This is a rough draft of a proposed article on generalized dihedral groups. Most of this content currently appears in the main dihedral groups article.

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.

${\displaystyle 0\rightarrow H\rightarrow \mathrm {Dih} (H)\rightarrow \mathbb {Z} _{2}\rightarrow 0}$

Elements of Dih(H) can be written as pairs (h, ε), where h ∈ H and ε = ±1, with the following rule for multiplication:

${\displaystyle (h,\epsilon )(h^{\prime },\epsilon ^{\prime })=(h+\epsilon h^{\prime },\epsilon \epsilon ^{\prime })}$

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

Examples

• Dih(Z_n) is the dihedral group D_n.
• 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 ∞∞ or type 22∞.
  • 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) ≅ H × Z₂.

See also

• dihedral group
• infinite dihedral group