Dihedral group

In group theory, a dihedral group is a group whose elements correspond to a closed set of rotations and reflections in the plane. The dihedral group with 2n elements is usually written as D_2n. It is generated by a single rotation r with order n, and a reflection f with order 2.

(Note that some authors use the notation D_n instead of Wikipedia's notation D_2n.)

The simplest dihedral group is D₄, which is generated by a rotation r of 180 degrees, and a 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.

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.

Whatever the order of the dihedral group, the rotation r and the reflection f always satisfy

r f = f r ^-1.

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.

Some equivalent definitions of D_2n are:

The symmetry group of a regular polygon with n sides (if n ≥ 3).
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}).
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)

The 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 will be the identity. If the rotation is not a rational multiple, then there is no such n; the resulting group is then called D_∞. It has presentation ({a,b}; {a², b²}}, and is isomorphic to a semidirect product of Z and C₂.

D_∞ can also be visualized as the automorphism group of the graph consisting of a path infinite to both sides.

Finally, if H is any non-trivial finite 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 order 2*order(H), a normal subgroup of index 2 isomorphic to H, and having an element f such that, for all x in H, f^-1 x f = x^-1.