For any abelian group H, the generalized dihedral group of H, written Dih(H), is the semidirect product of H and Z2, with Z2 acting on H by inverting elements.
Note that (h, 0) * (0,1) = (h,1), i.e. first the inversion and then the operation in H. Also (0, 1) * (h, t) = (−h, 1 + t); indeed (0,1) inverts h, and toggles t between "normal" (0) and "inverted" (1) (this combined operation is its own inverse).
The subgroup of Dih(H) of elements (h, 0) is a normal subgroup of index 2, isomorphic to H, while the elements (h, 1) are all their own inverse.
We have: Dih(H) is Abelian, with the semidirect product a direct product, if and only if all elements of H are their own inverse, i.e., an elementary abelian 2-group: etc.
Dih(Rn ) and its dihedral subgroups are disconnected topological groups.