In

_{0} < ''G''_{1} < ⋅⋅⋅ < ''G_{k}'' = ''G'' such that ''G''_{''j''−1} is normal in ''G_{j}'', and ''G_{j} ''/''G''_{''j''−1} is an abelian group, for ''j'' = 1, 2, …, ''k''.
Or equivalently, if its ^{(''n'')} = 1 is called the derived length of the solvable group ''G''.
For finite groups, an equivalent definition is that a solvable group is a group with a

_{3}. In fact, as the smallest simple non-abelian group is ''A''_{5}, (the

_{5} is not solvable — it has a composition series (and the _{5} and ''C''_{2}; and ''A''_{5} is not abelian. 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

^{a}q^{b}'' where ''p'' and ''q'' are

_{4} is an example of a finite solvable group that is not supersolvable.
If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups:
:

^{(''α'')} = ''G''^{(''α''+1)} is called the (transfinite) derived length of the group ''G'', and it has been shown that every ordinal is the derived length of some group .

Solvable groups as iterated extensions

{{DEFAULTSORT:Solvable Group Solvable groups, Properties of groups

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, more specifically in the field of group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, a solvable group or soluble group is a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

that can be constructed from abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s using extensions
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of ...

. Equivalently, a solvable group is a group whose derived series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

terminates in the trivial subgroupIn mathematics
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.
Motivation

Historically, the word "solvable" arose fromGalois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...

and the proof
Proof may refer to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Formal sciences
* Formal proof, a construct in proof theory
* Mathematical proof, a co ...

of the general unsolvability of quintic
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

equation. Specifically, a polynomial equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

is solvable in radicals if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

the corresponding Galois group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

is solvable (note this theorem holds only in characteristic
Characteristic (from the Greek word for a property, attribute or trait
Trait may refer to:
* Phenotypic trait in biology, which involve genes and characteristics of organisms
* Trait (computer programming), a model for structuring object-oriented ...

0). This means associated to a polynomial $f\; \backslash in\; F;\; href="/html/ALL/s/.html"\; ;"title="">$$F\; =\; F\_0\; \backslash subseteq\; F\_1\; \backslash subseteq\; F\_2\; \backslash subseteq\; \backslash cdots\; \backslash subseteq\; F\_m=K$such that # $F\_i\; =\; F\_;\; href="/html/ALL/s/alpha\_i.html"\; ;"title="alpha\_i">alpha\_i$

Example

For example, the smallest Galois field extension of $\backslash mathbb$ containing the element$a\; =\; \backslash sqrt;\; href="/html/ALL/s/.html"\; ;"title="">$gives a solvable group. It has associated field extensions

$\backslash mathbb\; \backslash subseteq\; \backslash mathbb(\backslash sqrt,\; \backslash sqrt)\; \backslash subseteq\; \backslash mathbb(\backslash sqrt,\; \backslash sqrt)\backslash left(e^\backslash sqrt;\; href="/html/ALL/s/.html"\; ;"title="">$giving a solvable group containing $\backslash mathbb/5$ (acting on the $e^$) and $\backslash mathbb/2\; \backslash times\; \backslash mathbb/2$ (acting on $\backslash sqrt\; +\; \backslash sqrt$).

Definition

A group ''G'' is called solvable if it has asubnormal seriesIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

whose factor group
A quotient group or factor group is a math
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...

s (quotient groups) are all abelian, that is, if there are subgroup
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...

s 1 = ''G''derived series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, the descending normal series
:$G\backslash triangleright\; G^\backslash triangleright\; G^\; \backslash triangleright\; \backslash cdots,$
where every subgroup is the commutator subgroup
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of the previous one, eventually reaches the trivial subgroup of ''G''. These two definitions are equivalent, since for every group ''H'' and every normal subgroup
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

''N'' of ''H'', the quotient ''H''/''N'' is abelian if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

''N'' includes the commutator subgroup of ''H''. The least ''n'' such that ''G''composition series In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...

all of whose factors are cyclic groups
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...

of prime
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

. This is equivalent because a finite group has finite composition length, and every simple
Simple or SIMPLE may refer to:
* Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...

abelian group is cyclic of prime order. The Jordan–Hölder theoremIn abstract algebra, a composition series provides a way to break up an algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...

guarantees that if one composition series has this property, then all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to ''n''th roots (radicals) over some field
Field may refer to:
Expanses of open ground
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* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
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* Meadow, a grassl ...

. The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s under addition is isomorphic
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

to Z itself, it has no composition series, but the normal series , with its only factor group isomorphic to Z, proves that it is in fact solvable.
Examples

Abelian groups

The basic example of solvable groups are abelian groups. They are trivially solvable since a subnormal series being given by just the group itself and the trivial group. But non-abelian groups may or may not be solvable.Nilpotent groups

More generally, allnilpotent group
In mathematics, specifically group theory, a nilpotent group ''G'' is a Group (mathematics), group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series termina ...

s are solvable. In particular, finite ''p''-groups are solvable, as all finite ''p''-groups are nilpotent.
Quaternion groups

In particular, thequaternion group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

is a solvable group given by the group extension$1\; \backslash to\; \backslash mathbb/2\; \backslash to\; Q\; \backslash to\; \backslash mathbb/2\; \backslash times\; \backslash mathbb/2\; \backslash to\; 1$where the kernel $\backslash mathbb/2$ is the subgroup generated by $-1$.

Group extensions

Group extension
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s form the prototypical examples of solvable groups. That is, if $G$ and $G\text{'}$ are solvable groups, then any extension$1\; \backslash to\; G\; \backslash to\; G\text{'}\text{'}\; \backslash to\; G\text{'}\; \backslash to\; 1$defines a solvable group $G\text{'}\text{'}$. In fact, all solvable groups can be formed from such group extensions.

Nonabelian group which is non-nilpotent

A small example of a solvable, non-nilpotent group is thesymmetric group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

''S''alternating group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of degree 5) it follows that ''every'' group with order less than 60 is solvable.
Finite groups of odd order

The celebratedFeit–Thompson theoremIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

states that every finite group of odd order is solvable. In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order.
Non-example

The group ''S''Jordan–Hölder theoremIn abstract algebra, a composition series provides a way to break up an algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...

states that every other composition series is equivalent to that one), giving factor groups isomorphic to ''A''polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s of degree ''n'' which are not solvable by radicals (Abel–Ruffini theorem
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

). This property is also used in complexity theory in the proof of Barrington's theorem.
Subgroups of GL_{2}

$B\; =\; \backslash left\backslash \; \backslash text\; U\; =\; \backslash left\backslash $ of $GL\_2(\backslash mathbb)$for some field $\backslash mathbb$. Then, the group quotient $B/U$ can be found by taking arbitrary elements in $B,U$, multiplying them together, and figuring out what structure this gives. So

$\backslash begin\; a\; \&\; b\; \backslash \backslash \; 0\; \&\; c\; \backslash end\; \backslash cdot\; \backslash begin\; 1\; \&\; d\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\; =\; \backslash begin\; a\; \&\; ad\; +\; b\; \backslash \backslash \; 0\; \&\; c\; \backslash end$Note the determinant condition on $GL\_2$ implies $ac\; \backslash neq\; 0$, hence $\backslash mathbb^\backslash times\; \backslash times\; \backslash mathbb^\backslash times\; \backslash subset\; B$ is a subgroup (which are the matrices where $b=0$). For fixed $a,b$, the linear equation $ad\; +\; b\; =\; 0$ implies $d\; =\; -b/a$, which is an arbitrary element in $\backslash mathbb$ since $b\; \backslash in\; \backslash mathbb$. Since we can take any matrix in $B$ and multiply it by the matrix

$\backslash begin\; 1\; \&\; d\; \backslash \backslash \; 0\; \&\; 1\; \backslash end$with $d\; =\; -b/a$, we can get a diagonal matrix in $B$. This shows the quotient group $B/U\; \backslash cong\; \backslash mathbb^\backslash times\; \backslash times\; \backslash mathbb^\backslash times$.

Remark

Notice that this description gives the decomposition of $B$ as $\backslash mathbb\; \backslash rtimes\; (\backslash mathbb^\backslash times\; \backslash times\; \backslash mathbb^\backslash times)$ where $(a,c)$ acts on $b$ by $(a,c)(b)\; =\; ab$. This implies $(a,c)(b\; +\; b\text{'})\; =\; (a,c)(b)\; +\; (a,c)(b\text{'})\; =\; ab\; +\; ab\text{'}$. Also, a matrix of the form$\backslash begin\; a\; \&\; b\; \backslash \backslash \; 0\; \&\; c\; \backslash end$corresponds to the element $(b)\; \backslash times\; (a,c)$ in the group.

Borel subgroups

For alinear algebraic group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

$G$ its Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski topology, Zariski closed and connected solvable group, solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' inve ...

is defined as a subgroup which is closed, connected, and solvable in $G$, and it is the maximal possible subgroup with these properties (note the second two are topological properties). For example, in $GL\_n$ and $SL\_n$ the group of upper-triangular, or lower-triangular matrices are two of the Borel subgroups. The example given above, the subgroup $B$ in $GL\_2$ is the Borel subgroup.
Borel subgroup in GL_{3}

$B\; =\; \backslash left\backslash ,\; \backslash text\; U\_1\; =\; \backslash left\backslash $Notice $B/U\_1\; \backslash cong\; \backslash mathbb^\backslash times\; \backslash times\; \backslash mathbb^\backslash times\; \backslash times\; \backslash mathbb^\backslash times$, hence the Borel group has the form

$U\backslash rtimes\; (\backslash mathbb^\backslash times\; \backslash times\; \backslash mathbb^\backslash times\; \backslash times\; \backslash mathbb^\backslash times)$

Borel subgroup in product of simple linear algebraic groups

In the product group $GL\_n\; \backslash times\; GL\_m$ the Borel subgroup can be represented by matrices of the form$\backslash begin\; T\; \&\; 0\; \backslash \backslash \; 0\; \&\; S\; \backslash end$where $T$ is an $n\backslash times\; n$ upper triangular matrix and $S$ is a $m\backslash times\; m$ upper triangular matrix.

Z-groups

Any finite group whose ''p''-Sylow subgroups are cyclic is asemidirect product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of two cyclic groups, in particular solvable. Such groups are called Z-groups.
OEIS values

Numbers of solvable groups with order ''n'' are (start with ''n'' = 0) :0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 12, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, ... Orders of non-solvable groups are :60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1020, 1080, 1092, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, ...Properties

Solvability is closed under a number of operations. * If ''G'' is solvable, and ''H'' is a subgroup of ''G'', then ''H'' is solvable. * If ''G'' is solvable, and there is ahomomorphism
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

from ''G'' onto
In , a surjective function (also known as surjection, or onto function) is a that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's is the of one element of its ...

''H'', then ''H'' is solvable; equivalently (by the first isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between Quotient (universal algebra), quotients, homomorphisms, and subobjects. Vers ...

), if ''G'' is solvable, and ''N'' is a normal subgroup of ''G'', then ''G''/''N'' is solvable.Rotman (1995),
* The previous properties can be expanded into the following "three for the price of two" property: ''G'' is solvable if and only if both ''N'' and ''G''/''N'' are solvable.
* In particular, if ''G'' and ''H'' are solvable, the direct productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

''G'' × ''H'' is solvable.
Solvability is closed under group extension
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

:
* If ''H'' and ''G''/''H'' are solvable, then so is ''G''; in particular, if ''N'' and ''H'' are solvable, their semidirect product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

is also solvable.
It is also closed under wreath product:
* If ''G'' and ''H'' are solvable, and ''X'' is a ''G''-set, then the wreath product
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...

of ''G'' and ''H'' with respect to ''X'' is also solvable.
For any positive integer ''N'', the solvable groups of derived length at most ''N'' form a subvariety
A subvariety (Latin: ''subvarietas'') in botanical nomenclature
Botanical nomenclature is the formal, scientific naming of plants. It is related to, but distinct from Alpha taxonomy, taxonomy. Plant taxonomy is concerned with grouping and classif ...

of the variety of groups, as they are closed under the taking of homomorphic images, subalgebraIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s, and (direct) products. The direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety.
Burnside's theorem

Burnside's theorem states that if ''G'' is afinite group
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...

of order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

''pprime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s, and ''a'' and ''b'' are non-negative
In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third ...

integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s, then ''G'' is solvable.
Related concepts

Supersolvable groups

As a strengthening of solvability, a group ''G'' is called supersolvable (or supersoluble) if it has an ''invariant'' normal series whose factors are all cyclic. Since a normal series has finite length by definition,uncountable
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

groups are not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated. The alternating group ''A''cyclic
Cycle or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in social scienc ...

< abelian < nilpotent
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

< supersolvable < polycyclic < solvable < finitely generated group
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...

.
Virtually solvable groups

A group ''G'' is called virtually solvable if it has a solvable subgroup of finite index. This is similar to virtually abelian. Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1.Hypoabelian

A solvable group is one whose derived series reaches the trivial subgroup at a ''finite'' stage. For an infinite group, the finite derived series may not stabilize, but the transfinite derived series always stabilizes. A group whose transfinite derived series reaches the trivial group is called a hypoabelian group, and every solvable group is a hypoabelian group. The first ordinal ''α'' such that ''G''See also

* Prosolvable group *Parabolic subgroupNotes

References

* *External links

*Solvable groups as iterated extensions

{{DEFAULTSORT:Solvable Group Solvable groups, Properties of groups