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
Occasionally (for example, at RoseHulman Institute of Technology, but not in Wikipedia), the opposite convention is used and the application of f after g is written as gf.
A transposition is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutations can be written as a product of transpositions; for instance, the permutation f from above can be written as f = (1 2)(2 5)(3 4). Since f can be written as a product of an odd number of transpositions, it is then called an odd permutation, whereas g 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,
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 S_{6} 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 S_{5}, (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.
Search Encyclopedia

Featured Article
