Editing Abel–Ruffini theorem (section)

===Solvable symmetric groups===
For {{math|''n'' > 4}}, the [[symmetric group]] <math>\mathcal S_n</math> of degree {{mvar|n}} has only the [[alternating group]] <math>\mathcal A_n</math> as a nontrivial [[normal subgroup]] (see {{slink|Symmetric group|Normal subgroups}}). For {{math|''n'' > 4}}, the alternating group <math>\mathcal A_n</math> is [[simple group|simple]] (that is, it does not have any nontrivial normal subgroup) and not [[abelian group|abelian]]. This implies that both <math>\mathcal A_n</math> and <math>\mathcal S_n</math> are not [[solvable group|solvable]] for {{math|''n'' > 4}}. Thus, the Abel–Ruffini theorem results from the existence of polynomials with a symmetric Galois group; this will be shown in the next section.

On the other hand, for {{math|''n'' ≤ 4}}, the symmetric group and all its subgroups are solvable. This explains the existence of the [[quadratic formula|quadratic]], [[cubic formula|cubic]], and [[quartic formula|quartic]] formulas, since a major result of [[Galois theory]] is that a [[polynomial equation]] has a [[solution in radicals]] if and only if its [[Galois group]] is solvable (the term "solvable group" takes its origin from this theorem).