Definitions

symmetric-group

Symmetric group

In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i.e., two such functions f and g can be composed to yield a new bijective function f circ g, defined by (f circ g)(x) = f(g(x)) for all x in X. Using this operation, SX forms a group. The operation is also written as fg (and sometimes, although not here, as gf).

Finite symmetric groups

Of particular importance is the symmetric group on the finite set

X = {1, ..., n},

denoted by Sn. The permutations of X form the set of bijective functions. The group Sn has order n! and is not abelian for n > 2. Similarly, the group Sn is solvable if and only if n ≤ 4. The remainder of this article will discuss Sn.

Subgroups of Sn are called permutation groups.

Composition of permutations

The group operation in a symmetric group is function composition; the symbol circ is often omitted, is demonstrated below. Let

f = (1 3)(4 5)=begin{pmatrix} 1 & 2 & 3 & 4 & 5 3 & 2 & 1 & 5 & 4end{pmatrix}
and
g = (1 2 5)(3 4)=begin{pmatrix} 1 & 2 & 3 & 4 & 5 2 & 5 & 4 & 3 & 1end{pmatrix} . (See permutation for an explanation of notation).
Applying f after g maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So composing f and g gives
fg = fcirc g = (1 2 4)(3 5)=begin{pmatrix} 1 & 2 &3 & 4 & 5 2 & 4 & 5 & 1 & 3end{pmatrix} .

A cycle of length L =k·m, taken to the k-th power, will decompose into k cycles of length m: For example (k=2, m=3),

(1~2~3~4~5~6)^2 = (1~3~5) (2~4~6) ~.

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 associativity, identity, and inverses. The operation of 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.

Transpositions

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 5)(1 2)(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.

To see this, consider the function P which maps a permutation to the number of pairs (i, j), i < j, for which f(j) < f(i). Intuitively, this is the total number of times each of the numbers 1,ldots,n has to cross over another number to arrive at its final position; it is independent of the way in which the permutation is written down. In fact, the parity of P(f) corresponds to the odd- or even-ness of the permutation f, as we will now show. It is fairly straightforward to prove that for a transposition (a~b), P(a~b) = 2(b-a-2)+1, which is odd. Furthermore, P(f(a~b)) has opposite parity to P(f) since we must either add or subtract P(a~b). Thus if we write f as a product f_1 f_2 ldots f_m of transpositions the parity of P(f) is the same as the parity of m. This shows that, given any two products of transpositions that yield the same permutation, their respective numbers of transpositions must have the same parity.

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:

operatorname{sgn}(f)=left{begin{matrix} +1, & mbox{if }fmbox { is even} -1, & mbox{if }f mbox{ is odd}. end{matrix}right.

With this definition,

sgn: Sn → {+1, –1}
is a group homomorphism ({+1, –1} is a group under multiplication, where +1 is e, the neutral element). The kernel of this homomorphism, i.e. the set of all even permutations, is called the alternating group An. It is a normal subgroup of Sn, and for n ≥ 2 it has n! / 2 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 (a~a+1). For instance, the permutation g from above can also be written as g = (4 5)(3 4)(4 5)(1 2)(2 3)(3 4)(4 5). The representation of a permutation as a product of adjacent transpositions is also not unique.

Cycles

A cycle 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. The permutation h defined by

h = begin{pmatrix} 1 & 2 & 3 & 4 & 5 4 & 2 & 1 & 3 & 5end{pmatrix}

is a cycle, since h(1) = 4, h(4) = 3 and h(3) = 1, leaving 2 and 5 untouched. We denote such a cycle by (1 4 3). The length of this cycle is three. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are disjoint if they move different elements. Disjoint cycles commute, e.g. in S6 we have (3 1 4)(2 5 6) = (2 5 6)(3 1 4). Every element of Sn can be written as a product of disjoint cycles; this representation is unique up to the order of the factors.

Conjugacy classes

The conjugacy classes of Sn correspond to the cycle structures 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:

k = begin{pmatrix} 1 & 2 & 3 & 4 & 5 1 & 4 & 3 & 2 & 5end{pmatrix}

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, i.e.

(2 4)(1 2 3)(4 5)(2 4)=(1 4 3)(2 5)

It is clear that such a permutation is not unique.

Examples

The group S3 is isomorphic to 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.

For a list of elements of S4, see Cycle notation. See cube for the proper rotations of a cube, which form a group isomorphic with S4. In this case, the permuted objects are the four diagonals of the cube.

Symmetric groups are Coxeter groups and reflection groups. They can be realized as a group of reflections with respect to hyperplanes x_i=x_j, 1leq i < j leq n. Braid groups Bn admit symmetric groups Sn as quotient groups.

Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group on the elements of G, as a group acts on itself faithfully by (left or right) multiplication.

Elements

Certain elements of the symmetric group are of particular interest.

The reverse permutation reverses the order of elements:

begin{pmatrix} 1 & 2 & cdots & n
n & n-1 & cdots & 1end{pmatrix} This is the longest element with respect to the Bruhat order, thinking of the symmetric group as a Coxeter group generated by the adjacent transpositions.

This is an involution, and consists of lfloor n/2 rfloor (non-adjacent) transpositions: (1,n)(2,n-1)cdots, or sum_{k=1}^{n-1} k = frac{n(n-1)}{2} adjacent transpositions: (n,n-1)(n-1,n-2)cdots(2,1)(n-1,n-2)(n-2,n-3)cdots, so it thus has sign:

mbox{sgn}(rho_n) = (-1)^{lfloor n/2 rfloor} =(-1)^{n(n-1)/2} = begin{cases}
+1 & n equiv 0,1 pmod{4} -1 & n equiv 2,3 pmod{4} end{cases} which is 4-periodic in n.

In S_{2n}, the perfect shuffle is the permutation that splits the set into 2 piles and interleaves them. Its sign is also (-1)^{lfloor n/2 rfloor}.

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.

Automorphism group

n mbox{Aut}(S_n) mbox{Out}(S_n)
nneq 2,6 S_n 1
n=2 1 1
n=6 S_6 rtimes C_2 C_2
For n neq 2, 6, S_n is a complete group: its center and outer automorphism group are both trivial.

For n = 2, the automorphism group is trivial, but S_2 is not trivial (it is isomorphic to C_2, which is abelian).

For n = 6, it has an outer automorphism of order 2: mbox{Out}(S_6)=C_2, and the automorphism group is a semidirect product: mbox{Aut}(S_6)=S_6 rtimes C_2.

See also

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