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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, D3 (sometimes alternatively denoted by D6) is 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, ...
of degree 3, or, in other words, the dihedral group of
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
6. It is isomorphic to the
symmetric group 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 ...
S3 of degree 3. It is also the smallest possible
non-abelian group In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ' ...
.. For the identification of D3 with S3, and the observation that this group is the smallest possible non-abelian group, se
p. 49
This page illustrates many group concepts using this group as example.


Symmetry groups

The dihedral group D3 is the
symmetry group 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 amb ...
of an
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
, 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 D3, every possible
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
of the triangle's vertices constitutes such a transformation, so that the group of these symmetries is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the symmetric group S3 of all permutations of three distinct elements. This is not the case for dihedral groups of higher orders. : The dihedral group D3 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): D3 *one with a 3-fold rotation axis in a plane of reflection (and hence also in two other planes of reflection): C3v :


Permutations of a set of three objects

Consider three colored blocks (red, green, and blue), initially placed in the order RGB. The
symmetric group 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 ...
S3 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 , 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
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s of the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
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 (RBG) * ''ba'' : RGB ↦ GBR or (RGB) * ''aba'' : RGB ↦ BGR or (RB) The notation in brackets is the
cycle notation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or p ...
. Note that the action ''aa'' has the effect , leaving the blocks as they were; so we can write . 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 In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
and closure (two of the necessary
group axioms In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. Th ...
); note for example that * (''ab'')''a'' = ''a''(''ba'') = ''aba'', and * (''ba'')''b'' = ''b''(''ab'') = ''bab''. The group is non-abelian since, for example, . Since it is built up from the basic actions ''a'' and ''b'', we say that the set '' generates'' it. The group has
presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
::\langle r, a \mid r^3 = 1, a^2 = 1, ara = r^ \rangle, also written \langle r, a \mid r^3, a^2, arar \rangle :or ::\langle a, b \mid a^2 = b^2 = (ab)^3 = 1 \rangle, also written \langle a, b \mid a^2, b^2, (ab)^3 \rangle where ''a'' and ''b'' are swaps and is a cyclic permutation. Note that the second presentation means that the group is a
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
. (In fact, all dihedral and symmetry groups are Coxeter groups.)


Summary of group operations

With the generators ''a'' and ''b'', we define the additional shorthands , and , 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 Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplic ...
: Note that non-equal non-identity elements only
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
if they are each other's inverse. Therefore, the group is
centerless In abstract algebra, the center of a group, , is the set of elements that commute with every element of . It is denoted , from German '' Zentrum,'' meaning ''center''. In set-builder notation, :. The center is a normal subgroup, . As a subgr ...
, i.e., the center of the group consists only of the identity element.


Conjugacy classes

We can easily distinguish three kinds of permutations of the three blocks, 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 of the group: *no change (), a group element of
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
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 Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
, but in general two group elements that have the same order need not be conjugate.


Subgroups

From Lagrange's theorem we know that any non-trivial
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of a group with 6 elements must have order 2 or 3. In fact the two
cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...
s of all three blocks, with the identity, form a subgroup of order 3,
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
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 the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic pr ...
A3. The left
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s and the right cosets of A3 coincide (as they do for any subgroup of index 2) and consist of A3 and the set of three swaps . The left cosets of are: * * * The right cosets of are: * * * Thus A3 is
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
, and the other three non-trivial subgroups are not. The
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
is isomorphic with ''C''2. G = \mathrm_3 \rtimes H, a
semidirect product 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: * an ''inner'' semidirect product is a particular way in wh ...
, 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 The Schur–Zassenhaus theorem is a theorem in group theory which states that if G is a finite group, and N is a normal subgroup whose order is coprime to the order of the quotient group G/N, then G is a semidirect product (or split extension) ...
. In terms of permutations the two group elements of 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 A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substance ...
". 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.


Semidirect products

\mathrm_3 \rtimes_\varphi \mathrm_2 is \mathrm_3 \times \mathrm_2 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 C3, 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 C3 and ''h''2 in C2. More concisely, :(n_1, h_1) * (n_2, h_2) = (n_1 + (-1)^ n_2, h_1 + h_2) for all ''n''1, ''n''2 in C3 and ''h''1, ''h''2 in C2. In a Cayley table: 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.


Group action

Consider ''D''3 in the geometrical way, as a
symmetry group 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 amb ...
of
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
of the plane, and consider the corresponding
group action 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 ...
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 ; 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 ; 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 ; this is not the case because there are reflections


Orbits and stabilizers

The
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
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'': :Gx = \left\ The orbits are and 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 . If ''Y'' is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of ''X'', we write ''GY'' for the set We call the subset ''Y'' ''invariant under G'' if (which is equivalent to . In that case, ''G'' also operates on ''Y''. The subset ''Y'' is called ''fixed under G'' if 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'': :G_x = \ If ''x'' is a reflection point , 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 . The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of this map is the orbit of ''x'' and the
coimage In algebra, the coimage of a homomorphism :f : A \rightarrow B is the quotient :\text f = A/\ker(f) of the domain by the kernel. The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies. ...
is the set of all left
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s of ''Gx''. The standard quotient theorem of set theory then gives a natural
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between and ''Gx''. Specifically, the bijection is given by . This result is known as the orbit-stabilizer theorem. In the two cases of a small orbit, the stabilizer is non-trivial. If two elements ''x'' and ''y'' belong to the same orbit, then their stabilizer subgroups, ''G''''x'' and ''G''''y'', are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. 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 Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma, the orbit-counting theorem, or the Lemma that is not Burnside's, is a result in group theory that is often useful in taking account of symmetry when ...
: :\left, X/G\=\frac\sum_\left, X^g\ 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: and Thus, the average is six, the number of orbits.


Representation theory

Up to isomorphism, this group has three irreducible complex unitary representations, which we will call I (the trivial representation), \rho_1 and \rho_2, 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 \mathbb^3 by permuting the entries of the vector, the fundamental representation. This representation is not irreducible, as it decomposes as a direct sum of I and \rho_2. I appears as the subspace of vectors of the form (\lambda, \lambda, \lambda), \lambda \in \mathbb and \rho_2 is the representation on its orthogonal complement, which are vectors of the form (\lambda_1, \lambda_2, -\lambda_1 -\lambda_2). The nontrivial one-dimensional representation \rho_1 arises through the groups \mathbb_2 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 (1^2 + 1^2 + 2^2 = 6, the order of the group), we see these must be all of the irreducible representations. 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 ), as elliptic transforms. This can be represented by matrices with entries 0 and ±1 (here written as
fractional linear transformation In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form :z \mapsto \frac , which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfo ...
s), known as the anharmonic group: * order 1: z * order 2: 1-z, 1/z, z/(z-1) * order 3: (z-1)/z, 1/(1-z) 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 mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ''a'i'' and ''b'j'' of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such is ...
S_3 \approx \mathrm(2, 2). In characteristic 3, this embedding stabilizes the point -1 = 1:1 since 2 = 1/2 = -1 (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 is isomorphic to the symmetric group on 4 elements, S4, the resulting embedding \mathrm_3 \hookrightarrow \mathrm_4 equals the stabilizer of the point -1.


See also

* Dihedral group of order 8


References

* {{citation, first=John B., last=Fraleigh, title=A First Course in Abstract Algebra, edition=5th, year=1993, publisher=Addison-Wesley, isbn=978-0-201-53467-2, pages=93–94


External links

*http://mathworld.wolfram.com/DihedralGroupD3.html Finite groups