| Algebraic structure →Group theory Group theory |
|---|
 |
Infinite dimensional Lie group
|
Inabstract algebra, thesymmetric group defined over anyset is thegroup whoseelements are all thebijections from the set to itself, and whosegroup operation is thecomposition of functions. In particular, the finite symmetric group defined over afinite set of symbols consists of thepermutations that can be performed on the symbols.[1] Since there are (factorial) such permutation operations, theorder (number of elements) of the symmetric group is.
Although symmetric groups can be defined oninfinite sets, this article focuses on the finite symmetric groups: their applications, their elements, theirconjugacy classes, afinite presentation, theirsubgroups, theirautomorphism groups, and theirrepresentation 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 asGalois theory,invariant theory, therepresentation theory of Lie groups, andcombinatorics.Cayley's theorem states that every group isisomorphic to asubgroup of the symmetric group on (theunderlying set of).
The symmetric group on a finite set is the group whose elements are all bijective functions from to and whose group operation is that offunction composition.[1] For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group ofdegree is the symmetric group on the set.
The symmetric group on a set is denoted in various ways, including,,,, and.[1] If is the set then the name may be abbreviated to,,, or.[1]
Symmetric groups on infinite sets behave quite differently from symmetric groups on finite sets, and are discussed in (Scott 1987, Ch. 11), (Dixon & Mortimer 1996, Ch. 8), and (Cameron 1999).
The symmetric group on a set of elements hasorder (thefactorial of).[2] It isabelian if and only if is less than or equal to 2.[3] For and (theempty set and thesingleton set), the symmetric groups aretrivial (they have order). The group Sn issolvable if and only if. This is an essential part of the proof of theAbel–Ruffini theorem that shows that for every there arepolynomials of degree which are not solvable by radicals, that is, the solutions cannot be expressed by performing a finite number of operations of addition,subtraction, multiplication, division and root extraction on the polynomial's coefficients.
The symmetric group on a set of sizen is theGalois group of the generalpolynomial of degreen and plays an important role inGalois theory. Ininvariant theory, the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-calledsymmetric functions. In therepresentation theory of Lie groups, therepresentation theory of the symmetric group plays a fundamental role through the ideas ofSchur functors.
In the theory ofCoxeter groups, the symmetric group is the Coxeter group of type An and occurs as theWeyl group of thegeneral linear group. Incombinatorics, the symmetric groups, their elements (permutations), and theirrepresentations provide a rich source of problems involvingYoung tableaux,plactic monoids, and theBruhat order.Subgroups of symmetric groups are calledpermutation groups and are widely studied because of their importance in understandinggroup actions,homogeneous spaces, andautomorphism groups ofgraphs, such as theHigman–Sims group and theHigman–Sims graph.
The elements of the symmetric group on a setX are thepermutations ofX.
The group operation in a symmetric group is function composition, denoted by the symbol ∘ or by simple juxtaposition. The compositionf ∘g of permutationsf andg, pronounced "f ofg", maps any elementx ofX tof(g(x)). Concretely, let (seepermutation for an explanation of notation):Applyingf afterg 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, composingf andg gives
Acycle of lengthL =k ·m, taken to thekth power, will decompose intok cycles of lengthm: For example, (k = 2,m = 3),
To check that the symmetric group on a setX is indeed agroup, it is necessary to verify the group axioms of closure, associativity, identity, and inverses.[4]
Atransposition 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 permutationg from above can be written asg = (1 2)(2 5)(3 4). Sinceg can be written as a product of an odd number of transpositions, it is then called anodd permutation, whereasf 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 thesign of a permutation:
With this definition,
is agroup homomorphism ({+1, −1} is a group under multiplication, where +1 is e, theneutral element). Thekernel of this homomorphism, that is, the set of all even permutations, is called thealternating group An. It is anormal subgroup of Sn, and forn ≥ 2 it hasn!/2 elements. The group Sn is thesemidirect product of An and any subgroup generated by a single transposition.
Furthermore, every permutation can be written as a product ofadjacent transpositions, that is, transpositions of the form(aa+1). For instance, the permutationg from above can also be written asg = (4 5)(3 4)(4 5)(1 2)(2 3)(3 4)(4 5). The sorting algorithmbubble sort is an application of this fact. The representation of a permutation as a product of adjacent transpositions is also not unique.
Acycle oflengthk is a permutationf for which there exists an elementx in {1, ...,n} such thatx,f(x),f2(x), ...,fk(x) =x are the only elements moved byf; it conventionally is required thatk ≥ 2 since withk = 1 the elementx itself would not be moved either. The permutationh defined by
is a cycle of length three, sinceh(1) = 4,h(4) = 3 andh(3) = 1, leaving 2 and 5 untouched. We denote such a cycle by(1 4 3), but it could equally well be written(4 3 1) or(3 1 4) by starting at a different point. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles aredisjoint if they have disjoint subsets of elements. Disjoint cyclescommute: for example, in S6 there is the equality(4 1 3)(2 5 6) = (2 5 6)(4 1 3). Every element of Sn can be written as a product of disjoint cycles; this representation is uniqueup 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 itsgenerators and relations.
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).
Theorder reversing permutation is the one given by:
This is the uniquemaximal element with respect to theBruhat order and thelongest element in the symmetric group with respect to generating set consisting of the adjacent transpositions(ii+1),1 ≤i ≤n − 1.
This is an involution, and consists of (non-adjacent) transpositions
so it thus has sign:
which is 4-periodic inn.
In S2n, theperfect shuffle is the permutation that splits the set into 2 piles and interleaves them. Its sign is also
Note that the reverse onn elements and perfect shuffle on 2n elements have the same sign; these are important to the classification ofClifford algebras, which are 8-periodic.
Theconjugacy classes of Sn correspond to thecycle types of permutations; that is, two elements of Sn are conjugate in Snif 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 tointeger partitions ofn: to the partitionμ = (μ1,μ2, ...,μk) with andμ1 ≥μ2 ≥ ... ≥μk, is associated the setCμ of permutations with cycles of lengthsμ1,μ2, ...,μk. ThenCμ is a conjugacy class of Sn, whose elements are said to be of cycle-type.
The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately.
Other than the trivial mapSn → C1 ≅ S0 ≅ S1 and the sign mapSn → S2, the most notable homomorphisms between symmetric groups, in order ofrelative dimension, are:
There are also a host of other homomorphismsSm → Sn wherem <n.
Forn ≥ 5, thealternating group An issimple, and the induced quotient is the sign map:An → Sn → S2 which is split by taking a transposition of two elements. Thus Sn is the semidirect productAn ⋊ S2, and has no other proper normal subgroups, as they would intersect An in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in An (and thus themselves be An or Sn).
Sn acts on its subgroup An by conjugation, and forn ≠ 6, Sn is the full automorphism group of An: Aut(An) ≅ Sn. Conjugation by even elements areinner automorphisms of An while theouter automorphism of An of order 2 corresponds to conjugation by an odd element. Forn = 6, there is anexceptional outer automorphism of An so Sn is not the full automorphism group of An.
Conversely, forn ≠ 6, Sn has no outer automorphisms, and forn ≠ 2 it has no center, so forn ≠ 2, 6 it is acomplete group, as discussed inautomorphism group, below.
Forn ≥ 5, Sn is analmost simple group, as it lies between the simple group An and its group of automorphisms.
Sn can be embedded into An+2 by appending the transposition(n + 1,n + 2) to all odd permutations, while embedding into An+1 is impossible forn > 1.
The symmetric group onn letters is generated by theadjacent transpositions that swapi andi + 1.[6] The collection generatesSn subject to the following relations:[7]
where 1 represents the identity permutation. This representation endows the symmetric group with the structure of aCoxeter group (and so also areflection group).
Other possible generating sets include the set of transpositions that swap1 andi for2 ≤i ≤n,[8] or more generally any set of transpositions that forms a connected graph,[9] and a set containing anyn-cycle and a2-cycle of adjacent elements in then-cycle.[10][11]
Asubgroup of a symmetric group is called apermutation group.
Thenormal subgroups of the finite symmetric groups are well understood. Ifn ≤ 2, Sn has at most 2 elements, and so has no nontrivial proper subgroups. Thealternating group of degreen is always a normal subgroup, a proper one forn ≥ 2 and nontrivial forn ≥ 3; forn ≥ 3 it is in fact the only nontrivial proper normal subgroup ofSn, except whenn = 4 where there is one additional such normal subgroup, which is isomorphic to theKlein four group.
The symmetric group on an infinite set does not have a subgroup of index 2, asVitali (1915[12]) proved that each permutation can be written as a product of three squares. (Any squared element must belong to the hypothesized subgroup of index 2, hence so must the product of any number of squares.) However it contains the normal subgroupS of permutations that fix all but finitely many elements, which is generated by transpositions. Those elements ofS that are products of an even number of transpositions form a subgroup of index 2 inS, called the alternating subgroupA. SinceA is even acharacteristic subgroup ofS, it is also a normal subgroup of the full symmetric group of the infinite set. The groupsA andS are the only nontrivial proper normal subgroups of the symmetric group on a countably infinite set. This was first proved byOnofri (1929[13]) and independentlySchreier–Ulam (1934[14]). For more details see (Scott 1987, Ch. 11.3). That result, often called the Schreier-Ulam theorem, is superseded by a stronger one which says that the nontrivial normal subgroups of the symmetric group on a set are 1) the even permutations with finite support and 2) for every cardinality the group of permutations with support less than (Dixon & Mortimer 1996, Ch. 8.1).
Themaximal subgroups ofSn fall into three classes: the intransitive, the imprimitive, and the primitive. The intransitive maximal subgroups are exactly those of the formSk × Sn–k for1 ≤k <n/2. The imprimitive maximal subgroups are exactly those of the formSk wr Sn/k, where2 ≤k ≤n/2 is a proper divisor ofn and "wr" denotes thewreath product. The primitive maximal subgroups are more difficult to identify, but with the assistance of theO'Nan–Scott theorem and theclassification of finite simple groups, (Liebeck, Praeger & Saxl 1988) gave a fairly satisfactory description of the maximal subgroups of this type, according to (Dixon & Mortimer 1996, p. 268).
TheSylow subgroups of the symmetric groups are important examples ofp-groups. They are more easily described in special cases first:
The Sylowp-subgroups of the symmetric group of degreep are just the cyclic subgroups generated byp-cycles. There are(p − 1)!/(p − 1) = (p − 2)! such subgroups simply by countinggenerators. Thenormalizer therefore has orderp⋅(p − 1) and is known as aFrobenius groupFp(p−1) (especially forp = 5), and is theaffine general linear group,AGL(1,p).
The Sylowp-subgroups of the symmetric group of degreep2 are thewreath product of two cyclic groups of orderp. For instance, whenp = 3, a Sylow 3-subgroup of Sym(9) is generated bya = (1 4 7)(2 5 8)(3 6 9) and the elementsx = (1 2 3),y = (4 5 6),z = (7 8 9), and every element of the Sylow 3-subgroup has the formaixjykzl for.
The Sylowp-subgroups of the symmetric group of degreepn are sometimes denoted Wp(n), and using this notation one has thatWp(n + 1) is the wreath product of Wp(n) and Wp(1).
In general, the Sylowp-subgroups of the symmetric group of degreen are a direct product ofai copies of Wp(i), where0 ≤ai ≤p − 1 andn =a0 + p⋅a1 + ... + pk⋅ak (the basep expansion ofn).
For instance,W2(1) = C2 and W2(2) = D8, thedihedral group of order 8, and so a Sylow 2-subgroup of the symmetric group of degree 7 is generated by{ (1,3)(2,4), (1,2), (3,4), (5,6) } and is isomorphic toD8 × C2.
These calculations are attributed to (Kaloujnine 1948) and described in more detail in (Rotman 1995, p. 176). Note however that (Kerber 1971, p. 26) attributes the result to an 1844 work ofCauchy, and mentions that it is even covered in textbook form in (Netto 1882, §39–40).
Atransitive subgroup of Sn is a subgroup whose action on {1, 2, ,..., n} istransitive. For example, the Galois group of a (finite)Galois extension is a transitive subgroup of Sn, for somen.
A subgroup ofSn that is generated by transpositions is called aYoung subgroup. They are all of the form where is aninteger partition ofn. These groups may also be characterized as theparabolic subgroups ofSn when it is viewed as areflection group.
Cayley's theorem states that every groupG is isomorphic to a subgroup of some symmetric group. In particular, one may take a subgroup of the symmetric group on the elements ofG, since every group acts on itself faithfully by (left or right) multiplication.
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 theleast common multiple of the lengths of its cycles. For example, in S5, one cyclic subgroup of order 5 is generated by (13254), whereas the largest cyclic subgroups of S5 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.[citation needed] For example, S5 has no subgroup of order 15 (a divisor of the order of S5), 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 Sn) is given byLandau's function.
| n | Aut(Sn) | Out(Sn) | Z(Sn) |
| n ≠ 2, 6 | Sn | C1 | C1 |
| n = 2 | C1 | C1 | S2 |
| n = 6 | S6 ⋊ C2 | C2 | C1 |
Forn ≠ 2, 6, Sn is acomplete group: itscenter andouter automorphism group are both trivial.
Forn = 2, 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.
Forn = 6, it has an outer automorphism of order 2:Out(S6) = C2, and the automorphism group is a semidirect productAut(S6) = S6 ⋊ C2.
In fact, for any setX of cardinality other than 6, every automorphism of the symmetric group onX is inner, a result first due to (Schreier & Ulam 1936) according to (Dixon & Mortimer 1996, p. 259).
Thegroup homology of Sn is quite regular and stabilizes: the first homology (concretely, theabelianization) is:
The first homology group is the abelianization, and corresponds to the sign map Sn → S2 which is the abelianization forn ≥ 2; forn < 2 the symmetric group is trivial. This homology is easily computed as follows: Sn is generated by involutions (2-cycles, which have order 2), so the only non-trivial mapsSn → Cp 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 mapsSn → S2 ≅ {±1} 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 Sn.
The second homology (concretely, theSchur multiplier) is:
This was computed in (Schur 1911), and corresponds to thedouble cover of the symmetric group, 2 · Sn.
Note that theexceptional 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 A6 and A7 extend to triple covers of S6 and S7 – but these are nothomological – the map does not change the abelianization of S4, and the triple covers do not correspond to homology either.
The homology "stabilizes" in the sense ofstable homotopy theory: there is an inclusion mapSn → Sn+1, and for fixedk, the induced map on homologyHk(Sn) →Hk(Sn+1) is an isomorphism for sufficiently highn. This is analogous to the homology of familiesLie groups stabilizing.
The homology of the infinite symmetric group is computed in (Nakaoka 1961), with the cohomology algebra forming aHopf algebra.
Therepresentation theory of the symmetric group is a particular case of therepresentation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, fromsymmetric function theory to problems ofquantum mechanics for a number ofidentical particles.
The symmetric group Sn has ordern!. Itsconjugacy classes are labeled bypartitions of n. Therefore, according to the representation theory of a finite group, the number of inequivalentirreducible representations, over thecomplex 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 ofn or equivalentlyYoung 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 theYoung symmetrizers acting on a space generated by theYoung tableaux of shape given by the Young diagram.
Over otherfields the situation can become much more complicated. If the fieldK hascharacteristic equal to zero or greater thann then byMaschke's theorem thegroup algebraKSn 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 ofmodules 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 calledSpecht 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 theirdimensions 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.
Learning materials related tothe S4 symmetric group at Wikiversity