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

In [[Mathematics|mathematics]], more specifically in the field of [[Group theory|group theory]], a '''solvable group''' or '''soluble group''' is a [[Group (mathematics)|group]] that can be constructed from [[Abelian group|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 [[Mathematical proof|proof]] of the general unsolvability of quintic equations. Specifically, a polynomial equation is solvable in [[Nth root|radicals]] [[If and only if|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 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|splitting field]] for <math>f(x)</math>

=== Example ===
For 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. It has 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>giving a solvable group of Galois extensions containing the following composition factors:

* <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 5<sup>th</sup> 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>.

, where <math>1</math> is the identity permutation. All of the defining group actions change a single extension while keeping all of the other extensions fixed. For example, an element of this group is the group action <math>fgh^3j^4 </math>. A general element in the group can be written as <math>f^ag^bh^nj^l,\ 0 \leq a, b \leq 1,\ 0 \leq n \leq 3,\ 0 \leq l \leq 4 </math> for a total of 80 elements.

It is worthwhile to note that this group is not [[Abelian group|abelian]] itself.  For example:

<math>hj(a) = h(e^{2i\pi/5}a) = e^{4i\pi/5}a </math>

<math>jh(a) = j(a) = e^{2i\pi/5}a </math>

In fact, in this group, <math>jh = hj^3 </math>. The solvable group is isometric to <math>(\mathbb{C}_5 \rtimes_\varphi \mathbb{C}_4) \times (\mathbb{C}_2 \times \mathbb{C}_2),\ \mathrm{where}\ \varphi_h(j) = hjh^{-1} = j^2 </math>, defined using the [[Semidirect product|semidirect product]] and [[Direct product of groups|direct product]] of the [[Cyclic group|cyclic groups]].  In the solvable group, <math>\mathbb{C}_4 </math> is not a normal subgroup.

==Definition==
A group ''G'' is called '''solvable''' if it has a subnormal series whose factor groups (quotient groups) are all [[Abelian group|abelian]], that is, if there are [[Subgroup|subgroup]]s 1 = ''G''<sub>0</sub> < ''G''<sub>1</sub> < ⋅⋅⋅ < ''G<sub>k</sub>'' = ''G'' such that ''G''<sub>''j''−1</sub> is [[Normal subgroup|normal]] in ''G<sub>j</sub>'', and ''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|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|normal subgroup]] ''N'' of ''H'', the quotient ''H''/''N'' is abelian [[If and only if|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|composition series]] all of whose factors are [[Cyclic group|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|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|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|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|symmetric group]] ''S''<sub>3</sub>. In fact, as the smallest simple non-abelian group is ''A''<sub>5</sub>, (the [[Alternating group|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|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|linear algebraic group]] <math>G</math> its [[Borel subgroup]] is defined as a subgroup which is closed, connected, and solvable in <math>G</math>, and it is the 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 group 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 the 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 ''p''-Sylow subgroups are cyclic is a [[Semidirect product|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'' 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.<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|group extension]]:
* If ''H'' and ''G''/''H'' are solvable, then so is ''G''; in particular, if ''N'' and ''H'' are solvable, their [[Semidirect product|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|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 [[Variety (universal algebra)|subvariety]] of the variety of groups, as they are closed under the taking of [[Homomorphism|homomorphic]] images, [[Subalgebra|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|finite group]] of [[Order (group theory)|order]] ''p<sup>a</sup>q<sup>b</sup>'' where ''p'' and ''q'' are [[Prime number|prime number]]s, and ''a'' and ''b''  are non-negative [[Integer|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|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|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==
* {{Citation | last1=Malcev | first1=A. I. | 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. |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:Solvable groups| ]]
[[Category:Properties of groups]]

{{Sourceattribution|Solvable group}}