# Permutation group

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Group whose operation is composition of permutations

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In [mathematics](/source/Mathematics), a **permutation group** is a [group](/source/Group_(mathematics)) *G* whose elements are [permutations](/source/Permutation) of a given [set](/source/Set_(mathematics)) *M* and whose [group operation](/source/Group_operation) is the [composition of permutations](/source/Function_composition) in *G* (which are thought of as [bijective functions](/source/Bijective_function) from the set *M* to itself). The group of *all* permutations of a set *M* is the [symmetric group](/source/Symmetric_group) of *M*, often written as Sym(*M*).[1] The term *permutation group* thus means a [subgroup](/source/Subgroup) of the [symmetric group](/source/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](/source/Cayley's_theorem), every group is [isomorphic](/source/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](/source/Group_action_(mathematics)). Group actions have applications in the study of [symmetries](/source/Symmetry), [combinatorics](/source/Combinatorics) and many other branches of [mathematics](/source/Mathematics), [physics](/source/Physics) and chemistry.

The popular puzzle [Rubik's Cube](/source/Rubik's_Cube) invented in 1974 by [Ernő Rubik](/source/Ern%C5%91_Rubik) has been used as an illustration of permutation groups.  Each rotation of a layer of the cube results in a [permutation](/source/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](/source/Rubik's_Cube_group).

## Basic properties and terminology

A *permutation group* is a [subgroup](/source/Subgroup) of a [symmetric group](/source/Symmetric_group); that is, its elements are [permutations](/source/Permutation) of a given set. It is thus a subset of a symmetric group that is [closed](/source/Closure_(mathematics)) under [composition](/source/Function_composition) of permutations, contains the [identity permutation](/source/Identity_permutation), and contains the [inverse permutation](/source/Inverse_element) of each of its elements.[2] A general property of finite groups implies that a finite nonempty subset of a symmetric group is a permutation group if and only if it is closed under permutation composition.[3]

The **degree** of a group of permutations of a [finite set](/source/Finite_set) is the [number of elements](/source/Cardinality) in the set. The **order** of a group (of any type) is the number of elements (cardinality) in the group. By [Lagrange's theorem](/source/Lagrange's_theorem_(group_theory)), the order of any finite permutation group of degree *n* must divide *n*! since *n*-[factorial](/source/Factorial) is the order of the symmetric group *S**n*.

## Notation

Main article: [Permutation § Notations](/source/Permutation#Notations)

Since permutations are [bijections](/source/Bijection) of a set, they can be represented by [Cauchy](/source/Augustin-Louis_Cauchy)'s *two-line notation*.[4] 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 σ {\displaystyle \sigma } is a permutation of the set M = { x 1 , x 2 , … , x n } {\displaystyle M=\{x_{1},x_{2},\ldots ,x_{n}\}} then,

- σ = ( x 1 x 2 x 3 ⋯ x n σ ( x 1 ) σ ( x 2 ) σ ( x 3 ) ⋯ σ ( x n ) ) . {\displaystyle \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

- σ = ( 1 2 3 4 5 2 5 4 3 1 ) ; {\displaystyle \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

- σ = ( 3 2 5 1 4 4 5 1 2 3 ) . {\displaystyle \sigma ={\begin{pmatrix}3&2&5&1&4\\4&5&1&2&3\end{pmatrix}}.}

Permutations are also often written in [cycle notation](/source/Cycle_notation) (*cyclic form*)[5] 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 cycle notation as σ = ( 125 ) ( 34 ) . {\displaystyle \sigma =(125)(34).}

## Composition of permutations–the group product

The product of two permutations is defined as their [composition](/source/Function_composition) as functions, so σ ⋅ π {\displaystyle \sigma \cdot \pi } is the function that maps any element *x* of the set to σ ( π ( x ) ) {\displaystyle \sigma (\pi (x))} . Note that the rightmost permutation is applied to the argument first, because of the way function composition is written.[6][7] 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](/source/Superscript), so the permutation σ {\displaystyle \sigma } acting on the element x {\displaystyle x} results in the image x σ {\displaystyle x^{\sigma }} . With this convention, the product is given by x σ ⋅ π = ( x σ ) π {\displaystyle x^{\sigma \cdot \pi }=(x^{\sigma })^{\pi }} .[8] [9] [10] 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 = ( 1 2 3 4 5 2 4 1 3 5 ) and Q = ( 1 2 3 4 5 5 4 3 2 1 ) , {\displaystyle 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:

- Q P = ( 1 2 3 4 5 5 4 3 2 1 ) ( 1 2 3 4 5 2 4 1 3 5 ) = ( 2 4 1 3 5 4 2 5 3 1 ) ( 1 2 3 4 5 2 4 1 3 5 ) = ( 1 2 3 4 5 4 2 5 3 1 ) . {\displaystyle 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 cycle notation, 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, the above product would be given by:

- Q ⋅ P = ( 15 ) ( 24 ) ⋅ ( 1243 ) = ( 1435 ) . {\displaystyle Q\cdot P=(15)(24)\cdot (1243)=(1435).}

Since function composition is [associative](/source/Associative), so is the product operation on permutations: ( σ ⋅ π ) ⋅ ρ = σ ⋅ ( π ⋅ ρ ) {\displaystyle (\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 σ π ρ {\displaystyle \sigma \pi \rho } ).

## Neutral element and inverses

The identity permutation, which maps every element of the set to itself, is the [neutral element](/source/Neutral_element) for this product. In two-line notation, the identity is

- ( 1 2 3 ⋯ n 1 2 3 ⋯ n ) . {\displaystyle {\begin{pmatrix}1&2&3&\cdots &n\\1&2&3&\cdots &n\end{pmatrix}}.}

In cycle notation, *e* = (1)(2)(3)...(*n*) which by convention is also denoted by just (1) or even ().[11]

Since [bijections](/source/Bijections) have [inverses](/source/Inverse_function), 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

- ( 1 2 3 4 5 2 5 4 3 1 ) − 1 = ( 2 5 4 3 1 1 2 3 4 5 ) = ( 1 2 3 4 5 5 1 4 3 2 ) . {\displaystyle {\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 ) . {\displaystyle (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 ) . {\displaystyle [(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](/source/Group_(mathematics)), Sym(*M*); a permutation group.

## Examples

Consider the following set *G*1 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.

*G*1 forms a group, since *aa* = *bb* = *e*, *ba* = *ab*, and *abab* = *e*. This permutation group is, as an [abstract group](/source/Abstract_group), the [Klein group](/source/Klein_group) *V*4.

As another example consider the [group of symmetries of a square](/source/Examples_of_groups#The_symmetry_group_of_a_square:_dihedral_group_of_order_8). 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 known, as an abstract group, as the [dihedral group](/source/Dihedral_group) of order 8.

## Group actions

Main article: [Group action (mathematics)](/source/Group_action_(mathematics))

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**.[12]

Let *G* be a [group](/source/Group_(mathematics)) and *M* a nonempty [set](/source/Set_(mathematics)). 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](/source/Identity_element) (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 pair of conditions can also be expressed as saying that the action induces a group homomorphism from *G* into *Sym*(*M*).[12] 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.[12] 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: *t*1 = 234, *t*2 = 134, *t*3 = 124 and *t*4 = 123. It also acts on the two diagonals: *d*1 = 13 and *d*2 = 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](/source/Orbit_(group_theory)) under the action of *G*.[13] Of the examples [above](#Examples), 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

Main article: [Primitive permutation group](/source/Primitive_permutation_group)

A permutation group *G* acting transitively on a non-empty finite set *M* is *imprimitive* if there is some nontrivial [set partition](/source/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](/source/Singleton_set) 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 imprimitive 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 primitive.

## Cayley's theorem

Main article: [Cayley's theorem](/source/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](/source/Regular_group_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](/source/Cayley's_theorem).

For example, consider the group *G*1 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 *G*1 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 *f*1 and *f*2 respectively, then we say that *G* and *H* are *permutation isomorphic* (or *[isomorphic](/source/Isomorphism) as permutation groups*) if there exists a [bijective map](/source/Bijection) *λ* : *X* → *Y* and a [group isomorphism](/source/Group_isomorphism) *ψ* : *G* → *H* such that

- *λ*(*f*1(*g*, *x*)) = *f*2(*ψ*(*g*), *λ*(*x*)) for all *g* in *G* and *x* in *X*.[14]

If *X* = *Y* this is equivalent to *G* and *H* being conjugate as subgroups of Sym(*X*).[15] The special case where *G* = *H* and *ψ* is the [identity map](/source/Identity_map) gives rise to the concept of *equivalent actions* of a group.[16]

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 *G*1 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](/source/Set_(mathematics)) *S*, the action may be extended naturally to the [Cartesian product](/source/Cartesian_product) *Sn* of *S*, consisting of *n*-tuples of elements of *S*: the action of an element *g* on the *n*-tuple (*s*1, ..., *s**n*) is given by

- *g*(*s*1, ..., *s**n*) = (*g*(*s*1), ..., *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*.[17][18] (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](/source/Model_theory), for example when considering [automorphisms](/source/Automorphism) in [countably categorical theories](/source/Countably_categorical_theory).[19]

## History

Main article: [History of group theory](/source/History_of_group_theory)

The study of [groups](/source/Group_(mathematics)) originally grew out of an understanding of permutation groups.[20] [Permutations](/source/Permutation) had themselves been intensively studied by [Lagrange](/source/Lagrange) in 1770 in his work on the algebraic solutions of polynomial equations. This subject flourished and by the mid 19th century a well-developed theory of permutation groups existed, codified by [Camille Jordan](/source/Camille_Jordan) in his book *Traité des Substitutions et des Équations Algébriques* of 1870. Jordan's book was, in turn, based on the papers that were left by [Évariste Galois](/source/%C3%89variste_Galois) in 1832.

When [Cayley](/source/Arthur_Cayley) introduced the concept of an [abstract group](/source/Abstract_group), it was not immediately clear whether or not this was a larger collection of objects than the known permutation groups (which had a definition different from the modern one). Cayley went on to prove that the two concepts were equivalent in Cayley's theorem.[21]

Another classical text containing several chapters on permutation groups is [Burnside](/source/William_Burnside)'s *Theory of Groups of Finite Order* of 1911.[22] The first half of the twentieth century was a fallow period in the study of group theory in general, but interest in permutation groups was revived in the 1950s by [H. Wielandt](/source/H._Wielandt) whose German lecture notes were reprinted as *Finite Permutation Groups* in 1964.[23]

## See also

- [2-transitive group](/source/2-transitive_group)

- [Rank 3 permutation group](/source/Rank_3_permutation_group)

- [Mathieu group](/source/Mathieu_group)

## Notes

1. **[^](#cite_ref-1)** The notations **S***M* and **S***M* are also used.

1. **[^](#cite_ref-2)** [Rotman 2006](#CITEREFRotman2006), p. 148, Definition of subgroup

1. **[^](#cite_ref-3)** [Rotman 2006](#CITEREFRotman2006), p. 149, Proposition 2.69

1. **[^](#cite_ref-4)** Wussing, Hans (2007), [*The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory*](https://books.google.com/books?id=Xp3JymnfAq4C&pg=PA94), Courier Dover Publications, p. 94, [ISBN](/source/ISBN_(identifier)) [9780486458687](https://en.wikipedia.org/wiki/Special:BookSources/9780486458687), Cauchy used his permutation notation—in which the arrangements are written one below the other and both are enclosed in parentheses—for the first time in 1815.

1. **[^](#cite_ref-5)** especially when the algebraic properties of the permutation are of interest.

1. **[^](#cite_ref-6)** [Biggs, Norman L.](/source/Norman_L._Biggs); White, A. T. (1979). *Permutation groups and combinatorial structures*. Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [0-521-22287-7](https://en.wikipedia.org/wiki/Special:BookSources/0-521-22287-7).

1. **[^](#cite_ref-7)** [Rotman 2006](#CITEREFRotman2006), p. 107 – note especially the footnote on this page.

1. **[^](#cite_ref-8)** [Dixon & Mortimer 1996](#CITEREFDixonMortimer1996), p. 3 – see the comment following Example 1.2.2

1. **[^](#cite_ref-9)** Cameron, Peter J. (1999). [*Permutation groups*](https://archive.org/details/permutationgroup0000came). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [0-521-65302-9](https://en.wikipedia.org/wiki/Special:BookSources/0-521-65302-9).

1. **[^](#cite_ref-10)** Jerrum, M. (1986). "A compact representation of permutation groups". *J. Algorithms*. **7** (1): 60–78. [doi](/source/Doi_(identifier)):[10.1016/0196-6774(86)90038-6](https://doi.org/10.1016%2F0196-6774%2886%2990038-6).

1. **[^](#cite_ref-11)** [Rotman 2006](#CITEREFRotman2006), p. 108

1. ^ [***a***](#cite_ref-Dixon96_12-0) [***b***](#cite_ref-Dixon96_12-1) [***c***](#cite_ref-Dixon96_12-2) [Dixon & Mortimer 1996](#CITEREFDixonMortimer1996), p. 5

1. **[^](#cite_ref-13)** [Artin 1991](#CITEREFArtin1991), p. 177

1. **[^](#cite_ref-14)** [Dixon & Mortimer 1996](#CITEREFDixonMortimer1996), p. 17

1. **[^](#cite_ref-15)** [Dixon & Mortimer 1996](#CITEREFDixonMortimer1996), p. 18

1. **[^](#cite_ref-16)** [Cameron 1994](#CITEREFCameron1994), p. 228

1. **[^](#cite_ref-17)** [Cameron, Peter J.](/source/Peter_Cameron_(mathematician)) (1990). *Oligomorphic permutation groups*. London Mathematical Society Lecture Note Series. Vol. 152. Cambridge: [Cambridge University Press](/source/Cambridge_University_Press). [ISBN](/source/ISBN_(identifier)) [0-521-38836-8](https://en.wikipedia.org/wiki/Special:BookSources/0-521-38836-8). [Zbl](/source/Zbl_(identifier)) [0813.20002](https://zbmath.org/?format=complete&q=an:0813.20002).

1. **[^](#cite_ref-18)** [Oligomorphic permutation groups](http://www.newton.ac.uk/files/preprints/ni08029.pdf) - Isaac Newton Institute preprint, Peter J. Cameron

1. **[^](#cite_ref-19)** Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1998). *Notes on infinite permutation groups*. Lecture Notes in Mathematics. Vol. 1698. Berlin: [Springer-Verlag](/source/Springer-Verlag). p. 83. [ISBN](/source/ISBN_(identifier)) [3-540-64965-4](https://en.wikipedia.org/wiki/Special:BookSources/3-540-64965-4). [Zbl](/source/Zbl_(identifier)) [0916.20002](https://zbmath.org/?format=complete&q=an:0916.20002).

1. **[^](#cite_ref-20)** [Dixon & Mortimer 1996](#CITEREFDixonMortimer1996), p. 28

1. **[^](#cite_ref-21)** [Cameron 1994](#CITEREFCameron1994), p. 226

1. **[^](#cite_ref-22)** Burnside, William (1955) [1911], *Theory of Groups of Finite Order* (2nd ed.), Dover

1. **[^](#cite_ref-23)** Wielandt, H. (1964), *Finite Permutation Groups*, Academic Press

## References

- [Artin, Michael](/source/Michael_Artin) (1991), *Algebra*, Prentice-Hall, [ISBN](/source/ISBN_(identifier)) [0-13-004763-5](https://en.wikipedia.org/wiki/Special:BookSources/0-13-004763-5)

- Cameron, Peter J. (1994), *Combinatorics: Topics, Techniques, Algorithms*, Cambridge University Press, [ISBN](/source/ISBN_(identifier)) [0-521-45761-0](https://en.wikipedia.org/wiki/Special:BookSources/0-521-45761-0)

- Dixon, John D.; Mortimer, Brian (1996), [*Permutation Groups*](https://archive.org/details/permutationgroup0000dixo), Graduate Texts in Mathematics 163), Springer-Verlag, [ISBN](/source/ISBN_(identifier)) [0-387-94599-7](https://en.wikipedia.org/wiki/Special:BookSources/0-387-94599-7)

- Rotman, Joseph J. (2006), *A First Course in Abstract Algebra with Applications* (3rd ed.), Pearson Prentice-Hall, [ISBN](/source/ISBN_(identifier)) [0-13-186267-7](https://en.wikipedia.org/wiki/Special:BookSources/0-13-186267-7)

## Further reading

- Akos Seress. *Permutation group algorithms*. Cambridge Tracts in Mathematics, 152. Cambridge University Press, Cambridge, 2003.

- Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller and Peter M. Neumann. *Notes on Infinite Permutation Groups*. Number 1698 in Lecture Notes in Mathematics. Springer-Verlag, 1998.

- [Peter J. Cameron](/source/Peter_Cameron_(mathematician)). *Permutation Groups*. LMS Student Text 45. Cambridge University Press, Cambridge, 1999.

- Peter J. Cameron. *Oligomorphic Permutation Groups*. Cambridge University Press, Cambridge, 1990.

## External links

- ["Permutation group"](https://www.encyclopediaofmath.org/index.php?title=Permutation_group), *[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)*, [EMS Press](/source/European_Mathematical_Society), 2001 [1994]

- Alexander Hulpke. GAP Data Library ["Transitive Permutation Groups"](http://www.gap-system.org/Datalib/trans.html) [Archived](https://web.archive.org/web/20210119030637/http://www.gap-system.org/Datalib/trans.html) 2021-01-19 at the [Wayback Machine](/source/Wayback_Machine).

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Adapted from the Wikipedia article [Permutation group](https://en.wikipedia.org/wiki/Permutation_group) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Permutation_group?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.