In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections 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 permutations that can be performed on the symbols. Since there are ( factorial) such permutation operations, the order (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, and combinatorics. Cayley's theorem states that every group is isomorphic to a subgroup of the symmetric group on (the underlying 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. 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 (the factorial of ). It is abelian 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 (they have order ). The group Sn 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 polynomials 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 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, 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 An and occurs as the Weyl group of the general linear group. In combinatorics, the symmetric groups, their elements (permutations), 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 spaces, 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 permutations of X.
Multiplication The group operation in a symmetric group is function composition, denoted by the symbol ∘ or by simple juxtaposition. The composition of permutations and , pronounced " of ", maps any element of to . Concretely, let (see permutation for an explanation of notation):
Applying after 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 and gives
A cycle of length , taken to the th power, will decompose into cycles of length : For example, (, ),
Verification of group axioms To check that the symmetric group on a set X is indeed a group, it is necessary to verify the group axioms of closure, associativity, identity, and inverses. The operation of function composition is closed in the set of permutations of the given set X. Function composition is always associative. The trivial bijection that assigns each element of X to itself serves as an identity for the group. Every bijection has an inverse function that undoes its action, and thus each element of a symmetric group does have an inverse which is a permutation too.
Transpositions, sign, and the alternating group
A transposition is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutation can be written as a product of transpositions; for instance, the permutation g from above can be written as g = (1 2)(2 5)(3 4). Since g can be written as a product of an odd number of transpositions, it is then called an odd permutation, whereas f is an even permutation.
The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd. There are several short proofs of the invariance of this parity of a permutation.
The product of two even permutations is even, the product of two odd permutations is even, and the product of one of each is odd. Thus we can define the sign of a permutation:
With this definition,
is a group homomorphism ({+1, −1} 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 An. It is a normal subgroup of Sn, and for it has elements. The group Sn is the semidirect product of An 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 {1, ..., n} such that x, f(x), f2(x), ..., fk(x) = x are the only elements moved by f; it conventionally 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: for example, in S6 there is the equality . Every element of Sn 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 {1, 2, ..., n} 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 S2n, 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 2n elements have the same sign; these are important to the classification of Clifford algebras, which are 8-periodic.
Conjugacy classes
The conjugacy classes of Sn correspond to the cycle types of permutations; that is, two elements of Sn are conjugate in Sn 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 Sn 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 Sn correspond to integer partitions of n: to the partition with and , is associated the set Cμ of permutations with cycles of lengths . Then Cμ is a conjugacy class of Sn, 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.
S0 and S1 The symmetric groups on the empty set and the singleton set are trivial, which corresponds to . In this case the alternating group agrees with the symmetric group, rather than being an index 2 subgroup, and the sign map is trivial. In the case of S0, its only member is the empty function.
S2 This group consists of exactly two elements: the identity and the permutation swapping the two points. It is a cyclic group and is thus abelian. In Galois theory, this corresponds to the fact that the quadratic formula gives a direct solution to the general quadratic polynomial after extracting only a single root. In invariant theory, the representation theory of the symmetric group on two points is quite simple and is seen as writing a function of two variables as a sum of its symmetric and anti-symmetric parts: Setting , and , one gets that . This process is known as symmetrization.
S3 S3 is the first nonabelian symmetric group. This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations. In Galois theory, the sign map from S3 to S2 corresponds to the resolving quadratic for a cubic polynomial, as discovered by Gerolamo Cardano, while the A3 kernel corresponds to the use of the discrete Fourier transform of order 3 in the solution, in the form of Lagrange resolvents.
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. This group also describes the symmetries under rotations and reflections of a regular tetrahedron, since these symmetries permute the four vertices of the tetrahedron. In this context, the subgroup A4 represents the rotations of the tetrahedron. These two symmetries of three-dimensional objects yield two three-dimensional irreducible representations of S4. S4 has a Klein four-group V as a proper normal subgroup, namely the even transpositions with quotient S3. In Galois theory, this map corresponds to the resolving cubic to a quartic polynomial, which allows the quartic to be solved by radicals, as established by Lodovico Ferrari. The Klein group can be understood in terms of the Lagrange resolvents 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 .