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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 groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, a solvable group or soluble group is a group 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 comm ...
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 * Ex ...
. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.


Motivation

Historically, the word "solvable" arose from Galois theory and the
proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a c ...
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 (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
is solvable in
radicals Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
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 bic ...
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 po ...
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 Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , which ...
/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 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 whose
factor group Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, s ...
s (quotient groups) are all
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
, that is, if there are subgroups 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, the descending normal series :G\triangleright G^\triangleright G^ \triangleright \cdots, where every subgroup is the commutator subgroup 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, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
''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 bic ...
''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 all of whose factors are cyclic groups 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 Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
. 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 natura ...
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. 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 languag ...
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 i ...
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 groups are solvable. In particular, finite ''p''-groups are solvable, as all finite ''p''-groups are nilpotent.


Quaternion groups

In particular, the quaternion group 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 extensions 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 ''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 pr ...
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 t ...
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 natura ...
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 exampl ...
s of degree ''n'' which are not solvable by radicals ( Abel–Ruffini theorem). 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 ...
G its Borel 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 In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
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.


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 morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
from ''G'' onto ''H'', then ''H'' is solvable; equivalently (by the first isomorphism theorem), 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 In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
''G'' × ''H'' is solvable. Solvability is closed under group extension: * If ''H'' and ''G''/''H'' are solvable, then so is ''G''; in particular, if ''N'' and ''H'' are solvable, their
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
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 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 is a taxonomic rank. They are rarely used to classify organisms. Plant taxonomy Subvariety is ranked: *below that of variety (''varietas'') *above that of form (''forma''). Subva ...
of the variety of groups, as they are closed under the taking of homomorphic images, subalgebras, 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 ma ...
of
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
''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
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 languag ...
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 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 so ...
<
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
<
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 cl ...
< supersolvable < polycyclic < solvable < finitely generated group.


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


Notes


References

* *


External links

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