# 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 2

*n*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 D_{4}, which is generated by a rotation *r* of 180 degrees, and a reflection *f* across the y-axis. The elements of D_{4} can then be represented as {*e*, *r*, *f*, *rf*}, where *e* is the identity or null transformation.

D_{4} 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 D_{8}, 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}.

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*}; {*r*^{n},*f*², (*rf*)²}). - The semidirect product of cyclic groups C
_{n}and C_{2}, with C_{2}acting on C_{n}by inversion (thus, D_{2n}always has a normal subgroup isomorphic to C_{n})

_{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 *r*^{n} 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*}; {*a*², *b*²}}, and is isomorphic to a semidirect product of **Z** and C_{2}.

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 C_{2}, 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}.