The smallest non-Abelian group has 6 elements. It is a dihedral group with notation D₃ or D₆ (unfortunately both are used) and the symmetric group of degree 3, with notation S₃. In mathematics, an abelian group is a commutative group, i. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In group theory, the dihedral groups are certain groups consisting of rotations (about the origin) and reflections (across axes through the origin) of the plane, the group operation being composition of these reflections and rotations. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

It is the symmetry group of an equilateral triangle. As opposed to the case of e.g. a square, all permutations of the vertices can be achieved by rotation and flipping over (or reflecting). The symmetry group of an object (e. ...

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

Permutations of a set of three objects

Consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the action "swap the first block and the second block", and let b be the action "swap the second block and the third block".

Cycle graph for the group. A loop specifies a series of powers of any element connected to the identity element (1). For example, the e-ba-ab loop reflects the fact that (ba)²=ab and (ba)³=e, as well as the fact that (ab)²=ba and (ab)³=e The other "loops" are roots of unity so that, for example a²=e.

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: I, the creator of this image, hereby release it into the public domain. ... In the mathematical field of graph theory a cycle graph or circle graph is a graph that consists of a cycle. ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection, from a finite set X onto itself. ... In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ...

• 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)

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; note for example that

• (ab)a = a(ba) = aba, and
• (ba)b = b(ab) = aba.

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. In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

The group has presentation In mathematics, one method of defining a group is by a presentation. ...

, also written

or

, also written

where a and b are swaps and r is a cyclic permutation.

Summary of group operations

With x, y, and z different blocks R, G, and B we have:

• (xyz)(xyz)=(xzy)
• (xyz)(xzy)=()
• (xyz)(xy)=(xz)
• (xy)(xyz)=(yz)
• (xy)(xy)=()
• (xy)(xz)=(xzy)

In the form of a Cayley table: A Cayley table is a representation of a product defined on a set G. It is a group-theoretic generalization of an addition or a multiplication table. ...

Note that non-equal non-identity elements only commute if they are each other's inverse. Therefore the group is centerless. In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if...

Conjugacy classes

We can easily distinguish three kinds of permutations of the three blocks, called conjugacy classes of the group: In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ...

• 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). In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...

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

Subgroups

From Lagrange's theorem we know that any non-trivial subgroup has 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. In mathematics, most commonly, Lagranges theorem states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. This can be shown using the concept of left cosets of H... In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... A cyclic permutation is a permutation that shifts all elements of given ordered set by a fixed offset, with the elements shifted off the end inserted back at the beginning in the same order, i. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...

The first-mentioned is {(),(RGB),(RBG)}, the alternating group A₃. In mathematics an alternating group is the group of even permutations of a finite set. ...

The left cosets and the right cosets of A₃ are both that subgroup itself and the three swaps. In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...

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

• that subgroup itself
• {(RB),(RGB)}
• {(GB),(RBG)}

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

• that subgroup itself
• {(RB),(RBG)}
• {(GB),(RGB)}

Thus A₃ is normal, and the other three non-trivial subgroups are not. The quotient group G/A₃ is isomorphic with C₂. In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ...

G = A₃ H, a semidirect product, where H is a subgroup of two elements: () and one of the three swaps. This is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ...

In terms of permutations the two group elements of G/A₃ are the set of even permutations and the set of odd permutations. In mathematics, the permutations of a finite set (i. ...

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". Mirror Image is an episode of the television series The Twilight Zone. ...

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

C_{3 φ} C₂ is C₃ × C₂ 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. This is the symbol for the semidirect product operation, until Wikipedia:TeX support for semidirect is implemented (rtimes is the standard LaTeX command, actually) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

Thus we get:

(n₁, 0) * (n₂, h₂) = (n₁ + n₂, h₂)

(n₁, 1) * (n₂, h₂) = (n₁ - n₂, 1 + h₂)

for all n₁, n₂ in C₃ and h₂ in C₂.

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 2x2 table, with 3x3 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 D₃ in the geometrical way, as 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. The symmetry group of an object (e. ... In mathematics, groups are often used to describe symmetries of objects. ...

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 an 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

Orbits and stabilizers

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: In mathematics, groups are often used to describe symmetries of objects. ...

The orbits are {0,10,20}, {1,9,11,19,21,29}, {2,8,12,18,22,28}, {3,8,13,18,23,28}, {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. A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...

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 G/G_x and Gx. Specifically, the bijection is given by hG_x |-> h·x. This result is known as the orbit-stabilizer theorem. In the two cases of a small orbit, the stabilizer is non-trivial. In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, particularly in algebra, the coimage of a homomorphism f: A → B is the quotient coim f = A/ker f of the domain and kernel. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ...

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 -20, reflection about 3, and rotation of 20. In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...

A result closely related to the orbit-stabilizer theorem is Burnside's lemma: Burnsides lemma, sometimes also called Burnsides counting theorem, Pólyas formula or Cauchy-Frobenius lemma, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. ...

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.

External link

• http://mathworld.wolfram.com/DihedralGroupD3.html