Solvable groups measure how far a group is from being built by abelian pieces. The classical link to radicals explains why group theory controls formulas for polynomial roots.
Commutators and derived series
Definition. The commutator of \(g,h\in G\) is \([g,h]=ghg^{-1}h^{-1}\). The commutator subgroup \([G,G]\) is generated by all commutators. The derived series is \[ G^{(0)}=G, \qquad G^{(n+1)}=[G^{(n)},G^{(n)}]. \]
Proposition. The quotient \(G/[G,G]\) is abelian, and every homomorphism from \(G\) to an abelian group factors uniquely through \(G/[G,G]\).
Solvable groups
Definition. A group \(G\) is solvable if \(G^{(n)}=1\) for some \(n\). Equivalently, \(G\) has a subnormal series whose successive quotients are abelian.
Example. Every abelian group is solvable. The symmetric group \(S_3\) is solvable because \(1\trianglelefteq A_3\trianglelefteq S_3\), with quotients \(A_3\cong C_3\) and \(S_3/A_3\cong C_2\).
Simple groups and obstruction
Proposition. A nonabelian simple group is not solvable.
Proof. If \(G\) is nonabelian simple, then \([G,G]\) is a nontrivial normal subgroup. Hence \([G,G]=G\), so the derived series never decreases.
Theorem. The alternating group \(A_5\) is simple and nonabelian. Therefore \(A_5\) and \(S_5\) are not solvable.
Extensions of solvable groups
Theorem. If \(N\trianglelefteq G\), and both \(N\) and \(G/N\) are solvable, then \(G\) is solvable.
Remark. This theorem is the main reason solvability is stable under building groups by layers. It also explains why composition factors matter.
Radical extensions
Definition. A radical extension is built by repeatedly adjoining elements \(\alpha\) such that \(\alpha^n\) lies in the previous field. A polynomial is solvable by radicals if its roots lie in such an extension of the base field.
Theorem. Over a field of characteristic zero, a polynomial is solvable by radicals only if its Galois group is solvable. Under standard hypotheses, the converse also holds.
The quintic boundary
Example. The general polynomial of degree \(5\) has Galois group \(S_5\). Since \(S_5\) is not solvable, there is no formula by radicals for the general quintic.
Remark. This statement does not mean every quintic is unsolvable by radicals. Special quintics may have smaller solvable Galois groups.
Derived series and closure properties
Proposition. A group \(G\) is solvable if and only if it has a subnormal series \[ \{e\}=G_r\triangleleft G_{r-1}\triangleleft\cdots\triangleleft G_0=G \] with abelian factors. Subgroups and quotients of solvable groups are solvable, and an extension of a solvable group by a solvable group is solvable.
Proof. The derived series \(G^{(0)}=G\), \(G^{(i+1)}=[G^{(i)},G^{(i)}]\) has abelian successive quotients. Conversely, commutators descend one step in any series with abelian factors, so \(G^{(i)}\subseteq G_i\). Derived series commute with homomorphic images and restrict into subgroup derived series, proving the closure statements. For extensions, splice the derived series of the normal subgroup with the pullback of a derived series of the quotient.
Why radicals produce solvable Galois groups
Theorem. Over a characteristic-zero field containing the required roots of unity, a polynomial solvable by radicals has solvable Galois group. Conversely, a polynomial with solvable Galois group is solvable by radicals after adjoining roots of unity.
Proof outline. Adjoining an \(n\)-th root when the \(n\)-th roots of unity are present gives a cyclic Galois extension after normal closure. A tower of radical extensions therefore yields a Galois group with a series of abelian quotients. Conversely, refine a solvable Galois group to a series with cyclic prime-order quotients. Galois correspondence gives a field tower whose steps are cyclic; Kummer theory realizes each step by adjoining a radical.
The symmetric group \(S_n\) is solvable for \(n\le4\), while \(S_n\) is not solvable for \(n\ge5\) because it contains the nonabelian simple group \(A_n\). Generic degree-\(n\) equations have Galois group \(S_n\), explaining the impossibility of a universal radical formula from degree five onward.