Solvable group

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In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

Contents

• 1 Motivation
  • 1.1 Example
• 2 Definition
• 3 Examples
  • 3.1 Abelian groups
  • 3.2 Nilpotent groups
    • 3.2.1 Quaternion groups
  • 3.3 Group extensions
  • 3.4 Nonabelian group which is non-nilpotent
  • 3.5 Finite groups of odd order
  • 3.6 Non-example
  • 3.7 Subgroups of GL₂
    • 3.7.1 Remark
  • 3.8 Borel subgroups
    • 3.8.1 Borel subgroup in GL₃
    • 3.8.2 Borel subgroup in product of simple linear algebraic groups
  • 3.9 Z-groups
• 4 OEIS values
• 5 Properties
• 6 Burnside's theorem
• 7 Related concepts
  • 7.1 Supersolvable groups
  • 7.2 Virtually solvable groups
  • 7.3 Hypoabelian
• 8 See also
• 9 Notes
• 10 References
• 11 External links

Motivation[edit]

Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable^[1] (note this theorem holds only in characteristic 0). This means associated to a polynomial ${\displaystyle f\in F[x]}$ there is a tower of field extensions

${\displaystyle F=F_{0}\subset F_{1}\subset F_{2}\subset \cdots \subset F_{m}=K}$

such that

1. ${\displaystyle F_{i}=F_{i-1}[\alpha _{i}]}$ where ${\displaystyle \alpha _{i}^{m_{i}}\in F_{i-1}}$, so ${\displaystyle \alpha _{i}}$ is a solution to the equation ${\displaystyle x^{m_{i}}-a}$ where ${\displaystyle a\in F_{i-1}}$
2. ${\displaystyle F_{m}}$ contains a splitting field for ${\displaystyle f(x)}$

Example[edit]

For example, the smallest Galois field extension of ${\displaystyle \mathbb {Q} }$ containing the element

${\displaystyle a={\sqrt[{5}]{{\sqrt {2}}+{\sqrt {3}}}}}$

gives a solvable group. It has associated field extensions

${\displaystyle \mathbb {Q} \subset \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\subset \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\left(e^{2\pi i/5}{\sqrt[{5}]{{\sqrt {2}}+{\sqrt {3}}}}\right)}$

giving a solvable group containing ${\displaystyle \mathbb {Z} /5}$ (acting on the ${\displaystyle e^{2\pi i/5}}$) and ${\displaystyle \mathbb {Z} /2\times \mathbb {Z} /2}$ (acting on ${\displaystyle {\sqrt {2}}+{\sqrt {3}}}$).

Definition[edit]

A group G is called solvable if it has a subnormal series whose factor groups (quotient groups) are all abelian, that is, if there are subgroups 1 = G₀ < G₁ < ⋅⋅⋅ < 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 derived series, the descending normal series

${\displaystyle G\triangleright G^{(1)}\triangleright G^{(2)}\triangleright \cdots ,}$

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 N of H, the quotient H/N is abelian if and only if 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 order. This is equivalent because a finite group has finite composition length, and every simple abelian group is cyclic of prime order. The Jordan–Hölder theorem 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 nth 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 integers under addition is isomorphic to Z itself, it has no composition series, but the normal series {0, Z}, with its only factor group isomorphic to Z, proves that it is in fact solvable.

Examples[edit]

Abelian groups[edit]

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[edit]

More generally, all nilpotent groups are solvable. In particular, finite p-groups are solvable, as all finite p-groups are nilpotent.

Quaternion groups[edit]

In particular, the quaternion group is a solvable group given by the group extension

${\displaystyle 1\to \mathbb {Z} /4\to Q\to \mathbb {Z} /2\to 1}$

where ${\displaystyle \mathbb {Z} /4}$ is the subgroup generated by ${\displaystyle i}$.

Group extensions[edit]

Group extensions form the prototypical examples of solvable groups. That is, if ${\displaystyle G}$ and ${\displaystyle G'}$ are solvable groups, then any extension

${\displaystyle 1\to G\to G''\to G'\to 1}$

defines a solvable group ${\displaystyle G''}$. In fact, all solvable groups can be formed from such group extensions.

Nonabelian group which is non-nilpotent[edit]

A small example of a solvable, non-nilpotent group is the symmetric group S₃. In fact, as the smallest simple non-abelian group is A₅, (the alternating group of degree 5) it follows that every group with order less than 60 is solvable.

Finite groups of odd order[edit]

The celebrated Feit–Thompson theorem 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[edit]

The group S₅ is not solvable — it has a composition series {E, A₅, S₅} (and the Jordan–Hölder theorem states that every other composition series is equivalent to that one), giving factor groups isomorphic to A₅ and C₂; and A₅ 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 polynomials 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[edit]

Consider the subgroups

${\displaystyle B=\left\{{\begin{bmatrix}*&*\\0&*\end{bmatrix}}\right\}{\text{, }}U=\left\{{\begin{bmatrix}1&*\\0&1\end{bmatrix}}\right\}}$ of ${\displaystyle GL_{2}(\mathbb {F} )}$

for some field ${\displaystyle \mathbb {F} }$. Then, the group quotient ${\displaystyle B/U}$ can be found by taking arbitrary elements in ${\displaystyle B,U}$, multiplying them together, and figuring out what structure this gives. So

${\displaystyle {\begin{bmatrix}a&b\\0&c\end{bmatrix}}\cdot {\begin{bmatrix}1&d\\0&1\end{bmatrix}}={\begin{bmatrix}a&ad+b\\0&c\end{bmatrix}}}$

Note the determinant condition on ${\displaystyle GL_{2}}$ implies ${\displaystyle ac\neq 0}$, hence ${\displaystyle \mathbb {F} ^{\times }\times \mathbb {F} ^{\times }\subset B}$ is a subgroup (which are the matrices where ${\displaystyle b=0}$). For fixed ${\displaystyle a,b}$, the linear equation ${\displaystyle ad+b=0}$ implies ${\displaystyle d=-b/a}$, which is an arbitrary element in ${\displaystyle \mathbb {F} }$ since ${\displaystyle b\in \mathbb {F} }$. Since we can take any matrix in ${\displaystyle B}$ and multiply it by the matrix

${\displaystyle {\begin{bmatrix}1&d\\0&1\end{bmatrix}}}$

with ${\displaystyle d=-b/a}$, we can get a diagonal matrix in ${\displaystyle B}$. This shows the quotient group ${\displaystyle B/U\cong \mathbb {F} ^{\times }\times \mathbb {F} ^{\times }}$.

Remark[edit]

Notice that this description gives the decomposition of ${\displaystyle B}$ as ${\displaystyle \mathbb {F} \rtimes (\mathbb {F} ^{\times }\times \mathbb {F} ^{\times })}$ where ${\displaystyle (a,c)}$ acts on ${\displaystyle b}$ by ${\displaystyle (a,c)(b)=ab}$. This implies ${\displaystyle (a,c)(b+b')=(a,c)(b)+(a,c)(b')=ab+ab'}$. Also, a matrix of the form

${\displaystyle {\begin{bmatrix}a&b\\0&c\end{bmatrix}}}$

corresponds to the element ${\displaystyle (b)\times (a,c)}$ in the group.

Borel subgroups[edit]

For a linear algebraic group ${\displaystyle G}$ its Borel subgroup is defined as a subgroup which is closed, connected, and solvable in ${\displaystyle G}$, and it is the maximal possible subgroup with these properties (note the second two are topological properties). For example, in ${\displaystyle GL_{n}}$ and ${\displaystyle SL_{n}}$ the group of upper-triangular, or lower-triangular matrices are two of the Borel subgroups. The example given above, the subgroup ${\displaystyle B}$ in ${\displaystyle GL_{2}}$ is the Borel subgroup.

Borel subgroup in GL₃[edit]

In ${\displaystyle GL_{3}}$ there are the subgroups

${\displaystyle B=\left\{{\begin{bmatrix}*&*&*\\0&*&*\\0&0&*\end{bmatrix}}\right\},{\text{ }}U_{1}=\left\{{\begin{bmatrix}1&*&*\\0&1&*\\0&0&1\end{bmatrix}}\right\}}$

Notice ${\displaystyle B/U_{1}\cong \mathbb {F} ^{\times }\times \mathbb {F} ^{\times }\times \mathbb {F} ^{\times }}$, hence the Borel group has the form

${\displaystyle U\rtimes (\mathbb {F} ^{\times }\times \mathbb {F} ^{\times }\times \mathbb {F} ^{\times })}$

Borel subgroup in product of simple linear algebraic groups[edit]

In the product group ${\displaystyle GL_{n}\times GL_{m}}$ the Borel subgroup can be represented by matrices of the form

${\displaystyle {\begin{bmatrix}T&0\\0&S\end{bmatrix}}}$

where ${\displaystyle T}$ is an ${\displaystyle n\times n}$ upper triangular matrix and ${\displaystyle S}$ is a ${\displaystyle m\times m}$ upper triangular matrix.

Z-groups[edit]

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-groups.

OEIS values[edit]

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, ... (sequence A201733 in the OEIS)

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, ... (sequence A056866 in the OEIS)

Properties[edit]

Solvability is closed under a number of operations.

• If G is solvable, and H is a subgroup of G, then H is solvable.^[2]
• If G is solvable, and there is a homomorphism 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.^[3]
• 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:

• 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 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, 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[edit]

Main article: Burnside's theorem

Burnside's theorem states that if G is a finite group of order p^aq^b where p and q are prime numbers, and a and b are non-negative integers, then G is solvable.

Related concepts[edit]

Supersolvable groups[edit]

Main article: supersolvable group

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 A₄ 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 < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group.

Virtually solvable groups[edit]

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[edit]

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 (Malcev 1949).

See also[edit]

• Prosolvable group
• Parabolic subgroup

Notes[edit]

1. ^ Milne. Field Theory (PDF). p. 45.
2. ^ Rotman (1995), Theorem 5.15, p. 102, at Google Books
3. ^ Rotman (1995), Theorem 5.16, p. 102, at Google Books

References[edit]

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Solvable group" – news · newspapers · books · scholar · JSTOR (January 2008) (Learn how and when to remove this template message)

• Malcev, A. I. (1949), "Generalized nilpotent algebras and their associated groups", Mat. Sbornik N.S., 25 (67): 347–366, MR 0032644
• Rotman, Joseph J. (1995), An Introduction to the Theory of Groups, Graduate Texts in Mathematics, 148 (4 ed.), Springer, ISBN 978-0-387-94285-8

External links[edit]

• OEIS sequence A056866 (Orders of non-solvable groups)
• Solvable groups as iterated extensions

Retrieved from "https://en.wikipedia.org/w/index.php?title=Solvable_group&oldid=990794502"

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• Properties of groups

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