 Generalizing this argument, coupled with the fact that "A""n" is a normal, maximal, non-abelian simple subgroup of "S""n" for "n" > 4, we see that "S""n" is not solvable for "n" > 4.This is a key step in the proof that for every "n" > 4 there are polynomials of degree "n" which are not solvable by radicals (Abel–Ruffini theorem).This property is also used in complexity theory in the proof of Barrington's theorem.