In abstract algebra
, the symmetric group defined over any set
is the group
are all the bijection
s 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
symbols consists of the permutation
s that can be performed on the
[Jacobson (2009), p. 31.]
Since there are
) such permutation operations, the order
(number of elements) of the symmetric group
Although symmetric groups can be defined on infinite set
s, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy class
es, a finite presentation
, their subgroup
s, their automorphism group
s, and their representation
theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.
The symmetric group is important to diverse areas of mathematics such as Galois theory
, invariant theory
, the representation theory of Lie groups
, and combinatorics
. Cayley's theorem
states that every group
to a subgroup
of the symmetric group on (the underlying set
Definition and first properties
The symmetric group on a finite set
is the group whose elements are all bijective functions from
and whose group operation is that of function composition
For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree
is the symmetric group on the set
The symmetric group on a set
is denoted in various ways, including
is the set
then the name may be abbreviated to
Symmetric groups on infinite sets behave quite differently from symmetric groups on finite sets, and are discussed in , , and .
The symmetric group on a set of
elements has order
). It is abelian
if and only if
is less than or equal to 2. For
(the empty set
and the singleton set
), the symmetric groups are trivial
(they have order
). The group S''n''
if and only if
. This is an essential part of the proof of the Abel–Ruffini theorem
that shows that for every
there are polynomial
s of degree
which are not solvable by radicals, that is, the solutions cannot be expressed by performing a finite number of operations of addition, subtraction, multiplication, division and root extraction on the polynomial's coefficients.
The symmetric group on a set of size ''n'' is the Galois group
of the general polynomial
of degree ''n'' and plays an important role in Galois theory
. In invariant theory
, the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called symmetric function
s. In the representation theory of Lie groups
, the representation theory of the symmetric group
plays a fundamental role through the ideas of Schur functor
s. In the theory of Coxeter group
s, the symmetric group is the Coxeter group of type A''n''
and occurs as the Weyl group
of the general linear group
. In combinatorics
, the symmetric groups, their elements (permutation
s), and their representations
provide a rich source of problems involving Young tableaux
, plactic monoid
s, and the Bruhat order
s of symmetric groups are called permutation group
s and are widely studied because of their importance in understanding group action
s, homogeneous space
s, and automorphism group
s of graph
s, such as the Higman–Sims group
and the Higman–Sims graph
The elements of the symmetric group on a set ''X'' are the permutation
s of ''X''.
The group operation in a symmetric group is function composition
, denoted by the symbol ∘ or simply by juxtaposition of the permutations. The composition of permutations ''f'' and ''g'', pronounced "''f'' of ''g''", maps any element ''x'' of ''X'' to ''f''(''g''(''x'')). Concretely, let (see permutation
for an explanation of notation):
Applying ''f'' after ''g'' maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So composing ''f'' and ''g'' gives
of length , taken to the ''k''-th power, will decompose into ''k'' cycles of length ''m'': For example, (, ),
Verification of group axioms
To check that the symmetric group on a set ''X'' is indeed a group
, it is necessary to verify the group axioms of closure, associativity, identity, and inverses.
# The operation of function composition
is closed in the set of permutations of the given set ''X''.
# Function composition
is always associative.
# The trivial bijection that assigns each element of ''X'' to itself serves as an identity for the group.
# Every bijection has an inverse function
that undoes its action, and thus each element of a symmetric group does have an inverse which is a permutation too.
Transpositions, sign, and the alternating group
A transposition is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutation can be written as a product of transpositions; for instance, the permutation ''g'' from above can be written as ''g'' = (1 2)(2 5)(3 4). Since ''g'' can be written as a product of an odd number of transpositions, it is then called an odd permutation
, whereas ''f'' is an even permutation.
The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd. There are several short proofs of the invariance of this parity of a permutation.
The product of two even permutations is even, the product of two odd permutations is even, and all other products are odd. Thus we can define the sign of a permutation:
With this definition,
is a group homomorphism
( is a group under multiplication, where +1 is e, the neutral element
). The kernel
of this homomorphism, that is, the set of all even permutations, is called the alternating group
. It is a normal subgroup
, and for it has elements. The group S''n''
is the semidirect product
and any subgroup generated by a single transposition.
Furthermore, every permutation can be written as a product of ''adjacent transposition
s'', that is, transpositions of the form . For instance, the permutation ''g'' from above can also be written as . The sorting algorithm bubble sort
is an application of this fact. The representation of a permutation as a product of adjacent transpositions is also not unique.
of ''length'' ''k'' is a permutation ''f'' for which there exists an element ''x'' in such that ''x'', ''f''(''x''), ''f''2
(''x''), ..., ''f''''k''
(''x'') = ''x'' are the only elements moved by ''f''; it is required that since with the element ''x'' itself would not be moved either. The permutation ''h'' defined by
is a cycle of length three, since , and , leaving 2 and 5 untouched. We denote such a cycle by , but it could equally well be written or by starting at a different point. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are ''disjoint'' if they move disjoint subsets of elements. Disjoint cycles commute
: for example, in S6
there is the equality . Every element of S''n''
can be written as a product of disjoint cycles; this representation is unique up to
the order of the factors, and the freedom present in representing each individual cycle by choosing its starting point.
Cycles admit the following conjugation property with any permutation
, this property is often used to obtain its generators and relations
Certain elements of the symmetric group of are of particular interest (these can be generalized to the symmetric group of any finite totally ordered set, but not to that of an unordered set).
The is the one given by:
This is the unique maximal element with respect to the Bruhat order
in the symmetric group with respect to generating set consisting of the adjacent transpositions , .
This is an involution, and consists of
so it thus has sign:
which is 4-periodic in ''n''.
, the ''perfect shuffle
'' is the permutation that splits the set into 2 piles and interleaves them. Its sign is also
Note that the reverse on ''n'' elements and perfect shuffle on 2''n'' elements have the same sign; these are important to the classification of Clifford algebra
s, which are 8-periodic.
The conjugacy class
es of S''n''
correspond to the cycle structures of permutations; that is, two elements of S''n''
are conjugate in S''n''
if and only if they consist of the same number of disjoint cycles of the same lengths. For instance, in S5
, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. A conjugating element of S''n''
can be constructed in "two line notation" by placing the "cycle notations" of the two conjugate permutations on top of one another. Continuing the previous example:
which can be written as the product of cycles, namely: (2 4).
This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, that is,
It is clear that such a permutation is not unique.
Low degree groups
The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately.
: The symmetric groups on the empty set
and the singleton set
are trivial, which corresponds to . In this case the alternating group agrees with the symmetric group, rather than being an index 2 subgroup, and the sign map is trivial. In the case of S0
, its only member is the empty function
: This group consists of exactly two elements: the identity and the permutation swapping the two points. It is a cyclic group
and is thus abelian
. In Galois theory
, this corresponds to the fact that the quadratic formula
gives a direct solution to the general quadratic polynomial
after extracting only a single root. In invariant theory
, the representation theory of the symmetric group on two points is quite simple and is seen as writing a function of two variables as a sum of its symmetric and anti-symmetric parts: Setting , and , one gets that . This process is known as symmetrization
is the first nonabelian symmetric group. This group is isomorphic to the dihedral group of order 6
, the group of reflection and rotation symmetries of an equilateral triangle
, since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations. In Galois theory, the sign map from S3
corresponds to the resolving quadratic for a cubic polynomial
, as discovered by Gerolamo Cardano
, while the A3
kernel corresponds to the use of the discrete Fourier transform
of order 3 in the solution, in the form of Lagrange resolvent
: The group S4
is isomorphic to the group of proper rotations about opposite faces, opposite diagonals and opposite edges, 9, 8 and 6
permutations, of the cube
. Beyond the group A4
has a Klein four-group
V as a proper normal subgroup
, namely the even transpositions with quotient S3
. In Galois theory
, this map corresponds to the resolving cubic to a quartic polynomial
, which allows the quartic to be solved by radicals, as established by Lodovico Ferrari
. The Klein group can be understood in terms of the Lagrange resolvent
s of the quartic. The map from S4
also yields a 2-dimensional irreducible representation, which is an irreducible representation of a symmetric group of degree ''n'' of dimension below , which only occurs for .
is the first non-solvable symmetric group. Along with the special linear group
and the icosahedral group
is one of the three non-solvable groups of order 120, up to isomorphism. S5
is the Galois group
of the general quintic equation
, and the fact that S5
is not a solvable group
translates into the non-existence of a general formula to solve quintic polynomial
s by radicals. There is an exotic inclusion map as a transitive subgroup
; the obvious inclusion map fixes a point and thus is not transitive. This yields the outer automorphism of S6
, discussed below, and corresponds to the resolvent sextic of a quintic.
: Unlike all other symmetric groups, S6
, has an outer automorphism
. Using the language of Galois theory
, this can also be understood in terms of Lagrange resolvents
. The resolvent of a quintic is of degree 6—this corresponds to an exotic inclusion map as a transitive subgroup (the obvious inclusion map fixes a point and thus is not transitive) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of S6
—see automorphisms of the symmetric and alternating groups
:Note that while A6
have an exceptional Schur multiplier
(a triple cover
) and that these extend to triple covers of S6
, these do not correspond to exceptional Schur multipliers of the symmetric group.
Maps between symmetric groups
Other than the trivial map and the sign map , the most notable homomorphisms between symmetric groups, in order of relative dimension
* corresponding to the exceptional normal subgroup ;
* (or rather, a class of such maps up to inner automorphism) corresponding to the outer automorphism of S6
* as a transitive subgroup, yielding the outer automorphism of S6
as discussed above.
There are also a host of other homomorphisms where .
Relation with alternating group
For , the alternating group
, and the induced quotient is the sign map: which is split by taking a transposition of two elements. Thus S''n''
is the semidirect product , and has no other proper normal subgroups, as they would intersect A''n''
in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in A''n''
(and thus themselves be A''n''
acts on its subgroup A''n''
by conjugation, and for , S''n''
is the full automorphism group of A''n''
) ≅ S''n''
. Conjugation by even elements are inner automorphism
s of A''n''
while the outer automorphism
of order 2 corresponds to conjugation by an odd element. For , there is an exceptional outer automorphism
is not the full automorphism group of A''n''
Conversely, for , S''n''
has no outer automorphisms, and for it has no center, so for it is a complete group
, as discussed in automorphism group
For , S''n''
is an almost simple group
, as it lies between the simple group A''n''
and its group of automorphisms.
can be embedded into A''n''+2
by appending the transposition to all odd permutations, while embedding into A''n''+1
is impossible for .
Generators and relations
The symmetric group on letters is generated by the adjacent transposition
that swap and . The collection
generates subject to the following relations:
where 1 represents the identity permutation. This representation endows the symmetric group with the structure of a Coxeter group
(and so also a reflection group
Other possible generating sets include the set of transpositions that swap and for , and a set containing any -cycle and a -cycle of adjacent elements in the -cycle.
of a symmetric group is called a permutation group
The normal subgroup
s of the finite symmetric groups are well understood. If , S''n''
has at most 2 elements, and so has no nontrivial proper subgroups. The alternating group
of degree ''n'' is always a normal subgroup, a proper one for and nontrivial for ; for it is in fact the only non-identity proper normal subgroup of S''n''
, except when where there is one additional such normal subgroup, which is isomorphic to the Klein four group
The symmetric group on an infinite set does not have a subgroup of index 2, as Vitali
(1915) proved that each permutation can be written as a product of three squares. However it contains the normal subgroup ''S'' of permutations that fix all but finitely many elements, which is generated by transpositions. Those elements of ''S'' that are products of an even number of transpositions form a subgroup of index 2 in ''S'', called the alternating subgroup ''A''. Since ''A'' is even a characteristic subgroup
of ''S'', it is also a normal subgroup of the full symmetric group of the infinite set. The groups ''A'' and ''S'' are the only non-identity proper normal subgroups of the symmetric group on a countably infinite set. This was first proved by Onofri
(1929) and independently Schreier
[Über die Permutationsgruppe der natürlichen Zahlenfolge. Studia Mathematica (1933) Vol. 4(1), p.134-141, 1933]
). For more details see or .
The maximal subgroup
s of the finite symmetric groups fall into three classes: the intransitive, the imprimitive, and the primitive. The intransitive maximal subgroups are exactly those of the form for . The imprimitive maximal subgroups are exactly those of the form Sym(''k'') wr Sym(''n''/''k'') where is a proper divisor of ''n'' and "wr" denotes the wreath product
acting imprimitively. The primitive maximal subgroups are more difficult to identify, but with the assistance of the O'Nan–Scott theorem
and the classification of finite simple groups
, gave a fairly satisfactory description of the maximal subgroups of this type according to .
The Sylow subgroup
s of the symmetric groups are important examples of ''p''-groups
. They are more easily described in special cases first:
The Sylow ''p''-subgroups of the symmetric group of degree ''p'' are just the cyclic subgroups generated by ''p''-cycles. There are such subgroups simply by counting generators. The normalizer
therefore has order and is known as a Frobenius group
(especially for ), and is the affine general linear group
The Sylow ''p''-subgroups of the symmetric group of degree ''p''2
are the wreath product
of two cyclic groups of order ''p''. For instance, when , a Sylow 3-subgroup of Sym(9) is generated by and the elements , and every element of the Sylow 3-subgroup has the form for .
The Sylow ''p''-subgroups of the symmetric group of degree ''p''''n''
are sometimes denoted W''p''
(''n''), and using this notation one has that is the wreath product of W''p''
(''n'') and W''p''
In general, the Sylow ''p''-subgroups of the symmetric group of degree ''n'' are a direct product of ''a''''i''
copies of W''p''
(''i''), where and (the base ''p'' expansion of ''n'').
For instance, , the dihedral group of order 8
, and so a Sylow 2-subgroup of the symmetric group of degree 7 is generated by and is isomorphic to .
These calculations are attributed to and described in more detail in . Note however that attributes the result to an 1844 work of Cauchy
, and mentions that it is even covered in textbook form in .
A transitive subgroup of S''n''
is a subgroup whose action on is transitive
. For example, the Galois group of a (finite
) Galois extension
is a transitive subgroup of S''n''
, for some ''n''.
states that every group ''G'' is isomorphic to a subgroup of some symmetric group. In particular, one may take a subgroup of the symmetric group on the elements of ''G'', since every group acts on itself faithfully by (left or right) multiplication.
For , S''n''
is a complete group
: its center
and outer automorphism group
are both trivial.
For , the automorphism group is trivial, but S2
is not trivial: it is isomorphic to C2
, which is abelian, and hence the center is the whole group.
For , it has an outer automorphism of order 2: , and the automorphism group is a semidirect product .
In fact, for any set ''X'' of cardinality other than 6, every automorphism of the symmetric group on ''X'' is inner, a result first due to according to .
The group homology
is quite regular and stabilizes: the first homology (concretely, the abelianization
The first homology group is the abelianization, and corresponds to the sign map S''n''
which is the abelianization for ''n'' ≥ 2; for ''n'' < 2 the symmetric group is trivial. This homology is easily computed as follows: S''n''
is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps are to S2
and all involutions are conjugate, hence map to the same element in the abelianization (since conjugation is trivial in abelian groups). Thus the only possible maps send an involution to 1 (the trivial map) or to −1 (the sign map). One must also show that the sign map is well-defined, but assuming that, this gives the first homology of S''n''
The second homology (concretely, the Schur multiplier
This was computed in , and corresponds to the double cover of the symmetric group
, 2 · S''n''
Note that the exceptional
low-dimensional homology of the alternating group (
corresponding to non-trivial abelianization, and
due to the exceptional 3-fold cover) does not change the homology of the symmetric group; the alternating group phenomena do yield symmetric group phenomena – the map
and the triple covers of A6
extend to triple covers of S6
– but these are not ''homological'' – the map
does not change the abelianization of S4
, and the triple covers do not correspond to homology either.
The homology "stabilizes" in the sense of stable homotopy
theory: there is an inclusion map , and for fixed ''k'', the induced map on homology is an isomorphism for sufficiently high ''n''. This is analogous to the homology of families Lie groups
The homology of the infinite symmetric group is computed in , with the cohomology algebra forming a Hopf algebra
The representation theory of the symmetric group
is a particular case of the representation theory of finite groups
, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function
theory to problems of quantum mechanics
for a number of identical particles
The symmetric group S''n''
has order ''n''!
. Its conjugacy class
es are labeled by partition
s of ''n''. Therefore, according to the representation theory of a finite group, the number of inequivalent irreducible representation
s, over the complex number
s, is equal to the number of partitions of ''n''. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representation by the same set that parametrizes conjugacy classes, namely by partitions of ''n'' or equivalently Young diagram
s of size ''n''.
Each such irreducible representation can be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the Young symmetrizer
s acting on a space generated by the Young tableau
x of shape given by the Young diagram.
Over other field
s the situation can become much more complicated. If the field ''K'' has characteristic
equal to zero or greater than ''n'' then by Maschke's theorem
the group algebra
is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary).
However, the irreducible representations of the symmetric group are not known in arbitrary characteristic. In this context it is more usual to use the language of module
s rather than representations. The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible. The modules so constructed are called ''Specht modules
'', and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their dimension
s are not known in general.
The determination of the irreducible modules for the symmetric group over an arbitrary field is widely regarded as one of the most important open problems in representation theory.
* Braid group
* History of group theory
* Signed symmetric group
and Generalized symmetric group
* Symmetric inverse semigroup
* Symmetric power
* Marcus du Sautoy: Symmetry, reality's riddle
(video of a talk)
br>Entries dealing with the Symmetric Group
Category:Finite reflection groups