Symmetric Group

The symmetric group on a finite set X is the group whose elements are all bijective functions from X to X and whose group operation is that of function composition. For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree n is the symmetric group on the set X = { 1, 2, ..., n }.

The symmetric group on a set X is denoted in various ways including S_X, Σ_X, and Sym(X). If X is the set { 1, 2, ..., n }, then the symmetric group on X is also denoted S_n, Σ_n, and Sym(n).

Symmetric groups on infinite sets behave quite differently than symmetric groups on finite sets, and are discussed in (Scott 1987, Ch. 11), (Dixon & Mortimer 1996, Ch. 8), and (Cameron 1999). This article concentrates on the finite symmetric groups.

The symmetric group on a set of n elements has order n! It is abelian if and only if n ≤ 2. For n = 0 and n = 1 (the empty set and the singleton set) the symmetric group is trivial (note that this agrees with 0! = 1! = 1), and in these cases the alternating group equals the symmetric group, rather than being an index two subgroup. The group S_n is solvable if and only if n ≤ 4. This is an essential part of the proof of the Abel–Ruffini theorem that shows that for every n > 4 there are polynomials of degree n which are not solvable by radicals, i.e., 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.