In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, the symmetric group defined over any
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is the
group whose
elements are all the
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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 of
symbols consists of the
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s that can be performed on the
symbols.
Since there are
(
factorial) such permutation operations, the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
(number of elements) of the symmetric group
is
.
Although symmetric groups can be defined on
infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their
conjugacy classes, a
finite presentation, their
subgroups, their
automorphism groups, 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
In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vect ...
, and
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
.
Cayley's theorem states that every group
is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to a
subgroup of the symmetric group on (the
underlying set
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
of)
.
Definition and first properties
The symmetric group on a finite set
is the group whose elements are all bijective functions from
to
and whose group operation is that of
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
.
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
,
,
,
, and
.
If
is the set
then the name may be abbreviated to
,
,
, or
.
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
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
(the
factorial of
). It is
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
if and only if
is less than or equal to 2. For
and
(the
empty set and the
singleton set), the symmetric groups are
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
(they have order
). The group S
''n'' is
solvable 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
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
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.
Applications
The symmetric group on a set of size ''n'' is the
Galois group of the general
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
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 functions. In the
representation theory of Lie groups
In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vect ...
, the
representation theory of the symmetric group plays a fundamental role through the ideas of
Schur functors.
In the theory of
Coxeter groups, the symmetric group is the Coxeter group of type A
''n'' and occurs as the
Weyl group of the
general linear group. In
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
, the symmetric groups, their elements (
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s), and their
representations provide a rich source of problems involving
Young tableaux,
plactic monoids, and the
Bruhat order.
Subgroups of symmetric groups are called
permutation groups and are widely studied because of their importance in understanding
group actions,
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements ...
s, and
automorphism groups of
graphs, such as the
Higman–Sims group and the
Higman–Sims graph.
Group properties and special elements
The elements of the symmetric group on a set ''X'' are the
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s of ''X''.
Multiplication
The group operation in a symmetric group is
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
, denoted by the symbol ∘ or simply by just a composition 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 mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
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
:
A
cycle 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
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
is closed in the set of permutations of the given set ''X''.
#
Function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
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
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total o ...
, 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 A
''n''. It is a
normal subgroup of S
''n'', and for it has elements. The group S
''n'' is the
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 is an application of this fact. The representation of a permutation as a product of adjacent transpositions is also not unique.
Cycles
A
cycle 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 have disjoint subsets of elements. Disjoint cycles
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
: for example, in S
6 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.
:
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:
:
This is the unique maximal element with respect to the
Bruhat order 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
(non-adjacent) transpositions
:
::
so it thus has sign:
:
which is 4-periodic in ''n''.
In S
2''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 algebras, which are 8-periodic.
Conjugacy classes
The
conjugacy classes of S
''n'' correspond to the
cycle types 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 S
5, (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 as (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.
Conjugacy classes of
correspond to
integer partition
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same part ...
s of
: to the partition
with
and
, is associated the set
of permutations with cycles of lengths
. Then
is a conjugacy class of
, whose elements are said to be of cycle-type
.
Low degree groups
The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately.
;S
0 and S
1: 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 S
0, its only member is the
empty function.
;S
2: 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
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
. In
Galois theory, this corresponds to the fact that the
quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...
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
In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables.
Similarly, antisymmetrization converts any function in n variables into an antisymmetric function.
Two variables
Let S ...
.
;S
3: S
3 is the first nonabelian symmetric group. This group is isomorphic to the
dihedral group of order 6
In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3, or, in other words, the dihedral group of order 6. It is isomorphic to the symmetric group S3 of degree 3. It is also the smallest possible non-abe ...
, 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 S
3 to S
2 corresponds to the resolving quadratic for a
cubic polynomial, 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) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
, while the A
3 kernel corresponds to the use of the
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
of order 3 in the solution, in the form of
Lagrange resolvents.
;S
4: 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, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
. Beyond the group
A4, S
4 has a
Klein four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one ...
V as a proper
normal subgroup, namely the even transpositions with quotient S
3. In
Galois theory, this map corresponds to the resolving cubic to a
quartic polynomial
In algebra, a quartic function is a function of the form
:f(x)=ax^4+bx^3+cx^2+dx+e,
where ''a'' is nonzero,
which is defined by a polynomial of degree four, called a quartic polynomial.
A ''quartic equation'', or equation of the fourth deg ...
, 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 resolvents of the quartic. The map from S
4 to S
3 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 .
;S
5: S
5 is the first non-solvable symmetric group. Along with the
special linear group and the
icosahedral group , S
5 is one of the three non-solvable groups of order 120, up to isomorphism. S
5 is the
Galois group of the general
quintic equation, and the fact that S
5 is not a
solvable group translates into the non-existence of a general formula to solve
quintic polynomials 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 S
6, discussed below, and corresponds to the resolvent sextic of a quintic.
;S
6: Unlike all other symmetric groups, S
6, 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 S
6—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 except ...
'' for details.
:Note that while A
6 and A
7 have an exceptional
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \ope ...
(a
triple cover) and that these extend to triple covers of S
6 and S
7, 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, are:
* corresponding to the exceptional normal subgroup ;
* (or rather, a class of such maps up to inner automorphism) corresponding to the outer automorphism of S
6.
* as a transitive subgroup, yielding the outer automorphism of S
6 as discussed above.
There are also a host of other homomorphisms where .
Relation with alternating group
For , the
alternating group 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 an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
s of A
''n'' while the
outer automorphism 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, 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
that swap and . The collection
generates subject to the following relations:
*
*
for
, and
*
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.
Subgroup structure
A
subgroup of a symmetric group is called a
permutation group.
Normal subgroups
The
normal subgroups 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 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
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
Onofri (1929) and independently
Schreier–
Ulam Ulam may refer to:
* ULAM, the ICAO airport code for Naryan-Mar Airport, Russia
* Ulam (surname)
* Ulam (salad), a type of Malay salad
* ''Ulam'', a Filipino term loosely translated to viand or side dish; see Tapa (Filipino cuisine)
* Ulam, the l ...
(1934
). For more details see or .
Maximal subgroups
The
maximal subgroup
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.
In group theory, a maximal subgroup ''H'' of a group ''G'' is a proper subgroup, such that no proper subgroup ''K'' contains ''H'' ...
s 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''-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
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. For example, the Galois group of a (
finite)
Galois extension is a transitive subgroup of S
''n'', for some ''n''.
Cayley's theorem
Cayley's theorem 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.
Cyclic subgroups
Cyclic groups are those that are generated by a single permutation. When a permutation is represented in cycle notation, the order of the cyclic subgroup that it generates is the
least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by ...
of the lengths of its cycles. For example, in S, one cyclic subgroup of order 5 is generated by (13254), whereas the largest cyclic subgroups of S are generated by elements like (123)(45) that have one cycle of length 3 and another cycle of length 2. This rules out many groups as possible subgroups of symmetric groups of a given size. For example, S has no subgroup of order 15 (a divisor of the order of S), because the only group of order 15 is the cyclic group. The largest possible order of a cyclic subgroup (equivalently, the largest possible order of an element in S) is given by
Landau's function In mathematics, Landau's function ''g''(''n''), named after Edmund Landau, is defined for every natural number ''n'' to be the largest order of an element of the symmetric group ''S'n''. Equivalently, ''g''(''n'') is the largest least common ...
.
Automorphism group
For , S
''n'' is a
complete group: its
center and
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a ...
are both trivial.
For , the automorphism group is trivial, but S
2 is not trivial: it is isomorphic to C
2, 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:
:
The first homology group is the abelianization, and corresponds to the sign map S
''n'' → S
2 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 S
2 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
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier \ope ...
) is:
:
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
extends to
and the triple covers of A
6 and A
7 extend to triple covers of S
6 and S
7 – but these are not ''homological'' – the map
does not change the abelianization of S
4, and the triple covers do not correspond to homology either.
The homology "stabilizes" in the sense of
stable homotopy
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the ...
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 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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
for a number of
identical particles.
The symmetric group S
''n'' has order ''n''
!. Its
conjugacy classes are labeled by
partitions of ''n''. Therefore, according to the representation theory of a finite group, the number of inequivalent
irreducible representations, over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
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 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
fields 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
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make gener ...
the
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
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
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
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
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.
See also
*
Braid group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
*
History of group theory
*
Signed symmetric group and
Generalized symmetric group
*
*
Symmetric inverse semigroup
*
Symmetric power
Notes
References
*
*
* .
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External links
*
*
*
Marcus du Sautoy: Symmetry, reality's riddle(video of a talk)
*
OEISbr>
Entries dealing with the Symmetric Group
{{DEFAULTSORT:Symmetric Group
Permutation groups
Symmetry
Finite reflection groups