Permutation Groups

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.

By Cayley's theorem, every group is isomorphic to some permutation group.

The way in which the elements of a permutation group permute the elements of the set is called its group action. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry.

The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation groups. Each rotation of a layer of the cube results in a permutation of the surface colors and is a member of the group. The permutation group of the cube is called the Rubik's cube group.

Basic properties and terminology

Being a subgroup of a symmetric group, all that is necessary for a set of permutations to satisfy the group axioms and be a permutation group is that it contain the identity permutation, the inverse permutation of each permutation it contains, and be closed under composition of its permutations. A general property of finite groups implies that a finite nonempty subset of a symmetric group is again a group if and only if it is closed under the group operation.

The degree of a group of permutations of a finite set is the number of elements in the set. The order of a group (of any type) is the number of elements (cardinality) in the group. By Lagrange's theorem, the order of any finite permutation group of degree n must divide n! since n-factorial is the order of the symmetric group S_n.

Notation

Since permutations are bijections of a set, they can be represented by Cauchy's two-line notation. This notation lists each of the elements of M in the first row, and for each element, its image under the permutation below it in the second row. If $\sigma$ is a permutation of the set $M=\{x_{1},x_{2},\ldots ,x_{n}\}$ then,

$\sigma ={\begin{pmatrix}x_{1}&x_{2}&x_{3}&\cdots &x_{n}\\\sigma (x_{1})&\sigma (x_{2})&\sigma (x_{3})&\cdots &\sigma (x_{n})\end{pmatrix}}.$

For instance, a particular permutation of the set {1, 2, 3, 4, 5} can be written as

$\sigma ={\begin{pmatrix}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}};$

this means that σ satisfies σ(1) = 2, σ(2) = 5, σ(3) = 4, σ(4) = 3, and σ(5) = 1. The elements of M need not appear in any special order in the first row, so the same permutation could also be written as

$\sigma ={\begin{pmatrix}3&2&5&1&4\\4&5&1&2&3\end{pmatrix}}.$

Permutations are also often written in cyclic notation (cyclic form) so that given the set M = {1, 2, 3, 4}, a permutation g of M with g(1) = 2, g(2) = 4, g(4) = 1 and g(3) = 3 will be written as (1, 2, 4)(3), or more commonly, (1, 2, 4) since 3 is left unchanged; if the objects are denoted by single letters or digits, commas and spaces can also be dispensed with, and we have a notation such as (124). The permutation written above in 2-line notation would be written in cyclic notation as $\sigma =(125)(34).$

Composition of permutations–the group product

The product of two permutations is defined as their composition as functions, so $\sigma \cdot \pi$ is the function that maps any element x of the set to $\sigma (\pi (x))$. Note that the rightmost permutation is applied to the argument first, because of the way function composition is written. Some authors prefer the leftmost factor acting first, but to that end permutations must be written to the right of their argument, often as a superscript, so the permutation $\sigma$ acting on the element $x$ results in the image $x^{\sigma }$. With this convention, the product is given by $x^{\sigma \cdot \pi }=(x^{\sigma })^{\pi }$. However, this gives a different rule for multiplying permutations. This convention is commonly used in the permutation group literature, but this article uses the convention where the rightmost permutation is applied first.

Since the composition of two bijections always gives another bijection, the product of two permutations is again a permutation. In two-line notation, the product of two permutations is obtained by rearranging the columns of the second (leftmost) permutation so that its first row is identical with the second row of the first (rightmost) permutation. The product can then be written as the first row of the first permutation over the second row of the modified second permutation. For example, given the permutations,

$P={\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}\quad {\text{ and }}\quad Q={\begin{pmatrix}1&2&3&4&5\\5&4&3&2&1\end{pmatrix}},$

the product QP is:

$QP={\begin{pmatrix}1&2&3&4&5\\5&4&3&2&1\end{pmatrix}}{\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}={\begin{pmatrix}2&4&1&3&5\\4&2&5&3&1\end{pmatrix}}{\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}={\begin{pmatrix}1&2&3&4&5\\4&2&5&3&1\end{pmatrix}}.$

The composition of permutations, when they are written in cyclic form, is obtained by juxtaposing the two permutations (with the second one written on the left) and then simplifying to a disjoint cycle form if desired. Thus, in cyclic notation the above product would be given by:

$Q\cdot P=(15)(24)\cdot (1243)=(1435).$

Since function composition is associative, so is the product operation on permutations: $(\sigma \cdot \pi )\cdot \rho =\sigma \cdot (\pi \cdot \rho )$. Therefore, products of two or more permutations are usually written without adding parentheses to express grouping; they are also usually written without a dot or other sign to indicate multiplication (the dots of the previous example were added for emphasis, so would simply be written as $\sigma \pi \rho$).

Neutral element and inverses

The identity permutation, which maps every element of the set to itself, is the neutral element for this product. In two-line notation, the identity is

${\begin{pmatrix}1&2&3&\cdots &n\\1&2&3&\cdots &n\end{pmatrix}}.$

In cyclic notation, e = (1)(2)(3)...(n) which by convention is also denoted by just (1) or even ().

Since bijections have inverses, so do permutations, and the inverse σ⁻¹ of σ is again a permutation. Explicitly, whenever σ(x)=y one also has σ⁻¹(y)=x. In two-line notation the inverse can be obtained by interchanging the two lines (and sorting the columns if one wishes the first line to be in a given order). For instance

${\begin{pmatrix}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}}^{-1}={\begin{pmatrix}2&5&4&3&1\\1&2&3&4&5\end{pmatrix}}={\begin{pmatrix}1&2&3&4&5\\5&1&4&3&2\end{pmatrix}}.$

To obtain the inverse of a single cycle, we reverse the order of its elements. Thus,

$(125)^{-1}=(521)=(152).$

To obtain the inverse of a product of cycles, we first reverse the order of the cycles, and then we take the inverse of each as above. Thus,

$[(125)(34)]^{-1}=(34)^{-1}(125)^{-1}=(43)(521)=(34)(152).$

Having an associative product, an identity element, and inverses for all its elements, makes the set of all permutations of M into a group, Sym(M); a permutation group.

Examples

Consider the following set G₁ of permutations of the set M = {1, 2, 3, 4}:

• e = (1)(2)(3)(4) = (1)
  • This is the identity, the trivial permutation which fixes each element.
• a = (1 2)(3)(4) = (1 2)
  • This permutation interchanges 1 and 2, and fixes 3 and 4.
• b = (1)(2)(3 4) = (3 4)
  • Like the previous one, but exchanging 3 and 4, and fixing the others.
• ab = (1 2)(3 4)
  • This permutation, which is the composition of the previous two, exchanges simultaneously 1 with 2, and 3 with 4.

G₁ forms a group, since aa = bb = e, ba = ab, and abab = e. This permutation group is isomorphic, as an abstract group, to the Klein group V₄.

As another example consider the group of symmetries of a square. Let the vertices of a square be labeled 1, 2, 3 and 4 (counterclockwise around the square starting with 1 in the top left corner). The symmetries are determined by the images of the vertices, that can, in turn, be described by permutations. The rotation by 90° (counterclockwise) about the center of the square is described by the permutation (1234). The 180° and 270° rotations are given by (13)(24) and (1432), respectively. The reflection about the horizontal line through the center is given by (12)(34) and the corresponding vertical line reflection is (14)(23). The reflection about the 1,3−diagonal line is (24) and reflection about the 2,4−diagonal is (13). The only remaining symmetry is the identity (1)(2)(3)(4). This permutation group is abstractly known as the dihedral group of order 8.

The Permutation Groups on a Set X, SX

If $\{1,2,...,n\}$ is the $n$-element set of positive integers and if $S_{n}$ denotes the set of all permutations on $\{1,2,...,n\}$ then $(S_{n},\circ )$ where $\circ$ is the operation of function composition defines a group called the symmetric group on $n$-elements.

Now suppose that $X$ is any set. We can analogously define a group on the set of permutations on $X$.

Definition: Let $X$ be any set and let $S_{X}$ denote the set of all permutations on $X$. Then $(S_{X},\circ )$ is called the Permutation Group on $X$.

Note that if $X=\{1,2,...,n\}$ then $S_{X}=S_{n}$ is the permutation group on $n$-elements. If $X=A=\{x_{1},x_{2},...,x_{n}\}$ is a general $n$-element set then $S_{X}=S_{A}$ is a symmetric group on a general $n$-element set $A$. Of course, what's more interesting is when $X$ is a countably infinite or uncountably infinite set.

For example, consider the set $X=\mathbb {Z}$. Then $S_{X}=S_{\mathbb {Z} }$ is the set of all permutations on $\mathbb {Z}$. For example, the following functions $\sigma :\mathbb {Z} \to \mathbb {Z} ,\delta :\mathbb {Z} \to \mathbb {Z} \in S_{\mathbb {Z} }$ are permutations on $\mathbb {Z}$ as you should verify:

${\begin{aligned}\quad \sigma (x)=x+1\end{aligned}}$

${\begin{aligned}\quad \delta (x)=x-1\end{aligned}}$

Then the composition $\sigma \circ \delta$ is given for all $x\in \mathbb {Z}$ by

${\begin{aligned}\quad (\sigma \circ \delta )(x)=\sigma (\delta (x))=\sigma (x-1)=(x-1)+1=x\end{aligned}}$

Therefore we see that $\sigma \circ \delta =i$ where $i$ is the identity permutation on $\mathbb {Z}$.

Group actions

In the above example of the symmetry group of a square, the permutations "describe" the movement of the vertices of the square induced by the group of symmetries. It is common to say that these group elements are "acting" on the set of vertices of the square. This idea can be made precise by formally defining a group action.

Let G be a group and M a nonempty set. An action of G on M is a function f: G × M → M such that

• f(1, x) = x, for all x in M (1 is the identity (neutral) element of the group G), and
• f(g, f(h, x)) = f(gh, x), for all g,h in G and all x in M.

This last condition can also be expressed as saying that the action induces a group homomorphism from G into Sym(M). Any such homomorphism is called a (permutation) representation of G on M.

For any permutation group, the action that sends (g, x) → g(x) is called the natural action of G on M. This is the action that is assumed unless otherwise indicated. In the example of the symmetry group of the square, the group's action on the set of vertices is the natural action. However, this group also induces an action on the set of four triangles in the square, which are: t₁ = 234, t₂ = 134, t₃ = 124 and t₄ = 123. It also acts on the two diagonals: d₁ = 13 and d₂ = 24.

Group element	Action on triangles	Action on diagonals
(1)	(1)	(1)
(1234)	(t1 t2 t3 t4)	(d1 d2)
(13)(24)	(t1 t3)(t2 t4)	(1)
(1432)	(t1 t4 t3 t2)	(d1 d2)
(12)(34)	(t1 t2)(t3 t4)	(d1 d2)
(14)(23)	(t1 t4)(t2 t3)	(d1 d2)
(13)	(t1 t3)	(1)
(24)	(t2 t4)	(1)

Transitive actions

The action of a group G on a set M is said to be transitive if, for every two elements s, t of M, there is some group element g such that g(s) = t. Equivalently, the set M forms a single orbit under the action of G. Of the examples above, the group {e, (1 2), (3 4), (1 2)(3 4)} of permutations of {1, 2, 3, 4} is not transitive (no group element takes 1 to 3) but the group of symmetries of a square is transitive on the vertices.

Primitive actions

A permutation group G acting transitively on a non-empty finite set M is imprimitive if there is some nontrivial set partition of M that is preserved by the action of G, where "nontrivial" means that the partition isn't the partition into singleton sets nor the partition with only one part. Otherwise, if G is transitive but does not preserve any nontrivial partition of M, the group G is primitive.

For example, the group of symmetries of a square is primitive on the vertices: if they are numbered 1, 2, 3, 4 in cyclic order, then the partition {{1, 3}, {2, 4}} into opposite pairs is preserved by every group element. On the other hand, the full symmetric group on a set M is always imprimitive.

Cayley's theorem

Any group G can act on itself (the elements of the group being thought of as the set M) in many ways. In particular, there is a regular action given by (left) multiplication in the group. That is, f(g, x) = gx for all g and x in G. For each fixed g, the function f_g(x) = gx is a bijection on G and therefore a permutation of the set of elements of G. Each element of G can be thought of as a permutation in this way and so G is isomorphic to a permutation group; this is the content of Cayley's theorem.

For example, consider the group G₁ acting on the set {1, 2, 3, 4} given above. Let the elements of this group be denoted by e, a, b and c = ab = ba. The action of G₁ on itself described in Cayley's theorem gives the following permutation representation:

f_e ↦ (e)(a)(b)(c)

f_a ↦ (ea)(bc)

f_b ↦ (eb)(ac)

f_c ↦ (ec)(ab).

Isomorphisms of permutation groups

If G and H are two permutation groups on sets X and Y with actions f₁ and f₂ respectively, then we say that G and H are permutation isomorphic (or isomorphic as permutation groups) if there exists a bijective map λ : X → Y and a group isomorphism ψ : G → H such that

λ(f₁(g, x)) = f₂(ψ(g), λ(x)) for all g in G and x in X

If X = Y this is equivalent to G and H being conjugate as subgroups of Sym(X). The special case where G = H and ψ is the identity map gives rise to the concept of equivalent actions of a group.

In the example of the symmetries of a square given above, the natural action on the set {1,2,3,4} is equivalent to the action on the triangles. The bijection λ between the sets is given by i ↦ t_i. The natural action of group G₁ above and its action on itself (via left multiplication) are not equivalent as the natural action has fixed points and the second action does not.

Oligomorphic groups

When a group G acts on a set S, the action may be extended naturally to the Cartesian product Sⁿ of S, consisting of n-tuples of elements of S: the action of an element g on the n-tuple (s₁, ..., s_n) is given by

g(s₁, ..., s_n) = (g(s₁), ..., g(s_n)).

The group G is said to be oligomorphic if the action on Sⁿ has only finitely many orbits for every positive integer n. (This is automatic if S is finite, so the term is typically of interest when S is infinite.)

The interest in oligomorphic groups is partly based on their application to model theory, for example when considering automorphisms in countably categorical theories.

Licensing

Content obtained and/or adapted from:

• Permutation group, Wikipedia under a CC BY-SA license
• The Permutation Groups on a Set X, SX, mathonline.wikidot.com under a CC BY-SA license