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 o g, defined by (f o 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, but not in Wikipedia, as gf).

Of particular importance is the case of a finite set X = {1,...,n}, which we write as Sn. The remainder of this article will discuss Sn. The elements of Sn are called permutations; there are n! of them. The group Sn is abelian if and only if n ≤ 2.

Subgroups of Sn are called permutation groups.

The rule of composition in the symmetric group is demonstrated below: Let

[itex] f = (1\ 3)(2)(4\ 5)=\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 2 & 1 & 5 & 4\end{bmatrix} [itex]

and

[itex] g = (1\ 2\ 5)(3\ 4)=\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{bmatrix} [itex]

Applying f after g maps 1 to 2, and then to itself; 2 to 5 to 4; 3 to 4 to 5, and so on. So composing f and g gives

[itex] fg = (1\ 2\ 4)(3\ 5)=\begin{bmatrix} 1 & 2 &3 & 4 & 5 \\ 2 & 4 & 5 & 1 & 3\end{bmatrix} [itex].

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.

To see this, consider the function which maps a permutation to an integer corresponding to the number of pairs (i,j), i<j, for which f(j)<f(i). Note that for any transposition composed with f, the parity of this number changes.

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 signature of a permutation:

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

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 has n! / 2 elements. The group Sn is the semidirect product of An and any subgroup generated by a single transposition.

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 f shown above is a cycle, since f(1) = 4, f(4) = 3 and f(3) = 1. 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.

The conjugacy classes of Sn correspond to the cycle structures of permutations; that is, two elements of Sn are conjugate 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.

Braid groups are generalizations of symmetric groups.

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