# Solvable group

A **solvable group** (also written **soluble group**) is a group that can be built up from abelian groups by a finite chain of extensions: equivalently, a group with a finite subnormal series whose successive quotient groups are all abelian<sup>[1](https://encyclopediaofmath.org/wiki/Solvable_group)</sup><sup> • </sup><sup>[2](https://faculty.niu.edu/math_beachy/aaol/structure.shtml)</sup>. The concept arose in [Galois theory](https://www.edgepedia.org/galois-theory), where the solvability of a polynomial equation by radicals corresponds exactly to the solvability of its [Galois group](https://www.edgepedia.org/galois-group)<sup>[1](https://encyclopediaofmath.org/wiki/Solvable_group)</sup>. Solvable groups form one of the central classes in group theory, sitting strictly between abelian groups and general groups.

| Key fact | Detail |
|---|---|
| Definition | A group with a finite subnormal series whose factor groups are all abelian; equivalently, its derived series reaches the trivial subgroup<sup>[1](https://encyclopediaofmath.org/wiki/Solvable_group)</sup><sup> • </sup><sup>[3](https://math.libretexts.org/Workbench/Group_Theory_4e_%28Milne%29/06%3A_Subnormal_Series_Solvable_and_Nilpotent_Groups/6.02%3A_Solvable_groups)</sup> |
| Derived series | \( G^{(0)} = G \), \( G^{(i)} = [G^{(i-1)}, G^{(i-1)}] \); the group is solvable when \( G^{(k)} = 1 \) for some \( k \)<sup>[3](https://math.libretexts.org/Workbench/Group_Theory_4e_%28Milne%29/06%3A_Subnormal_Series_Solvable_and_Nilpotent_Groups/6.02%3A_Solvable_groups)</sup> |
| Derived length | The length of the shortest solvable series, equivalently the number of steps until the derived series terminates<sup>[1](https://encyclopediaofmath.org/wiki/Solvable_group)</sup> |
| Nilpotent implies solvable | \( D_n(G) \subseteq C_{2^n}(G) \), so every nilpotent group is solvable; the upper triangular group \( T(n, A) \) is solvable but not in general nilpotent<sup>[4](https://math-jh.com/en/math/group_theory/series_of_groups)</sup><sup> • </sup><sup>[5](https://math.mit.edu/~hrm/palestine/bourbaki-algebra-1i.pdf)</sup> |
| First nonsolvable examples | \( S_n \) is not solvable for \( n \geq 5 \), because \( A_n \) is simple for \( n \geq 5 \)<sup>[2](https://faculty.niu.edu/math_beachy/aaol/structure.shtml)</sup> |
| Feit–Thompson theorem | Every finite group of odd order is solvable<sup>[3](https://math.libretexts.org/Workbench/Group_Theory_4e_%28Milne%29/06%3A_Subnormal_Series_Solvable_and_Nilpotent_Groups/6.02%3A_Solvable_groups)</sup> |
| Galois connection | A polynomial is solvable by radicals if and only if its Galois group is solvable<sup>[6](https://academic.oup.com/book/45622/chapter-abstract/394863601)</sup> |

## Definition and the derived series

The most economical definition uses the **derived series**. The derived (or commutator) subgroup \( G' = [G, G] \) is the subgroup generated by all commutators \( [a, b] = a^{-1}b^{-1}ab \); it is the smallest normal subgroup of \( G \) with abelian quotient<sup>[2](https://faculty.niu.edu/math_beachy/aaol/structure.shtml)</sup><sup> • </sup><sup>[3](https://math.libretexts.org/Workbench/Group_Theory_4e_%28Milne%29/06%3A_Subnormal_Series_Solvable_and_Nilpotent_Groups/6.02%3A_Solvable_groups)</sup>. Iterating gives the derived series

\[ G = G^{(0)} \supseteq G^{(1)} \supseteq G^{(2)} \supseteq \cdots, \qquad G^{(i)} = [G^{(i-1)}, G^{(i-1)}]. \]

Each term \( G^{(i)} \) is a characteristic subgroup of \( G \), hence normal<sup>[7](https://math.ucla.edu/~sharifi/notes/algebra-ch07.html)</sup>. A group \( G \) is solvable if \( G^{(k)} = 1 \) for some \( k \)<sup>[3](https://math.libretexts.org/Workbench/Group_Theory_4e_%28Milne%29/06%3A_Subnormal_Series_Solvable_and_Nilpotent_Groups/6.02%3A_Solvable_groups)</sup>. The length of the derived series, or equivalently of the shortest solvable series, is called the **derived length** or degree of solvability<sup>[1](https://encyclopediaofmath.org/wiki/Solvable_group)</sup>.

The derived-series definition is equivalent to the series definition. If \( G = G_0 \triangleright G_1 \triangleright \cdots \triangleright G_s = 1 \) is any series with abelian factors, an induction on the series shows \( G^{(i)} \subseteq G_i \), so \( G^{(s)} = 1 \); conversely, when the derived series terminates it is itself a solvable series, and it is the shortest one<sup>[3](https://math.libretexts.org/Workbench/Group_Theory_4e_%28Milne%29/06%3A_Subnormal_Series_Solvable_and_Nilpotent_Groups/6.02%3A_Solvable_groups)</sup>. A third characterization: \( G \) is solvable of derived length at most \( n \) if and only if it has a normal subgroup \( A \) with \( A \) abelian and \( G/A \) solvable of derived length at most \( n-1 \), so a solvable group is obtained by extending the trivial group by \( n \) abelian groups<sup>[4](https://math-jh.com/en/math/group_theory/series_of_groups)</sup>.

## Basic properties

Solvability is well behaved under the standard group constructions<sup>[7](https://math.ucla.edu/~sharifi/notes/algebra-ch07.html)</sup>:

- Every subgroup of a solvable group is solvable.
- Every quotient (homomorphic image) of a solvable group is solvable.
- If \( N \) is a normal subgroup of \( G \) and both \( N \) and \( G/N \) are solvable, then \( G \) is solvable.

These closure properties mean the solvable groups form a class closed under extensions, which is what makes the derived length a useful measure of complexity.

**Relation to nilpotency.** The derived series descends at least as fast as the lower central series: \( D_n(G) \subseteq C_{2^n}(G) \), so every nilpotent group is solvable<sup>[4](https://math-jh.com/en/math/group_theory/series_of_groups)</sup>. The converse fails: the upper triangular group \( T(n, A) \) over a commutative ring \( A \) is solvable but not in general nilpotent<sup>[5](https://math.mit.edu/~hrm/palestine/bourbaki-algebra-1i.pdf)</sup>.

## Finite solvable groups

For finite groups the abelian-factor definition can be sharpened. A finite group is solvable if and only if it has a chain of subgroups in which each factor is cyclic of prime order<sup>[2](https://faculty.niu.edu/math_beachy/aaol/structure.shtml)</sup>. Equivalently, a group with a composition series is solvable exactly when it is finite and all of its Jordan–Hölder factors are cyclic of prime order<sup>[7](https://math.ucla.edu/~sharifi/notes/algebra-ch07.html)</sup>.

Finite solvable groups satisfy a converse to Lagrange's theorem: for any factorization \( n = n_1 n_2 \) of the group order into relatively prime factors, there exists a subgroup of order \( n_1 \), and any two such subgroups are conjugate (Hall's theorem)<sup>[1](https://encyclopediaofmath.org/wiki/Solvable_group)</sup>. If the order of a finite group is divisible by only two primes, the group is solvable<sup>[1](https://encyclopediaofmath.org/wiki/Solvable_group)</sup>. Every finite group of odd order is solvable, the [Feit–Thompson theorem](https://www.edgepedia.org/feit-thompson-theorem)<sup>[3](https://math.libretexts.org/Workbench/Group_Theory_4e_%28Milne%29/06%3A_Subnormal_Series_Solvable_and_Nilpotent_Groups/6.02%3A_Solvable_groups)</sup>.

The **solvable radical** \( \mathrm{Rad}(G) \) of a finite group is its unique maximal solvable normal subgroup; the quotient \( G/\mathrm{Rad}(G) \) is semisimple, meaning it has no nontrivial solvable normal subgroup<sup>[8](https://people.cs.uchicago.edu/~laci/papers/soda11.pdf)</sup>. There is also a commutator-based criterion: a group of finite order is soluble precisely when any 8 elements drawn from the same conjugacy class always generate a subgroup that is soluble.<sup>[9](https://arxiv.org/abs/math/0610983)</sup>

## Examples and nonexamples

- Every abelian group is solvable, with derived series \( G^{(i)} = 1 \) for \( i \geq 1 \)<sup>[7](https://math.ucla.edu/~sharifi/notes/algebra-ch07.html)</sup>.
- Every finite \( p \)-group is solvable<sup>[2](https://faculty.niu.edu/math_beachy/aaol/structure.shtml)</sup>, and all groups of order \( pq \) for distinct primes \( p \) and \( q \) are solvable<sup>[7](https://math.ucla.edu/~sharifi/notes/algebra-ch07.html)</sup>.
- \( S_4 \) is solvable, via the series \( S_4 \supset A_4 \supset V_4 \supset \{e\} \) with abelian factors<sup>[10](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_%28Judson%29/13%3A_The_Structure_of_Groups/13.02%3A_Solvable_Groups)</sup>.
- \( S_n \) is not solvable for \( n \geq 5 \): \( A_n \) is simple for \( n \geq 5 \), and the derived series of \( S_n \) stabilizes at \( A_n \), \( S_n \supset A_n \supset A_n \supset \cdots \)<sup>[2](https://faculty.niu.edu/math_beachy/aaol/structure.shtml)</sup><sup> • </sup><sup>[3](https://math.libretexts.org/Workbench/Group_Theory_4e_%28Milne%29/06%3A_Subnormal_Series_Solvable_and_Nilpotent_Groups/6.02%3A_Solvable_groups)</sup>. Nonabelian simple groups are never solvable, since their derived series is constant at the whole group<sup>[7](https://math.ucla.edu/~sharifi/notes/algebra-ch07.html)</sup>.

## Solvable groups in Galois theory

The term "solvable group" arose in Galois theory in connection with the solvability of algebraic equations by radicals<sup>[1](https://encyclopediaofmath.org/wiki/Solvable_group)</sup>. After Cardano, formulas in the coefficients using the arithmetic operations and root extraction were known for equations of degree 1 through 4. Building on Lagrange's investigation of permutations of the roots, Ruffini in 1803 and Abel in 1826 proved that for degree 5 such a formula does not exist in general<sup>[6](https://academic.oup.com/book/45622/chapter-abstract/394863601)</sup>. Galois introduced the concept of a permutation group attached to an equation and developed a method for deciding which equations can be solved by radicals<sup>[6](https://academic.oup.com/book/45622/chapter-abstract/394863601)</sup>.

## Matrix groups and geometry

Solvable groups play a structural role for matrix groups. The Lie–Kolchin theorem states that a solvable group of matrices defined over an algebraically closed field contains a finite-index subgroup that is conjugate to some subgroup of the triangular group.<sup>[1](https://encyclopediaofmath.org/wiki/Solvable_group)</sup> Every connected solvable [Lie group](https://www.edgepedia.org/lie-group), and every solvable matrix group connected in the [Zariski topology](https://www.edgepedia.org/zariski-topology), has a nilpotent commutator subgroup<sup>[1](https://encyclopediaofmath.org/wiki/Solvable_group)</sup>. In geometric group theory, the Milnor–Wolf theorem states that solvable groups of polynomial growth are virtually nilpotent; [Terence Tao](https://www.edgepedia.org/terence-tao), a mathematician at the University of California, Los Angeles, strengthened this in 2009 by showing that only one large ball in the group needs to have polynomial size<sup>[12](https://arxiv.org/abs/0906.3535)</sup>.

## Computation

Solvable groups are the most tractable finite groups for computation, and algorithms specialized to them are a core part of computational group theory<sup>[13](https://www.mathnet.ru/links/e163b9e3521e081945706f70dfc2eac0/pdm736.pdf)</sup>. The GAP computer algebra system tests solvability with a dedicated property, `IsSolvableGroup`, and uses solvability to select faster algorithms, for example for computing the order of a permutation group<sup>[14](https://docs.gap-system.org/doc/tut/manual.pdf)</sup>. GAP's solvable groups library contains the 1045 solvable groups of size between 2 and 100<sup>[15](https://www.math.rwth-aachen.de/homes/GAP/WWW2/Gap3/Manual3/C037S007.htm)</sup>, and the Small Groups library stores precomputed solvability data for groups of order at most 2000<sup>[16](https://docs.gap-system.org/pkg/smallgrp/doc/manual.pdf)</sup>.

## References

1. [Solvable group, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Solvable_group)
2. [Structure of Groups, Abstract Algebra Online (Beachy), Northern Illinois University](https://faculty.niu.edu/math_beachy/aaol/structure.shtml)
3. [6.2: Solvable groups, Group Theory (Milne), Mathematics LibreTexts](https://math.libretexts.org/Workbench/Group_Theory_4e_%28Milne%29/06%3A_Subnormal_Series_Solvable_and_Nilpotent_Groups/6.02%3A_Solvable_groups)
4. [Series of a Group, Blackbox](https://math-jh.com/en/math/group_theory/series_of_groups)
5. [Algebraic Structures, Derived Series, Solvable Groups (§6.4), MIT](https://math.mit.edu/~hrm/palestine/bourbaki-algebra-1i.pdf)
6. [The Insolvability of the Quintic by Radicals, in Jesper Lützen, A History of Mathematical Impossibility, Oxford University Press (2022)](https://academic.oup.com/book/45622/chapter-abstract/394863601)
7. [Abstract Algebra, Chapter 7: Topics in group theory, Romyar Sharifi, UCLA](https://math.ucla.edu/~sharifi/notes/algebra-ch07.html)
8. [László Babai et al., Algorithms for Group Isomorphism via Group Extensions and Cohomology, SIAM Journal on Computing](https://people.cs.uchicago.edu/~laci/papers/soda11.pdf)
9. [A commutator description of the solvable radical of a finite group, arXiv](https://arxiv.org/abs/math/0610983)
10. [13.2: Solvable Groups, Abstract Algebra: Theory and Applications (Judson), Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_%28Judson%29/13%3A_The_Structure_of_Groups/13.02%3A_Solvable_Groups)
11. [Felix Klein's teaching of Galois theory, Historia Mathematica (2023)](https://doi.org/10.1016/j.hm.2023.05.003)
12. [Terence Tao (2009), Freiman's theorem for solvable groups, arXiv](https://arxiv.org/abs/0906.3535)
13. [Algorithmic theory of solvable groups, Прикладная дискретная математика](https://www.mathnet.ru/links/e163b9e3521e081945706f70dfc2eac0/pdm736.pdf)
14. [GAP: A Tutorial, GAP documentation](https://docs.gap-system.org/doc/tut/manual.pdf)
15. [GAP Manual 37.7: The Solvable Groups Library](https://www.math.rwth-aachen.de/homes/GAP/WWW2/Gap3/Manual3/C037S007.htm)
16. [SmallGrp: The GAP Small Groups Library](https://docs.gap-system.org/pkg/smallgrp/doc/manual.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Group theory › Finite groups and classification › Solvable and nilpotent finite groups*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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