TheInfoList  In
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, the symmetric group defined over any set is the
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
whose elements are all the
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... s from the set to itself, and whose
group operation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
is the
composition of functions In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. In particular, the finite symmetric group $\mathrm_n$ defined over a
finite set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of $n$ symbols consists of the
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ... s that can be performed on the $n$ symbols.Jacobson (2009), p. 31. Since there are $n!$ ($n$
factorial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
) such permutation operations, the
order Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
(number of elements) of the symmetric group $\mathrm_n$ is $n!$. Although symmetric groups can be defined on
infinite set In set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch ...
s, this article focuses on the finite symmetric groups: their applications, their elements, their
conjugacy class In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
es, a
finite presentation In mathematics, a presentation is one method of specifying a group (mathematics), group. A presentation of a group ''G'' comprises a set ''S'' of generating set of a group, generators—so that every element of the group can be written as a produ ...
, their
subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
s, their
automorphism group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s, and their
representation Representation may refer to: Law and politics *Representation (politics) Political representation is the activity of making citizens "present" in public policy making processes when political actors act in the best interest of citizens. This def ...
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 In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
,
invariant theory Invariant theory is a branch of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), ...
, the
representation theory of Lie groups In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, and
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
.
Cayley's theorem In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
states that every group $G$ is
isomorphic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
to a
subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
of the symmetric group on (the
underlying set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of) $G$.

# Definition and first properties

The symmetric group on a finite set $X$ is the group whose elements are all bijective functions from $X$ to $X$ and whose group operation is that of
function composition In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
. For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree $n$ is the symmetric group on the set $X = \$. The symmetric group on a set $X$ is denoted in various ways, including $\mathrm_X$, $\mathfrak_X$, $\Sigma_X$, $X!$, and $\operatorname\left(X\right)$. If $X$ is the set $\$ then the name may be abbreviated to $\mathrm_n$, $\mathfrak_n$, $\Sigma_n$, or $\operatorname\left(n\right)$. 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 $n$ elements has
order Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
$n!$ (the
factorial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
of $n$). It is abelian if and only if $n$ is less than or equal to 2. For $n=0$ and $n=1$ (the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ... and the
singleton set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
), the symmetric groups are trivial (they have order $0! = 1! = 1$). The group S''n'' is solvable if and only if $n \leq 4$. This is an essential part of the proof of the
Abel–Ruffini theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
that shows that for every $n > 4$ there are
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... s of degree $n$ 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.

# Applications

The symmetric group on a set of size ''n'' is the
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of the general
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... of degree ''n'' and plays an important role in
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
. In
invariant theory Invariant theory is a branch of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), ...
, the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called
symmetric function In mathematics, a Function (mathematics), function of n variables is symmetric if its value is the same no matter the order of its Argument of a function, arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric fun ...
s. In the
representation theory of Lie groups In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, the
representation theory of the symmetric group In mathematics, 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 symm ...
plays a fundamental role through the ideas of
Schur functor In mathematics, especially in the field of representation theory, Schur functors are certain functors from the category (mathematics), category of module (mathematics), modules over a fixed commutative ring to itself. They generalize the constructi ...
s. In the theory of
Coxeter group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, the symmetric group is the Coxeter group of type A''n'' and occurs as the
Weyl group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of the
general linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. In
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
, the symmetric groups, their elements (
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ... s), and their
representations ''Representations'' is an interdisciplinary journal in the humanities Humanities are academic disciplines An academic discipline or academic field is a subdivision of knowledge Knowledge is a familiarity or awareness, of someone o ...
provide a rich source of problems involving
Young tableaux In mathematics, a Young tableau (; plural: tableaux) is a combinatorics, combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric group, symmetric ...
,
plactic monoidIn mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified with semistandard Young tableau, Young tableaux. It was discovered by (who called it the t ...
s, and the
Bruhat orderIn mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion o ...
.
Subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
s of symmetric groups are called
permutation group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s and are widely studied because of their importance in understanding
group action In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s,
homogeneous space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, and
automorphism group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s of
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
s, such as the
Higman–Sims group In the area of modern algebra known as group theory, the Higman–Sims group HS is a sporadic simple group of Order (group theory), order :   29⋅32⋅53⋅7⋅11 = 44352000 : ≈ 4. The Schur multiplier has order 2, th ...
and the Higman–Sims graph.

# Group properties and special elements

The elements of the symmetric group on a set ''X'' are the
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ... s of ''X''.

## Multiplication

The group operation in a symmetric group is
function composition In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, 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 In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ... for an explanation of notation): : $f = \left(1\ 3\right)\left(4\ 5\right)=\begin 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4\end$ : $g = \left(1\ 2\ 5\right)\left(3\ 4\right)=\begin 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end.$ 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 : $fg = f\circ g = \left(1\ 2\ 4\right)\left(3\ 5\right)=\begin 1 & 2 &3 & 4 & 5 \\ 2 & 4 & 5 & 1 & 3\end.$ A
cycle Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...
of length , taken to the ''k''th power, will decompose into ''k'' cycles of length ''m'': For example, (, ), : $\left(1~2~3~4~5~6\right)^2 = \left(1~3~5\right) \left(2~4~6\right).$

## Verification of group axioms

To check that the symmetric group on a set ''X'' is indeed a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, it is necessary to verify the group axioms of closure, associativity, identity, and inverses. # The operation of
function composition In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
is closed in the set of permutations of the given set ''X''. #
Function composition In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
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: :$\operatornamef = \begin +1, & \textf\mbox \\ -1, & \textf \text. \end$ With this definition, :$\operatorname\colon \mathrm_n \rightarrow \\$ is a
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... ( is a group under multiplication, where +1 is e, the
neutral element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
). The
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of this homomorphism, that is, the set of all even permutations, is called the
alternating group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
A''n''. It is a
normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
of S''n'', and for it has elements. The group S''n'' is the
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product of groups, direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product ...
of A''n'' and any subgroup generated by a single transposition. Furthermore, every permutation can be written as a product of '' adjacent transpositions'', that is, transpositions of the form . For instance, the permutation ''g'' from above can also be written as . The sorting algorithm
bubble sort Bubble sort, sometimes referred to as sinking sort, is a simple sorting algorithm In computer science, a sorting algorithm is an algorithm that puts elements of a List (computing), list in a certain Total order, order. The most frequently used o ...
is an application of this fact. The representation of a permutation as a product of adjacent transpositions is also not unique.

## Cycles

A
cycle Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...
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 :$h = \begin 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 1 & 3 & 5\end$ 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 Two mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 $\sigma$, this property is often used to obtain its generators and relations. :$\sigma\begin a & b & c & \ldots \end\sigma^=\begin\sigma\left(a\right) & \sigma\left(b\right) & \sigma\left(c\right) & \ldots\end$

## Special elements

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: :$\begin 1 & 2 & \cdots & n\\ n & n-1 & \cdots & 1\end.$ This is the unique maximal element with respect to the
Bruhat orderIn mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion o ...
and the longest element in the symmetric group with respect to generating set consisting of the adjacent transpositions , . This is an involution, and consists of $\lfloor n/2 \rfloor$ (non-adjacent) transpositions :$\left(1\,n\right)\left(2\,n-1\right)\cdots,\text\sum_^ k = \frac\text$ :: $\left(n\,n-1\right)\left(n-1\,n-2\right)\cdots\left(2\,1\right)\left(n-1\,n-2\right)\left(n-2\,n-3\right)\cdots,$ so it thus has sign: :$\mathrm\left(\rho_n\right) = \left(-1\right)^ =\left(-1\right)^ = \begin +1 & n \equiv 0,1 \pmod\\ -1 & n \equiv 2,3 \pmod \end$ which is 4-periodic in ''n''. In S2''n'', the '' perfect shuffle'' is the permutation that splits the set into 2 piles and interleaves them. Its sign is also $\left(-1\right)^.$ 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 In mathematics, a Clifford algebra is an algebra over a field, algebra generated by a vector space with a quadratic form, and is a Unital algebra, unital associative algebra. As algebra over a field, ''K''-algebras, they generalize the real nu ...
s, which are 8-periodic.

# Conjugacy classes

The
conjugacy class In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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: :$k = \begin 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 3 & 2 & 5\end$ 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, :$\left(2~4\right)\circ\left(1~2~3\right)\left(4~5\right)\circ\left(2~4\right)=\left(1~4~3\right)\left(2~5\right).$ 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. ;S0 and S1: The symmetric groups on the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ... and the
singleton set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. ;S2: This group consists of exactly two elements: the identity and the permutation swapping the two points. It is a
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ... and is thus abelian. In
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
, this corresponds to the fact that the
quadratic formula In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ... gives a direct solution to the general
quadratic polynomial In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
after extracting only a single root. In
invariant theory Invariant theory is a branch of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), ...
, 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 In mathematics, symmetrization is a process that converts any Function (mathematics), function in n variables to a symmetric function in n variables. Similarly, antisymmetrization converts any function in n variables into an Antisymmetric relation, ...
. ;S3: S3 is the first nonabelian symmetric group. This group is isomorphic to the
dihedral group of order 6 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, the group of reflection and rotation symmetries of an
equilateral triangle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ... , 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 to S2 corresponds to the resolving quadratic for a
cubic polynomial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, as discovered by
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576 (O. S.)) was an Italian polymath A polymath ( el, πολυμαθής, , "having learned much"; ... , while the A3 kernel corresponds to the use of the
discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...
of order 3 in the solution, in the form of
Lagrange resolvent In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group ''G'' is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a ratio ...
s. ;S4: 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ... . Beyond the group A4, S4 has a
Klein four-group In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three ...
V as a proper
normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
, namely the even transpositions with quotient S3. In
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
, this map corresponds to the resolving cubic to a
quartic polynomial In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysi ...
, which allows the quartic to be solved by radicals, as established by
Lodovico Ferrari Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Ital ...
. The Klein group can be understood in terms of the
Lagrange resolvent In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group ''G'' is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a ratio ...
s of the quartic. The map from S4 to S3 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 . ;S5: S5 is the first non-solvable symmetric group. Along with the
special linear group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and the
icosahedral group A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is th ...
, S5 is one of the three non-solvable groups of order 120, up to isomorphism. S5 is the
Galois group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of the general
quintic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
, and the fact that S5 is not a
solvable group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
translates into the non-existence of a general formula to solve
quintic polynomial In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ... 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. ;S6: Unlike all other symmetric groups, S6, has an
outer automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. Using the language of
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
, this can also be understood in terms of
Lagrange resolvents In Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reduci ...
. 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 In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exception ...
'' for details. :Note that while A6 and A7 have an exceptional
Schur multiplierSchur is a German or Jewish surname. Notable people with the surname include: * Alexander Schur (born 1971), German footballer * Dina Feitelson-Schur (1926–1992), Israeli educator * Friedrich Schur (1856-1932), German mathematician * Fritz Schur ...
(a triple cover) and that these extend to triple covers of S6 and S7, 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 In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension. In linear algebra, given a quotient space (linear algebra), quotient map V \to Q, the difference dim ''V'' − dim ''Q'' is the relativ ...
, are: * 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
A''n'' is
simple Simple or SIMPLE may refer to: * Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
, 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'' or S''n''). S''n'' acts on its subgroup A''n'' by conjugation, and for , S''n'' is the full automorphism group of A''n'': Aut(A''n'') ≅ S''n''. Conjugation by even elements are
inner automorphism In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s of A''n'' while the
outer automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
of A''n'' of order 2 corresponds to conjugation by an odd element. For , there is an exceptional outer automorphism of A''n'' so S''n'' 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, automorphism group, below. For , S''n'' is an almost simple group, as it lies between the simple group A''n'' and its group of automorphisms. S''n'' 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 transpositions $\sigma_i = \left(i, i + 1\right)$ that swap and . The collection $\sigma_1, \ldots, \sigma_$ generates subject to the following relations: *$\sigma_i^2 = 1,$ *$\sigma_i\sigma_j = \sigma_j\sigma_i$ for $, i-j, > 1$, and *$\left(\sigma_i\sigma_\right)^3 =1,$ where 1 represents the identity permutation. This representation endows the symmetric group with the structure of a
Coxeter group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
(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.

# Subgroup structure

A
subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
of a symmetric group is called a
permutation group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.

## Normal subgroups

The
normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of degree ''n'' is always a normal subgroup, a proper one for and nontrivial for ; for it is in fact the only nontrivial 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 Giuseppe_Vitali, 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 nontrivial proper normal subgroups of the symmetric group on a countably infinite set. This was first proved by Luigi_Onofri, Onofri (1929) and independently J%C3%B3zef_Schreier, Schreier–Stanislaw_Ulam, Ulam (1934Über die Permutationsgruppe der natürlichen Zahlenfolge. Studia Mathematica (1933) Vol. 4(1), p.134–141, 1933). For more details see or .

## Maximal subgroups

The maximal subgroups of S''n'' 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 , where is a proper divisor of ''n'' and "wr" denotes the wreath product. 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 .

## Sylow subgroups

The Sylow subgroups of the symmetric groups are important examples of p-group, ''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''(1). 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 Augustin-Louis Cauchy, Cauchy, and mentions that it is even covered in textbook form in .

## Transitive subgroups

A transitive subgroup of S''n'' is a subgroup whose action on is transitive action, transitive. For example, the Galois group of a (finite extension, finite) Galois extension is a transitive subgroup of S''n'', for some ''n''.

## Cayley's theorem

Cayley's theorem In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
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.

# Automorphism group

For , S''n'' is a complete group: its center (group theory), 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 .

# Homology

The group homology of S''n'' is quite regular and stabilizes: the first homology (concretely, the abelianization) is: :$H_1\left(\mathrm_n,\mathbf\right) = \begin 0 & n < 2\\ \mathbf/2 & n \geq 2.\end$ The first homology group is the abelianization, and corresponds to the sign map S''n'' → S2 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 multiplierSchur is a German or Jewish surname. Notable people with the surname include: * Alexander Schur (born 1971), German footballer * Dina Feitelson-Schur (1926–1992), Israeli educator * Friedrich Schur (1856-1932), German mathematician * Fritz Schur ...
) is: :$H_2\left(\mathrm_n,\mathbf\right) = \begin 0 & n < 4\\ \mathbf/2 & n \geq 4.\end$ This was computed in , and corresponds to the covering groups of the alternating and symmetric groups, double cover of the symmetric group, 2 · S''n''. Note that the exceptional object, exceptional low-dimensional homology of the alternating group ($H_1\left(\mathrm_3\right)\cong H_1\left(\mathrm_4\right) \cong \mathrm_3,$ corresponding to non-trivial abelianization, and $H_2\left(\mathrm_6\right)\cong H_2\left(\mathrm_7\right) \cong \mathrm_6,$ 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 $\mathrm_4 \twoheadrightarrow \mathrm_3$ extends to $\mathrm_4 \twoheadrightarrow \mathrm_3,$ and the triple covers of A6 and A7 extend to triple covers of S6 and S7 – but these are not ''homological'' – the map $\mathrm_4 \twoheadrightarrow \mathrm_3$ 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 stabilizing. The homology of the infinite symmetric group is computed in , with the cohomology algebra forming a Hopf algebra.

# Representation theory

The
representation theory of the symmetric group In mathematics, 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 symm ...
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 In mathematics, a Function (mathematics), function of n variables is symmetric if its value is the same no matter the order of its Argument of a function, arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric fun ...
theory to problems of quantum mechanics for a number of identical particles. The symmetric group S''n'' has order ''n''!. Its
conjugacy class In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
es are labeled by integer partition, partitions of ''n''. Therefore, according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, 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 diagrams 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 symmetrizers acting on a space generated by the Young tableaux of shape given by the Young diagram. Over other Field (mathematics), fields the situation can become much more complicated. If the field ''K'' has characteristic (algebra), characteristic equal to zero or greater than ''n'' then by Maschke's theorem the group ring, group algebra ''K''S''n'' 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 (mathematics), modules 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 (vector space), dimensions 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

# References

* * * . * * * * * * * * *