en>BenFrantzDale: /* Special cases */

2011-08-02T13:41:30Z

Special cases

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In [[mathematics]], the '''generalized dihedral groups''' are a family of [[Group (mathematics)|groups]] with algebraic structures similar to that of the [[dihedral group]]s. They include the finite dihedral groups, the [[infinite dihedral group]], and the [[orthogonal group]] ''O''(2).

==Definition==
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., <math>\mathrm{Dih}(H) = H \rtimes_\phi Z_2</math> with φ(0) the identity and φ(1) inversion.

Thus we get:
:(''h''1, 0) * (''h''2, ''t''2) = (''h''1 + ''h''2, ''t''2)
:(''h''1, 1) * (''h''2, ''t''2) = (''h''1 − ''h''2, 1 + ''t''2)
for all ''h''1, ''h''2 in ''H'' and ''t''2 in Z2.

(Writing Z2 multiplicatively, we have (''h''1, ''t''1) * (''h''2, ''t''2) = (''h''1 + ''t''1''h''2, ''t''1''t''2) .)

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 of a subgroup|index]] 2, isomorphic to ''H'', while the elements (''h'', 1) are all their own inverse.

The [[conjugacy class]]es are:
*the sets {(''h'',0 ), (−''h'',0 )}
*the sets {(''h'' + ''k'' + ''k'', 1) | ''k'' in ''H'' }

Thus for every subgroup ''M'' of ''H'', the corresponding set of elements (''m'',0) is also a normal subgroup. We have:
::Dih(''H'') ''/'' ''M'' = Dih ( ''H / M'' )

==Examples==
*Dih''n'' = Dih('''Z'''''n'') (the [[dihedral group]]s)
**For even ''n'' there are two sets {(''h'' + ''k'' + ''k'', 1) | ''k'' in ''H'' }, and each generates a normal subgroup of type Dih''n /'' 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 (the rotations of both are the same). However, they are isomorphic as abstract groups.
**For odd ''n'' there is only one set {(''h'' + ''k'' + ''k'', 1) | ''k'' in ''H'' }
*Dih∞ = Dih('''Z''') (the [[infinite dihedral group]]); there are two sets {(''h'' + ''k'' + ''k'', 1) | ''k'' in ''H'' }, 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 (the translations of both are the same: by even numbers). However, they are isomorphic as abstract groups.
*Dih(S1), or [[orthogonal group]] O(2,'''R'''), or O(2): the isometry group of a [[circle]], or equivalently, the group of isometries in 2D that keep the origin fixed. The rotations form the [[circle group]] S1, or equivalently SO(2,'''R'''), also written SO(2), and '''R'''/'''Z''' ; it is also the multiplicative group of [[complex number]]s of [[absolute value]] 1. In the latter case one of the reflections (generating the others) is [[Complex conjugate|complex conjugation]]. There are no proper normal subgroups with reflections. The discrete normal subgroups are cyclic groups of order ''n'' for all positive integers ''n''. The quotient groups are isomorphic with the same group Dih(S1).
*Dih('''R'''''n'' ): the group of isometries of '''R'''''n'' consisting of all translations and inversion in all points; for ''n'' = 1 this is the [[Euclidean group]] [[Symmetry groups in one dimension|E(1)]]; for ''n'' > 1 the group Dih('''R'''''n'' ) is a proper subgroup of E(''n'' ), i.e. it does not contain all isometries.
*''H'' can be any subgroup of '''R'''''n'', e.g. a discrete subgroup; in that case, if it extends in ''n'' directions it is a [[lattice (group)|lattice]].
**Discrete subgroups of Dih('''R'''2 ) which contain translations in one direction are of [[frieze group]] type <math>\infty\infty</math> and 22<math>\infty</math>.
**Discrete subgroups of Dih('''R'''2 ) which contain translations in two directions are of [[wallpaper group]] type p1 and p2.
**Discrete subgroups of Dih('''R'''3 ) which contain translations in three directions are [[space group]]s of the [[triclinic]] [[crystal system]].

==Properties==
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]] [[p-group|2-group]]:
*Dih('''Z'''1) = Dih1 = '''Z'''2
*Dih('''Z'''2) = Dih2 = '''Z'''2 × '''Z'''2 ([[Klein four-group]])
*Dih(Dih2) = Dih2 × '''Z'''2 = '''Z'''2 × '''Z'''2 × '''Z'''2
etc.

==Topology==
Dih('''R'''''n'' ) and its dihedral subgroups are disconnected [[topological group]]s. Dih('''R'''''n'' ) consists of two [[Connected space|connected]] components: the [[identity component]] isomorphic to '''R'''''n'', 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:
*Dih∞ as the isometry group of '''Z'''
*Dih∞ as a 2-dimensional isometry group generated by a rotation by an irrational number of turns, and a reflection

Both topological groups are [[totally disconnected space|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).

[[Category:Group theory]]

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en>BenFrantzDale: /* Special cases */