solvable group

TheInfoList

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 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 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 ...
.

# Motivation

Historically, the word "solvable" arose from
Galois 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 ...
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
$F = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m=K$
such that #

## Example

For example, the smallest Galois field extension of $\mathbb$ containing the element
gives a solvable group. It has associated field extensions
giving a solvable group containing $\mathbb/5$ (acting on the $e^$) and $\mathbb/2 \times \mathbb/2$ (acting on $\sqrt + \sqrt$).

# Definition

A group ''G'' is called solvable if it has a
subnormal 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''0 < ''G''1 < ⋅⋅⋅ < ''Gk'' = ''G'' such that ''G''''j''−1 is normal in ''Gj'', and ''Gj ''/''G''''j''−1 is an abelian group, for ''j'' = 1, 2, …, ''k''. Or equivalently, if its
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 :$G\triangleright G^\triangleright G^ \triangleright \cdots,$ 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''(''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
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 * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * 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 ...
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, all
nilpotent 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, the
quaternion 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
$1 \to \mathbb/2 \to Q \to \mathbb/2 \times \mathbb/2 \to 1$
where the kernel $\mathbb/2$ 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
$1 \to G \to G\text{'}\text{'} \to G\text{'} \to 1$
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 the
symmetric 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''3. In fact, as the smallest simple non-abelian group is ''A''5, (the
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 celebrated
Feit–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''5 is not solvable — it has a composition series (and 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 ...
states that every other composition series is equivalent to that one), giving factor groups isomorphic to ''A''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
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 GL2

Consider the subgroups
$B = \left\ \text U = \left\$ of $GL_2\left(\mathbb\right)$
for some field $\mathbb$. 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
$\begin a & b \\ 0 & c \end \cdot \begin 1 & d \\ 0 & 1 \end = \begin a & ad + b \\ 0 & c \end$
Note the determinant condition on $GL_2$ implies $ac \neq 0$, hence $\mathbb^\times \times \mathbb^\times \subset B$ 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 $\mathbb$ since $b \in \mathbb$. Since we can take any matrix in $B$ and multiply it by the matrix
$\begin 1 & d \\ 0 & 1 \end$
with $d = -b/a$, we can get a diagonal matrix in $B$. This shows the quotient group $B/U \cong \mathbb^\times \times \mathbb^\times$.

### Remark

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

## Borel subgroups

For a
linear 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 GL3

In $GL_3$ there are the subgroups
$B = \left\, \text U_1 = \left\$
Notice $B/U_1 \cong \mathbb^\times \times \mathbb^\times \times \mathbb^\times$, hence the Borel group has the form
$U\rtimes \left(\mathbb^\times \times \mathbb^\times \times \mathbb^\times\right)$

### Borel subgroup in product of simple linear algebraic groups

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

## Z-groups

Any finite group whose ''p''-Sylow subgroups are cyclic is a
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 ...
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 a
homomorphism 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 a
finite 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 ...
''paqb'' where ''p'' and ''q'' are
prime 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''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: :
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''(''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 .

* Prosolvable group *Parabolic subgroup

* *