The symmetric group on a set of n elements has order n ! (the factorial of n ). It is abelian if and only if n is less than or equal to 2. For n = 0 and n = 1 (the empty set and the singleton set), the symmetric groups are trivial (they have order 0 ! = 1 ! = 1 ). 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, that is, 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.