User:Jim.belk/Generalized Dihedral Group Draft

In mathematics, the generalized dihedral group Dih(H) associated to an abelian group H is the semidirect product of H and a cyclic group of order 2, the latter acting on the former by negation.


 * $$0 \rightarrow H \rightarrow \mathrm{Dih}(H) \rightarrow \mathbb{Z}_2 \rightarrow 0$$

Elements of Dih(H) can be written as pairs (h, &epsilon;), where h &isin; H and &epsilon; = &plusmn;1, with the following rule for multiplication:


 * $$(h,\epsilon)(h^\prime,\epsilon^\prime) = (h + \epsilon h^\prime, \epsilon \epsilon^\prime)$$

Note that each element of the form (h, –1) is its own inverse.

Examples

 * Dih(Zn) is the dihedral group Dn.
 * Dih(Z) is the infinite dihedral group.
 * If S1 denotes the circle group, then Dih(S1) is the orthogonal group O(2).
 * More generally, Dih(SO(n)) is the orthogonal group O(n).
 * Dih(R) is the full isometry group of the line.
 * Dih(Rn) is the point reflection group consisting of all translations and point reflections of Rn.
 * If H is a lattice in Rn, then Dih(H) the subgroup of elements of Dih(Rn) that leave the lattice invariant.
 * If H is a one-dimensional lattice in R2, then Dih(H) is a frieze group of type &infin;&infin; or type 22&infin;.
 * If H is a two-dimensional lattice in R2, then Dih(H) is a wallpaper group type p1 and p2.
 * If H is a three-dimensional lattice in R3, then Dih(H) is the space group of a triclinic crystal system.
 * If H has exponent 2, then Dih(H) &cong; H &times; Z2.