In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal groupO.
Definition
For any abelian groupH, the generalized dihedral group of H, written Dih, is the semidirect product of H and Z2, with Z2 acting on H by inverting elements. I.e., with φ the identity and φ inversion. Thus we get: for all h1, h2 in H and t2 in Z2. * = Note that * =, i.e. first the inversion and then the operation in H. Also * = ; indeed inverts h, and toggles t between "normal" and "inverted" . The subgroup of Dih of elements is a normal subgroup of index 2, isomorphic to H, while the elements are all their own inverse. The conjugacy classes are:
the sets
the sets
Thus for every subgroup M of H, the corresponding set of elements is also a normal subgroup. We have:
Examples
Dihn = Dih
*For even n there are two sets, and each generates a normal subgroup of type Dihn / 2. As subgroups of the isometry group of the set of vertices of a regular n-gon they are different: the reflections in one subgroup all have two fixed points, while none in the other subgroup has. However, they are isomorphic as abstract groups.
*For odd n there is only one set
Dih∞ = Dih ; there are two sets, and each generates a normal subgroup of type Dih∞. As subgroups of the isometry group of Z they are different: the reflections in one subgroup all have a fixed point, the mirrors are at the integers, while none in the other subgroup has, the mirrors are in between. However, they are isomorphic as abstract groups.
Dih: the group of isometries of Rnconsisting of all translations and inversion in all points; for n = 1 this is the Euclidean group E; for n > 1 the group Dih is a proper subgroup of E, i.e. it does not contain all isometries.
H can be any subgroup of Rn, e.g. a discrete subgroup; in that case, if it extends in n directions it is a lattice.
Dih and its dihedral subgroups are disconnected topological groups. Dih consists of two connected components: the identity component isomorphic to Rn, and the component with the reflections. Similarly O consists of two connected components: the identity component isomorphic to the circle group, and the component with the reflections. For the group Dih∞ we can distinguish two cases:
Both topological groups are totally disconnected, but in the first case the components are open, while in the second case they are not. Also, the first topological group is a closed subgroup of Dih but the second is not a closed subgroup of O.
