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 D2n. 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 Dn instead of Wikipedia's notation D2n.)
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 D2n is greater than 4, the operations of rotation and reflection in general do not commute and D2n 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.
Some equivalent definitions of D2n 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 Cn and C2, with C2 acting on Cn by inversion (thus, D2n always has a normal subgroup isomorphic to Cn)
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.