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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 group O(2). Dihedral groups play an important role in group theory, geometry, and chemistry.
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. I.e., with φ(0) the identity and φ(1) inversion.
Thus we get:
for all h1, h2 in H and t2 in Z2.
(Writing Z2 multiplicatively, we have (h1, t1) * (h2, t2) = (h1 + t1h2, t1t2) .)
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.
The conjugacy classes are:
Thus for every subgroup M of H, the corresponding set of elements (m,0) is also a normal subgroup. 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. Dih(Rn ) consists of two connected components: the identity component isomorphic to Rn, and the component with the reflections. Similarly O(2) 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 (singleton) components are open, while in the second case they are not. Also, the first topological group is a closed subgroup of Dih(R) but the second is not a closed subgroup of O(2).
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In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups.
