_2n is a subgroup of the symmetric group S_n. The properties of the dihedral groups D_2n with n ≥ 3 depend on whether n is even or odd. For example, the center of D_2n consists only of the identity if n is odd, but contains the element r^n/2 if n is even. All the reflections are conjugate to each other in case n is odd, but they fall into two conjugacy classes if n is even. This corresponds to the geometrical fact that every symmetry axis of a regular n-gon passes through a vertex and an opposite side if n is odd, but half of them pass through opposite sides and half pass through opposite vertices if n is even. If m divides n, then D_2m is a subgroup of D_2n. The total number of subgroups of D_2n (n ≥ 3), is equal to d(n) + σ(n), where d(n) is the number of positive divisors of n and σ(n) is the sum of the positive divisors of n. In addition to the finite dihedral groups, there is the infinite dihedral group D_∞. Every dihedral group is generated by a rotation r and a reflection; if the rotation is a rational multiple of a full rotation, then there is some integer n such that rⁿ is the identity, and we have a finite dihedral group. If the rotation is not a rational multiple of a full rotation, then there is no such n and the resulting group has infinitely many elements and is called D_∞. It has presentations

({r,f}; {f ², (rf)²}), or
({a,b}; {a², b²}),

and is isomorphic to a semidirect product of Z and C₂, and to the free product C₂ * C₂. It can also be visualized as the automorphism group of the graph consisting of a path infinite to both sides. Finally, if H is any abelian group, we can speak of the generalized dihedral group of H (sometimes written Dih(H)). This group is a semidirect product of H and C₂, with C₂ acting on H by inverting elements. Dih(H) has a normal subgroup of index 2 isomorphic to H, and contains in addition an element f of order 2 such that, for all x in H, x f = f x^-1. Clearly, we have D_2n = Dih(C_n) and D_∞ = Dih(Z). The symmetry group of a straight line is isomorphic to Dih(R) and the symmetry group of a circle is Dih(S¹) (where S¹ denotes the multiplicative group of complex numbers of absolute value 1).

Dihedral Group

See Also