This article may be too technical for most readers to understand.(April 2015) |

In mathematics, **D _{3}** (sometimes alternatively denoted by

- Symmetry groups
- Permutations of a set of three objects
- Summary of group operations
- Conjugacy classes
- Subgroups
- Semidirect products
- Group action
- Orbits and stabilizers
- Representation theory
- See also
- References
- External links

This page illustrates many group concepts using this group as example.

The dihedral group D_{3} is the symmetry group of an equilateral triangle, that is, it is the set of all transformations such as reflection, rotation, and combinations of these, that leave the shape and position of this triangle fixed. In the case of D_{3}, every possible permutation of the triangle's vertices constitutes such a transformation, so that the group of these symmetries is isomorphic to the symmetric group S_{3} of all permutations of three distinct elements. This is not the case for dihedral groups of higher orders.

The dihedral group D_{3} is isomorphic to two other symmetry groups in three dimensions:

- one with a 3-fold rotation axis and a perpendicular 2-fold rotation axis (hence three of these): D
_{3} - one with a 3-fold rotation axis in a plane of reflection (and hence also in two other planes of reflection): C
_{3v}

Consider three colored blocks (red, green, and blue), initially placed in the order RGB. The symmetric group S_{3} is then the group of all possible rearrangements of these blocks. If we denote by *a* the action "swap the first two blocks", and by *b* the action "swap the last two blocks", we can write all possible permutations in terms of these two actions.

In multiplicative form, we traditionally write *xy* for the combined action "first do *y*, then do *x*"; so that *ab* is the action RGB ↦ RBG ↦ BRG, i.e., "take the last block and move it to the front". If we write *e* for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:

*e*: RGB ↦ RGB or ()*a*: RGB ↦ GRB or (RG)*b*: RGB ↦ RBG or (GB)*ab*: RGB ↦ BRG or (RGB)*ba*: RGB ↦ GBR or (RBG)*aba*: RGB ↦ BGR or (RB)

The notation in brackets is the cycle notation.

Note that the action *aa* has the effect RGB ↦ GRB ↦ RGB, leaving the blocks as they were; so we can write *aa* = *e*. Similarly,

*bb*=*e*,- (
*aba*)(*aba*) =*e*, and - (
*ab*)(*ba*) = (*ba*)(*ab*) =*e*;

so each of the above actions has an inverse.

By inspection, we can also determine associativity and closure (two of the necessary group axioms); note for example that

- (
*ab*)*a*=*a*(*ba*) =*aba*, and - (
*ba*)*b*=*b*(*ab*) =*bab*.

The group is non-abelian since, for example, *ab* ≠ *ba*. Since it is built up from the basic actions *a* and *b*, we say that the set {*a*, *b*}* generates * it.

The group has presentation

- , also written

- or
- , also written

where *a* and *b* are swaps and *r* = *ab* is a cyclic permutation. Note that the second presentation means that the group is a Coxeter group. (In fact, all dihedral and symmetry groups are Coxeter groups.)

With the generators *a* and *b*, we define the additional shorthands *c* := *aba*, *d* := *ab* and *f* := *ba*, so that *a, b, c, d, e*, and *f* are all the elements of this group. We can then summarize the group operations in the form of a Cayley table:

* | e | a | b | c | d | f |
---|---|---|---|---|---|---|

e | e | a | b | c | d | f |

a | a | e | d | f | b | c |

b | b | f | e | d | c | a |

c | c | d | f | e | a | b |

d | d | c | a | b | f | e |

f | f | b | c | a | e | d |

Note that non-equal non-identity elements only commute if they are each other's inverse. Therefore, the group is centerless, i.e., the center of the group consists only of the identity element.

We can easily distinguish three kinds of permutations of the three blocks, the conjugacy classes of the group:

- no change (), a group element of order 1
- interchanging two blocks: (RG), (RB), (GB), three group elements of order 2
- a cyclic permutation of all three blocks: (RGB), (RBG), two group elements of order 3

For example, (RG) and (RB) are both of the form (*x**y*); a permutation of the letters R, G, and B (namely (GB)) changes the notation (RG) into (RB). Therefore, if we apply (GB), then (RB), and then the inverse of (GB), which is also (GB), the resulting permutation is (RG).

Note that conjugate group elements always have the same order, but in general two group elements that have the same order need not be conjugate.

From Lagrange's theorem we know that any non-trivial subgroup of a group with 6 elements must have order 2 or 3. In fact the two cyclic permutations of all three blocks, with the identity, form a subgroup of order 3, index 2, and the swaps of two blocks, each with the identity, form three subgroups of order 2, index 3. The existence of subgroups of order 2 and 3 is also a consequence of Cauchy's theorem.

The first-mentioned is { (), (RGB), (RBG) }, the alternating group A_{3}.

The left cosets and the right cosets of A_{3} coincide (as they do for any subgroup of index 2) and consist of A_{3} and the set of three swaps { (RB), (RG), (BG)}.

The left cosets of { (), (RG) } are:

- { (), (RG) }
- { (RB), (RGB) }
- { (GB), (RBG) }

The right cosets of { (RG), () } are:

- { (RG), () }
- { (RBG), (RB) }
- { (RGB), (GB) }

Thus A_{3} is normal, and the other three non-trivial subgroups are not. The quotient group *G* / *A*_{3} is isomorphic with *C*_{2}.

, a semidirect product, where *H* is a subgroup of two elements: () and one of the three swaps. This decomposition is also a consequence (particular case) of the Schur–Zassenhaus theorem.

In terms of permutations the two group elements of *G* / A_{3} are the set of even permutations and the set of odd permutations.

If the original group is that generated by a 120°-rotation of a plane about a point, and reflection with respect to a line through that point, then the quotient group has the two elements which can be described as the subsets "just rotate (or do nothing)" and "take a mirror image".

Note that for the symmetry group of a *square*, an uneven permutation of vertices does *not* correspond to taking a mirror image, but to operations not allowed for *rectangles*, i.e. 90° rotation and applying a diagonal axis of reflection.

is if both *φ*(0) and *φ*(1) are the identity. The semidirect product is isomorphic to the dihedral group of order 6 if *φ*(0) is the identity and *φ*(1) is the non-trivial automorphism of C_{3}, which inverses the elements.

Thus we get:

- (
*n*_{1}, 0) * (*n*_{2},*h*_{2}) = (*n*_{1}+*n*_{2},*h*_{2}) - (
*n*_{1}, 1) * (*n*_{2},*h*_{2}) = (*n*_{1}−*n*_{2}, 1 +*h*_{2})

for all *n*_{1}, *n*_{2} in C_{3} and *h*_{2} in C_{2}. More concisely,

for all *n*_{1}, *n*_{2} in C_{3} and *h*_{1}, *h*_{2} in C_{2}.

In a Cayley table:

00 | 10 | 20 | 01 | 11 | 21 | |
---|---|---|---|---|---|---|

00 | 00 | 10 | 20 | 01 | 11 | 21 |

10 | 10 | 20 | 00 | 11 | 21 | 01 |

20 | 20 | 00 | 10 | 21 | 01 | 11 |

01 | 01 | 21 | 11 | 00 | 20 | 10 |

11 | 11 | 01 | 21 | 10 | 00 | 20 |

21 | 21 | 11 | 01 | 20 | 10 | 00 |

Note that for the second digit we essentially have a 2×2 table, with 3×3 equal values for each of these 4 cells. For the first digit the left half of the table is the same as the right half, but the top half is different from the bottom half.

For the *direct* product the table is the same except that the first digits of the bottom half of the table are the same as in the top half.

Consider *D*_{3} in the geometrical way, as a symmetry group of isometries of the plane, and consider the corresponding group action on a set of 30 evenly spaced points on a circle, numbered 0 to 29, with 0 at one of the reflexion axes.

This section illustrates group action concepts for this case.

The action of *G* on *X* is called

*transitive*if for any two*x*,*y*in*X*there exists a*g*in*G*such that*g*·*x*=*y*; this is not the case*faithful*(or*effective*) if for any two different*g*,*h*in*G*there exists an*x*in*X*such that*g*·*x*≠*h*·*x*; this is the case, because, except for the identity, symmetry groups do not contain elements that "do nothing"*free*if for any two different*g*,*h*in*G*and all*x*in*X*we have*g*·*x*≠*h*·*x*; this is not the case because there are reflections

The orbit of a point *x* in *X* is the set of elements of *X* to which *x* can be moved by the elements of *G*. The orbit of *x* is denoted by *Gx*:

The orbits are {0, 10, 20},{1, 9, 11, 19, 21, 29},{2, 8, 12, 18, 22, 28},{3, 7, 13, 17, 23, 27},{4, 6, 14, 16, 24, 26}, and {5, 15, 25}. The points within an orbit are "equivalent". If a symmetry group applies for a pattern, then within each orbit the color is the same.

The set of all orbits of *X* under the action of *G* is written as *X* / *G*.

If *Y* is a subset of *X*, we write *GY* for the set { *g* · *y* : *y* ∈ *Y* and *g* ∈ *G* }. We call the subset *Y**invariant under G* if *GY* = *Y* (which is equivalent to *GY* ⊆ *Y*). In that case, *G* also operates on *Y*. The subset *Y* is called *fixed under G* if *g* · *y* = *y* for all *g* in *G* and all *y* in *Y*. The union of e.g. two orbits is invariant under *G*, but not fixed.

For every *x* in *X*, we define the **stabilizer subgroup** of *x* (also called the **isotropy group** or **little group**) as the set of all elements in *G* that fix *x*:

If *x* is a reflection point (0, 5, 10, 15, 20, or 25), its stabilizer is the group of order two containing the identity and the reflection in *x*. In other cases the stabilizer is the trivial group.

For a fixed *x* in *X*, consider the map from *G* to *X* given by *g* ↦ *g* · *x*. The image of this map is the orbit of *x* and the coimage is the set of all left cosets of *G _{x}*. The standard quotient theorem of set theory then gives a natural bijection between

If two elements *x* and *y* belong to the same orbit, then their stabilizer subgroups, *G*_{x} and *G*_{y}, are isomorphic. More precisely: if *y* = *g* · *x*, then *G*_{y} = *gG*_{x}*g*^{−1}. In the example this applies e.g. for 5 and 25, both reflection points. Reflection about 25 corresponds to a rotation of 10, reflection about 5, and rotation of −10.

A result closely related to the orbit-stabilizer theorem is Burnside's lemma:

where *X*^{g} is the set of points fixed by *g*. I.e., the number of orbits is equal to the average number of points fixed per group element.

For the identity all 30 points are fixed, for the two rotations none, and for the three reflections two each: {0, 15},{5, 20}, and {10, 25}. Thus, the average is six, the number of orbits.

Up to isomorphism, this group has three irreducible complex unitary representations, which we will call (the trivial representation), and , where the subscript indicates the dimension. By its definition as a permutation group over the set with three elements, the group has a representation on by permuting the entries of the vector, the fundamental representation. This representation is not irreducible, as it decomposes as a direct sum of and . appears as the subspace of vectors of the form and is the representation on its orthogonal complement, which are vectors of the form . The nontrivial one-dimensional representation arises through the groups grading: The action is multiplication by the sign of the permutation of the group element. Every finite group has such a representation since it is a subgroup of a cyclic group by its regular action. Counting the square dimensions of the representations (, the order of the group), we see these must be all of the irreducible representations.^{ [2] }

A 2-dimensional irreducible linear representation yields a 1-dimensional projective representation (i.e., an action on the projective line, an embedding in the Möbius group PGL(2, **C**)), as elliptic transforms. This can be represented by matrices with entries 0 and ±1 (here written as fractional linear transformations), known as the anharmonic group:

- order 1:
- order 2:
- order 3:

and thus descends to a representation over any field, which is always faithful/injective (since no two terms differ only by only a sign). Over the field with two elements, the projective line has only 3 points, and this is thus the exceptional isomorphism In characteristic 3, this embedding stabilizes the point since (in characteristic greater than 3 these points are distinct and permuted, and are the orbit of the harmonic cross-ratio). Over the field with three elements, the projective line has 4 elements, and since PGL(2, 3) is isomorphic to the symmetric group on 4 elements, S_{4}, the resulting embedding equals the stabilizer of the point .

In mathematics, especially in category theory and homotopy theory, a **groupoid** generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In mathematics, a **group action** on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group *acts* on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.

In mathematics, a **permutation group** is a group *G* whose elements are permutations of a given set *M* and whose group operation is the composition of permutations in *G* (which are thought of as bijective functions from the set *M* to itself). The group of *all* permutations of a set *M* is the symmetric group of *M*, often written as Sym(*M*). The term *permutation group* thus means a subgroup of the symmetric group. If *M* = {1, 2, ..., *n*} then Sym(*M*) is usually denoted by S_{n}, and may be called the *symmetric group on n letters*.

In group theory, the **symmetry group** of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object *X* is *G* = Sym(*X*).

In abstract algebra, the **symmetric group** defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are such permutation operations, the order of the symmetric group is .

In mathematics, specifically in group theory, the concept of a **semidirect product** is a generalization of a direct product. There are two closely related concepts of semidirect product:

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, geometry, and chemistry.

In group theory, **Cayley's theorem**, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group. More specifically, G is isomorphic to a subgroup of the symmetric group whose elements are the permutations of the underlying set of G. Explicitly,

Some elementary **examples of groups** in mathematics are given on Group (mathematics). Further examples are listed here.

In mathematics, the **orthogonal group** in dimension *n*, denoted O(*n*), is the group of distance-preserving transformations of a Euclidean space of dimension *n* that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the **general orthogonal group**, by analogy with the general linear group. Equivalently, it is the group of *n*×*n* orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.

In mathematics, specifically group theory, the **index** of a subgroup *H* in a group *G* is the number of left cosets of *H* in *G*, or equivalently, the number of right cosets of *H* in *G*. The index is denoted or or . Because *G* is the disjoint union of the left cosets and because each left coset has the same size as *H*, the index is related to the orders of the two groups by the formula

A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.

In mathematics, a **Cayley graph**, also known as a **Cayley color graph**, **Cayley diagram**, **group diagram**, or **color group** is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem, and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing families of expander graphs.

In geometry, the **cross-ratio**, also called the **double ratio** and **anharmonic ratio**, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points *A*, *B*, *C* and *D* on a line, their cross ratio is defined as

In group theory, a field of mathematics, a **double coset** is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let *G* be a group, and let *H* and *K* be subgroups. Let *H* act on *G* by left multiplication and let *K* act on *G* by right multiplication. For each *x* in *G*, the **( H, K)-double coset of x** is the set

The concept of **system of imprimitivity** is used in mathematics, particularly in algebra and analysis, both within the context of the theory of group representations. It was used by George Mackey as the basis for his theory of induced unitary representations of locally compact groups.

A **one-dimensional symmetry group** is a mathematical group that describes symmetries in one dimension (1D).

In mathematics, especially in the area of algebra known as group theory, the **holomorph** of a group is a group that simultaneously contains the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. In group theory, for a group , the holomorph of denoted can be described as a semidirect product or as a permutation group.

In mathematics, and especially in geometry, an object has **icosahedral symmetry** if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron and the rhombic triacontahedron.

In Galois theory, a discipline within the field of abstract algebra, a **resolvent** for a permutation group *G* is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial *p* and has, roughly speaking, a rational root if and only if the Galois group of *p* is included in *G*. More exactly, if the Galois group is included in *G*, then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are

- ↑ Kubo, Jisuke (2008), "The dihedral group as a family group",
*Quantum field theory and beyond*, World Sci. Publ., Hackensack, NJ, pp. 46–63, doi:10.1142/9789812833556_0004, MR 2588575 . For the identification of D_{3}with S_{3}, and the observation that this group is the smallest possible non-abelian group, see p. 49. - ↑ <span class="citation mathworld" id="Reference-Mathworld-Dihedral Group D
_{3}"> Weisstein, Eric W. "Dihedral Group D_{3}".*MathWorld*.

- Fraleigh, John B. (1993),
*A First Course in Abstract Algebra*(5th ed.), Addison-Wesley, pp. 93–94, ISBN 978-0-201-53467-2

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