generalized dihedral group
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In mathematics, the generalized dihedral groups are a family of
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
with algebraic structures similar to that of the
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
s. They include the finite dihedral groups, the
infinite dihedral group In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, ''p1m1'', ...
, and the orthogonal group ''O''(2). Dihedral groups play an important role in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
,
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, and chemistry.


Definition

For any
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
''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., \mathrm(H) = H \rtimes_\phi Z_2 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 2, isomorphic to ''H'', while the elements (''h'', 1) are all their own inverse. The
conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ...
es are: *the sets *the sets 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 In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
s) **For even ''n'' there are two sets , 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 *Dih = Dih(Z) (the
infinite dihedral group In mathematics, the infinite dihedral group Dih∞ is an infinite group with properties analogous to those of the finite dihedral groups. In two-dimensional geometry, the infinite dihedral group represents the frieze group symmetry, ''p1m1'', ...
); there are two sets , 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 A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
, 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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s of absolute value 1. In the latter case one of the reflections (generating the others) is 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 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 Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
. **Discrete subgroups of Dih(R2 ) which contain translations in one direction are of
frieze group In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. Frieze patterns can be classified into seven types according to their symmetrie ...
type \infty\infty and 22\infty. **Discrete subgroups of Dih(R2 ) which contain translations in two directions are of
wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformati ...
type p1 and p2. **Discrete subgroups of Dih(R3 ) which contain translations in three directions are
space group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it uncha ...
s of the
triclinic 180px, Triclinic (a ≠ b ≠ c and α ≠ β ≠ γ ) In crystallography, the triclinic (or anorthic) crystal system is one of the 7 crystal systems. A crystal system is described by three basis vectors. In the triclinic system, the crystal i ...
crystal system In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their poin ...
.


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
2-group In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of ''n''-groups. In some of the literature, 2-groups are also called gr-categories or groupal ...
: *Dih(Z1) = Dih1 = Z2 *Dih(Z2) = Dih2 = Z2 × Z2 (
Klein four-group In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. ...
) *Dih(Dih2) = Dih2 × Z2 = Z2 × Z2 × Z2 etc.


Topology

Dih(R''n'' ) and its dihedral subgroups are disconnected
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s. Dih(R''n'' ) consists of two
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
components: the
identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity compo ...
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 In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
, 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).


References

{{reflist Group theory