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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically in the field of
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a solvable group or soluble group is a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
that can be constructed from
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s using
extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...
. Equivalently, a solvable group is a group whose
derived series In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group (mathematics), group is the subgroup (mathematics), subgroup generating set of a group, generated by all the commutators of the group. Th ...
terminates in the
trivial subgroup In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
.


Motivation

Historically, the word "solvable" arose from
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
and the proof of the general unsolvability of
quintic In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
equation. Specifically, a
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation' ...
is solvable in radicals
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
the corresponding
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
is solvable (note this theorem holds only in characteristic 0). This means associated to a polynomial f \in F /math> there is a tower of field extensions
F = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m=K
such that # F_i = F_ alpha_i/math> where \alpha_i^ \in F_, so \alpha_i is a solution to the equation x^ - a where a \in F_ # F_m contains a
splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...
for f(x)


Example

For example, the smallest Galois field extension of \mathbb containing the element
a = \sqrt /math>
gives a solvable group. It has associated field extensions
\mathbb \subseteq \mathbb(\sqrt, \sqrt) \subseteq \mathbb(\sqrt, \sqrt)\left(e^\sqrt right)
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 series In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups: :1 = A_0 \leq A_1 \leq \cdots \leq A_n = G where 1 is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simple ...
whose factor groups (quotient groups) are all abelian, that is, if there are
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s 1 = ''G''0 < ''G''1 < ⋅⋅⋅ < ''Gk'' = ''G'' such that ''G''''j''−1 is
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
in ''Gj'', and ''Gj ''/''G''''j''−1 is an abelian group, for ''j'' = 1, 2, …, ''k''. Or equivalently, if its
derived series In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group (mathematics), group is the subgroup (mathematics), subgroup generating set of a group, generated by all the commutators of the group. Th ...
, the descending normal series :G\triangleright G^\triangleright G^ \triangleright \cdots, where every subgroup is the
commutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
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 ''N'' of ''H'', the quotient ''H''/''N'' is abelian
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
''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, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natura ...
all of whose factors are
cyclic groups In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...
of
prime A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...
order. 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 theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natur ...
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 grass ...
. The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s under addition is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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 is formed 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 that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intui ...
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 group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. It is given by the group presentation :\mathrm_8 ...
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, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\ove ...
s form the prototypical examples of solvable groups. That is, if G and G' are solvable groups, then any extension
1 \to G \to G'' \to G' \to 1
defines a solvable group G''. 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, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
''S''3. In fact, as the smallest simple non-abelian group is ''A''5, (the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic prop ...
of degree 5) it follows that ''every'' group with order less than 60 is solvable.


Finite groups of odd order

The
Feit–Thompson theorem In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by . History conjectured that every nonabelian finite simple group has even order. suggested using ...
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 theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natur ...
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, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s of degree ''n'' which are not solvable by radicals (
Abel–Ruffini theorem In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means th ...
). 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(\mathbb)
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 (\mathbb^\times \times \mathbb^\times) where (a,c) acts on b by (a,c)(b) = ab . This implies (a,c)(b + b') = (a,c)(b) + (a,c)(b') = ab + ab' . Also, a matrix of the form
\begin a & b \\ 0 & c \end
corresponds to the element (b) \times (a,c) in the group.


Borel subgroups

For a
linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n wh ...
G its
Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
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 (\mathbb^\times \times \mathbb^\times \times \mathbb^\times)


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 of two cyclic groups, in particular solvable. Such groups are called
Z-group In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups: * in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic. * in the s ...
s.


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, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from ''G''
onto In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
''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 quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
), 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 product ''G'' × ''H'' is solvable. Solvability is closed under
group extension In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\ove ...
: * If ''H'' and ''G''/''H'' are solvable, then so is ''G''; in particular, if ''N'' and ''H'' are solvable, their semidirect product 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, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used i ...
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 of the variety of groups, as they are closed under the taking of homomorphic images,
subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operat ...
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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
of order ''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 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 because the only ways ...
s, and ''a'' and ''b'' are
non-negative In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
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, cycles, 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 s ...
< abelian <
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the class ...
<
supersolvable In mathematics, a group (mathematics), group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvable group, solvability. Definition ...
< polycyclic < solvable <
finitely generated group In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses o ...
.


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 .


See also

* Prosolvable group *
Parabolic subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...


Notes


References

* *


External links

*
Solvable groups as iterated extensions
{{DEFAULTSORT:Solvable Group Properties of groups