The symmetric group on a set of formula_2 elements has order formula_4 (the factorial of formula_2). It is abelian if and only if formula_2 is less than or equal to 2. For formula_31 and formula_32 (the empty set and the singleton set), the symmetric groups are trivial (they have order formula_33). The group S"n" is solvable if and only if formula_34. This is an essential part of the proof of the Abel–Ruffini theorem that shows that for every formula_35 there are polynomials of degree formula_2 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.