A solvable group is a group that can be assembled from abelian groups by finitely many extensions. Equivalently, its noncommutativity disappears after repeatedly taking commutator subgroups. Solvability is a central organizing principle in finite-group theory and Galois theory: finite \(p\)-groups, nilpotent groups, and finite groups whose order involves only two primes are solvable, whereas the alternating group \(A_5\) is the smallest nonsolvable finite group. [@isaacs2008]

Definition and equivalent formulations

Set \(G^{(0)}=G\) and define recursively

\[ G^{(i+1)}=[G^{(i)},G^{(i)}]. \]

This characteristic series is the derived series. If \(d\) is the least integer such that \(G^{(d)}=1\), then \(d\) is the derived length of \(G\), denoted \(\operatorname{dl}(G)\). Every nontrivial abelian group has derived length one.

For a finite group there is a third characterization: \(G\) is solvable exactly when every composition factor is cyclic of prime order. The Jordan–Hölder theorem makes this independent of the selected composition series. The statement cannot be transferred verbatim to arbitrary infinite groups, since a solvable infinite group need not possess a finite composition series.

Closure properties and extensions

Solvability is stable under the standard finite constructions:

  • every subgroup of a solvable group is solvable;
  • every homomorphic image, hence every quotient, of a solvable group is solvable;
  • if \(N\triangleleft G\) and both \(N\) and \(G/N\) are solvable, then \(G\) is solvable;
  • a finite direct product of solvable groups is solvable;
  • semidirect products and finite wreath products of solvable groups are solvable.

This argument is also the practical way to estimate derived length: find a manageable normal subgroup and divide the calculation into a kernel problem and a quotient problem. Arbitrary infinite direct products require care. If the derived lengths of the factors are unbounded, no finite derived step kills all coordinates simultaneously, so the full product is not solvable.

Minimal normal subgroups of finite solvable groups

This result drives induction throughout finite solvable-group theory. After choosing \(M\), the quotient \(G/M\) is smaller and remains solvable, while \(M\) is a linear object to which module theory, coprime action, and Maschke averaging apply. It explains why many finite solvable-group arguments eventually become linear algebra over a finite field.

The product of two solvable normal subgroups of a finite group is solvable. It follows that every finite group has a unique largest solvable normal subgroup, its solvable radical \(R(G)\). The quotient \(G/R(G)\) has no nontrivial solvable normal subgroup. One has \(R(G)=G\) for a solvable group and \(R(G)=1\) for a nonabelian simple group.

Hall subgroups in finite solvable groups

Let \(\pi\) be a set of primes. A Hall \(\pi\)-subgroup of \(G\) has order divisible only by primes in \(\pi\) and index divisible by no prime in \(\pi\). Such a subgroup need not exist in an arbitrary finite group, but finite solvable groups have a complete Hall theory.

The proof proceeds by induction on \(|G|\). Choose an elementary abelian minimal normal subgroup \(M\), construct a Hall subgroup in \(G/M\), and study its inverse image. If the prime of \(M\) belongs to \(\pi\), include \(M\) in the lift; otherwise use Schur–Zassenhaus in the coprime extension to select a complement. Conjugacy and containment lift through the same kernel–quotient analysis. The full proof and the complement mechanism belong to Hall subgroup and Schur–Zassenhaus theorem. [@isaacs2008]

Examples and counterexamples

Every finite \(p\)-group is nilpotent and therefore solvable. Burnside's \(p^a q^b\) theorem says more: a finite group whose order is divisible by at most two primes is solvable. Chapter 7 of the source develops a group-theoretic proof. The Feit–Thompson odd-order theorem states that every finite group of odd order is solvable, but its proof lies far beyond the elementary closure theory in this article. [@feit-thompson1963]

Why the word “solvable” appears in Galois theory

The terminology comes from polynomial equations. Under the standard hypotheses in characteristic zero, a polynomial is solvable by radicals if and only if its Galois group is solvable. Adjoining one radical produces an abelian cyclic layer after the required roots of unity are present; a finite tower of radical extensions corresponds on the group side to a series with abelian factors. A general quintic can have Galois group \(S_5\). Since \(S_5\) has the nonabelian simple composition factor \(A_5\), no universal radical formula for quintics exists. [@rotman1995]

This use of “solvable” does not assert that one can efficiently enumerate the group or solve an arbitrary equation written in the group. Solvability is a structural statement that noncommutativity can be removed by finitely many abelian quotients; computational complexity and data representation are separate questions.

For finite groups one has strict containments

\[ \text{cyclic} \subsetneq\text{abelian} \subsetneq\text{nilpotent} \subsetneq\text{supersolvable} \subsetneq\text{solvable}. \]

Nonabelian finite \(p\)-groups show that abelian is strictly smaller than nilpotent; \(S_3\) separates nilpotent from supersolvable; \(A_4\) separates supersolvable from solvable. Prime-local structure in solvable groups is organized by Hall subgroups, while coprime extensions are controlled by the Schur–Zassenhaus theorem. The largest nilpotent normal subgroup is the Fitting subgroup. On the nonsolvable side, components, the layer, and the generalized Fitting subgroup replace abelian layers with quasisimple ones.

The definitions, closure arguments, minimal-normal-subgroup theorem, and exercise consequences above follow primarily from Chapter 3 of Isaacs. Rotman and the cited Wikipedia revision provide broader historical and Galois-theoretic context. [@wikipedia-solvable]