Editing Solvable group

{{Short description|Group with subnormal series where all factors are abelian}}
{{Group theory sidebar |Basics}}

In [[mathematics]], more specifically in the field of [[group theory]], a '''solvable group''' or '''soluble group''' is a [[group (mathematics)|group]] that can be constructed from [[abelian group]]s using [[Group extension|extensions]].  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 of the general unsolvability of [[quintic]] equations. Specifically, a [[polynomial equation]] is solvable in [[Nth root|radicals]] if and only if the corresponding [[Galois group]] is solvable<ref>{{Cite book|last=Milne|url=https://www.jmilne.org/math/CourseNotes/FT.pdf|title=Field Theory|pages=45}}</ref> (note this theorem holds only in [[characteristic of a field|characteristic]] 0). This means associated to a polynomial <math>f \in F[x]</math> there is a tower of field extensions<blockquote><math>F = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m=K</math></blockquote>such that

# <math>F_i = F_{i-1}[\alpha_i]</math> where <math>\alpha_i^{m_i} \in F_{i-1}</math>, so <math>\alpha_i</math> is a solution to the equation <math>x^{m_i} - a</math> where <math>a \in F_{i-1}</math>
# <math>F_m</math> contains a [[splitting field]] for <math>f(x)</math>

=== Example ===
The smallest Galois field extension of <math>\mathbb{Q}</math> containing the element<blockquote><math>a = \sqrt[5]{\sqrt{2} + \sqrt{3}}</math></blockquote>gives a solvable group. The associated field extensions<blockquote><math>\mathbb{Q} 
\subseteq \mathbb{Q}(\sqrt{2})  
\subseteq \mathbb{Q}(\sqrt{2}, \sqrt{3})  
\subseteq \mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}\right)
\subseteq \mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}, a\right)</math></blockquote>give a solvable group of Galois extensions containing the following [[composition factor]]s (where <math>1</math> is the identity permutation).

* <math>\mathrm{Aut}\left(\mathbb{Q(\sqrt{2})}\right/\mathbb{Q}) \cong \mathbb{Z}/2 </math> with group action <math>f\left(\pm\sqrt{2}\right) = \mp\sqrt{2}, \ f^2 = 1</math>, and [[Minimal polynomial (field theory)|minimal polynomial]] <math>x^2 - 2</math>
* <math>\mathrm{Aut}\left(\mathbb{Q(\sqrt{2},\sqrt{3})}\right/\mathbb{Q(\sqrt{2})}) \cong \mathbb{Z}/2 </math> with group action <math>g\left(\pm\sqrt{3}\right) = \mp\sqrt{3} ,\ g^2 = 1</math>, and minimal polynomial <math>x^2 - 3</math>
* <math>\mathrm{Aut}\left(
\mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}\right)/
\mathbb{Q}(\sqrt{2}, \sqrt{3})
\right)
\cong \mathbb{Z}/4 </math> with group action <math>h^n\left(e^{2im\pi/5}\right) = e^{2(n+1)mi\pi/5} , \  0 \leq n \leq 3, \ h^4 = 1</math>, and minimal polynomial <math>x^4 + x^3+x^2+x+1 = (x^5 - 1)/(x-1)</math> containing the 5th roots of unity excluding <math>1</math>
* <math>\mathrm{Aut}\left(
\mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}, a\right)/
\mathbb{Q}(\sqrt{2}, \sqrt{3})\left(e^{2i\pi/ 5}\right)
\right)
\cong \mathbb{Z}/5 </math> with group action <math>j^l(a) = e^{2li\pi/5}a, \ j^5 = 1</math>, and minimal polynomial <math>x^5 - \left(\sqrt{2} + \sqrt{3}\right)</math>

Each of the defining group actions (for example, <math>fgh^3j^4 </math>) changes a single extension while keeping all of the other extensions fixed. The 80 group actions are the set <math>\{f^ag^bh^nj^l,\ 0 \leq a, b \leq 1,\ 0 \leq n \leq 3,\ 0 \leq l \leq 4 \}</math>.

This group is not [[Abelian group|abelian]].  For example, <math>hj(a) = h(e^{2i\pi/5}a) = e^{4i\pi/5}a </math>, whilst <math>jh(a) = j(a) = e^{2i\pi/5}a</math>, and in fact, <math>jh = hj^3</math>. 

It is isomorphic to <math>(\mathbb{Z}_5 \rtimes_\varphi \mathbb{Z}_4) \times (\mathbb{Z}_2 \times \mathbb{Z}_2) </math>, where <math>\varphi_h(j) = hjh^{-1} = j^2 </math>, defined using the [[semidirect product]] and [[Direct product of groups|direct product]] of the [[cyclic group]]s.  <math>\mathbb{Z}_4 </math> is not a normal subgroup.

==Definition==
A group ''G'' is called '''solvable''' if it has a [[subnormal series]] whose [[factor group]]s (quotient groups) are all [[Abelian group|abelian]], that is, if there are [[subgroup]]s
:<math>1 = G_0\triangleleft G_1 \triangleleft \cdots \triangleleft G_k=G</math>
meaning that ''G''<sub>''j''−1</sub> is [[normal subgroup|normal]] in ''G<sub>j</sub>'', such that ''G<sub>j</sub>&thinsp;''/''G''<sub>''j''−1</sub> is an abelian group, for ''j'' = 1, 2, ..., ''k''.

Or equivalently, if its [[derived series]], the descending normal series
:<math>G\triangleright G^{(1)}\triangleright G^{(2)} \triangleright \cdots,</math>
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''<sup>(''n'')</sup> = 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 group]]s of [[prime number|prime]] [[order (group theory)|order]]. This is equivalent because a finite group has finite composition length, and every [[simple group|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 ''n''th roots (radicals) over some [[Field (mathematics)|field]]. The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group '''Z''' of [[integer]]s under addition is [[group isomorphism|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 ==

=== 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]]s are solvable. In particular, finite [[p-group|''p''-groups]] are solvable, as all finite [[p-group|''p''-groups]] are nilpotent.

==== Quaternion groups ====
In particular, the [[quaternion group]] is a solvable group given by the group extension<blockquote><math>1 \to \mathbb{Z}/2 \to Q \to \mathbb{Z}/2 \times \mathbb{Z}/2 \to 1</math></blockquote>where the kernel <math>\mathbb{Z}/2</math> is the subgroup generated by <math>-1</math>.

=== Group extensions ===
[[Group extension]]s form the prototypical examples of solvable groups. That is, if <math>G</math> and <math>G'</math> are solvable groups, then any extension<blockquote><math>1 \to G \to G'' \to G' \to 1</math></blockquote>defines a solvable group <math>G''</math>. In fact, all solvable groups can be formed from such group extensions.

=== Non-abelian group which is non-nilpotent ===
A small example of a solvable, non-nilpotent group is the [[symmetric group]] ''S''<sub>3</sub>. In fact, as the smallest simple non-abelian group is ''A''<sub>5</sub>, (the [[alternating group]] of degree 5) it follows that ''every'' group with order less than 60 is solvable.

=== Finite groups of odd order ===
The [[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 ===
The group ''S''<sub>5</sub> is not solvable &mdash; it has a composition series {E, ''A''<sub>5</sub>, ''S''<sub>5</sub>} (and the [[Jordan–Hölder theorem]] states that every other composition series is equivalent to that one), giving factor groups isomorphic to ''A''<sub>5</sub> and ''C''<sub>2</sub>; and ''A''<sub>5</sub> is not abelian. Generalizing this argument, coupled with the fact that ''A''<sub>''n''</sub> is a normal, maximal, non-abelian simple subgroup of ''S''<sub>''n''</sub> for ''n'' > 4, we see that ''S''<sub>''n''</sub> is not solvable for ''n'' > 4. This is a key step in the proof that for every ''n'' > 4 there are [[polynomial]]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 [[NC (complexity)|Barrington's theorem]].

=== Subgroups of GL<sub>2</sub> ===
Consider the subgroups<blockquote><math>B = \left\{ \begin{bmatrix}
* & * \\
0 & *
\end{bmatrix} \right\} \text{, }
U = \left\{ \begin{bmatrix}
1 & * \\
0 & 1
\end{bmatrix} \right\}</math> of <math>GL_2(\mathbb{F})</math></blockquote>for some field <math>\mathbb{F}</math>. Then, the group quotient <math>B/U</math> can be found by taking arbitrary elements in <math>B,U</math>, multiplying them together, and figuring out what structure this gives. So<blockquote><math>\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}

</math></blockquote>Note the determinant condition on <math>GL_2

</math> implies <math>ac \neq 0

</math>, hence <math>\mathbb{F}^\times \times \mathbb{F}^\times \subset B

</math> is a subgroup (which are the matrices where <math>b=0

</math>). For fixed <math>a,b

</math>, the linear equation <math>ad + b = 0

</math> implies <math>d = -b/a

</math>, which is an arbitrary element in <math>\mathbb{F}

</math> since <math>b \in \mathbb{F}

</math>. Since we can take any matrix in <math>B

</math> and multiply it by the matrix<blockquote><math>\begin{bmatrix}
1 & d \\
0 & 1
\end{bmatrix}

</math></blockquote>with <math>d = -b/a

</math>, we can get a diagonal matrix in <math>B

</math>. This shows the quotient group <math>B/U \cong \mathbb{F}^\times \times \mathbb{F}^\times</math>.

==== Remark ====
Notice that this description gives the decomposition of <math>B

</math> as <math>\mathbb{F} \rtimes (\mathbb{F}^\times \times \mathbb{F}^\times)

</math> where <math>(a,c)

</math> acts on <math>b

</math> by <math>(a,c)(b) = ab

</math>. This implies <math>(a,c)(b + b') = (a,c)(b) + (a,c)(b') = ab + ab'

</math>. Also, a matrix of the form<blockquote><math>\begin{bmatrix}
a & b \\
0 & c
\end{bmatrix}</math></blockquote>corresponds to the element <math>(b) \times (a,c)</math> in the group.

=== Borel subgroups ===
For a [[linear algebraic group]] <math>G</math>, a [[Borel subgroup]] is defined as a subgroup which is closed, connected, and solvable in <math>G</math>, and is a maximal possible subgroup with these properties (note the first two are topological properties). For example, in <math>GL_n</math> and <math>SL_n</math> the groups of upper-triangular, or lower-triangular matrices are two of the Borel subgroups. The example given above, the subgroup <math>B</math> in <math>GL_2</math>, is a Borel subgroup.

==== Borel subgroup in GL<sub>3</sub> ====
In <math>GL_3</math> there are the subgroups<blockquote><math>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\}</math></blockquote>Notice <math>B/U_1 \cong \mathbb{F}^\times \times \mathbb{F}^\times \times \mathbb{F}^\times</math>, hence the Borel group has the form<blockquote><math>U\rtimes
(\mathbb{F}^\times \times \mathbb{F}^\times \times \mathbb{F}^\times)

</math></blockquote>

==== Borel subgroup in product of simple linear algebraic groups ====
In the product group <math>GL_n \times GL_m</math> the Borel subgroup can be represented by matrices of the form<blockquote><math>\begin{bmatrix}
T & 0 \\
0 & S
\end{bmatrix}</math></blockquote>where <math>T</math> is an <math>n\times n</math> upper triangular matrix and <math>S</math> is a <math>m\times m</math> upper triangular matrix.

=== Z-groups ===
Any finite group whose [[Sylow group|''p''-Sylow subgroups]] are cyclic is a [[semidirect product]] of two cyclic groups, in particular solvable.  Such groups are called [[Z-group]]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, ... {{OEIS|id=A201733}}

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, ... {{OEIS|id=A056866}}

== Properties ==

Solvability is closed under a number of operations.

* If ''G'' is solvable, and ''H'' is a subgroup of ''G'', then ''H'' is solvable.<ref>Rotman (1995), {{Google books|id=lYrsiaHSHKcC|page=102|text=Every subgroup H of a solvable group G is itself solvable|title=Theorem 5.15}}</ref>
* If ''G'' is solvable, and there is a [[group homomorphism|homomorphism]] from ''G'' [[surjective|onto]] ''H'', then ''H'' is solvable; equivalently (by the [[Isomorphism theorem#First isomorphism theorem|first isomorphism theorem]]), if ''G'' is solvable, and ''N'' is a normal subgroup of ''G'', then ''G''/''N'' is solvable.<ref>Rotman (1995), {{Google books|id=lYrsiaHSHKcC|page=102|text=Every quotient of a solvable group is solvable|title=Theorem 5.16}}</ref>
* 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 of groups|direct product]] ''G'' &times; ''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 [[Algebraic variety|subvariety]] of the variety of groups, as they are closed under the taking of [[homomorphism|homomorphic]] images, [[subalgebra]]s, and [[Direct product of groups|(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==
{{main|Burnside's theorem}}
Burnside's theorem states that if ''G''  is a [[finite group]] of [[Order (group theory)|order]] ''p<sup>a</sup>q<sup>b</sup>'' where ''p'' and ''q'' are [[prime number]]s, and ''a'' and ''b''  are [[negative and positive numbers|non-negative]] [[integer]]s, then ''G'' is solvable.

==Related concepts==

===Supersolvable groups===
{{main|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 group|finitely generated]], and an abelian group is supersolvable if and only if it is finitely generated. The alternating group ''A''<sub>4</sub> 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 group|cyclic]] < [[abelian group|abelian]] < [[nilpotent group|nilpotent]] < [[supersolvable group|supersolvable]] < [[polycyclic group|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 '''[[perfect core|hypoabelian group]]''', and every solvable group is a hypoabelian group.  The first ordinal ''α'' such that ''G''<sup>(''α'')</sup> = ''G''<sup>(''α''+1)</sup> 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 {{harv|Malcev|1949}}.

===p-solvable===
A finite group is p-solvable for some prime p if every factor in the composition series is a [[p-group]] or has order prime to p. A finite group is solvable iff it is p-solvable for every p. 
<ref>{{cite web|title =p-solvable-groups|url=https://groupprops.subwiki.org/wiki/P-solvable_group |website=Group props wiki}}</ref>

==See also==

* [[Prosolvable group]]

==Notes==
{{Reflist}}

==References==
{{more citations needed|date=January 2008}}
* {{Citation | last1=Malcev | first1=A. I. | author1-link=Anatoly Maltsev|title=Generalized nilpotent algebras and their associated groups | mr=0032644  | year=1949 | journal= [[Matematicheskii Sbornik|Mat. Sbornik]] |series=New Series | volume=25 | issue=67 | pages=347–366}}
* {{Citation |last1=Rotman |first1=Joseph J. |author-link1= Joseph J. Rotman |title=An Introduction to the Theory of Groups |edition=4 |series=Graduate Texts in Mathematics |volume=148 |year=1995 |publisher=Springer |isbn=978-0-387-94285-8  }}

==External links==
* {{OEIS el|sequencenumber=A056866|name=Orders of non-solvable groups|formalname=Orders of non-solvable groups, i.e., numbers that are not solvable numbers}}
*[https://math.stackexchange.com/questions/3523759/can-any-solvable-finite-group-be-obtained-from-abelian-groups-and-combinations-o Solvable groups as iterated extensions]

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[[Category:Properties of groups]]