Symmetric group (original) (raw)

In mathematics, the symmetric group on a set X, denoted by S_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, S_X_ 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 S_n_. The remainder of this article will discuss S_n_. The elements of S_n_ are called permutations; there are n of them. The group S_n_ is abelian if and only if n ≤ 2.

Subgroups of S n are called permutation groups.

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

and

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

.

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. 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:

With this definition,

sgn: S_n_ → {+1,-1}

is a group homomorphism ({+1,-1} is a group under multiplication). The kernel of this homomorphism, i.e. the set of all even permutations, is called the alternating group A_n_. It is a normal subgroup of S_n_ and has n! / 2 elements. The group S_n_ is the semidirect product of A_n_ 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), f_2(x), ..., f k(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 S_n can be written as a product of disjoint cycles; this representation is unique up to the order of the factors.

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