Solvable group

Inmathematics,more specifically in the field ofgroup theory,asolvable grouporsoluble groupis agroupthat can be constructed fromabelian groupsusingextensions.Equivalently, a solvable group is a group whosederived seriesterminates in thetrivial subgroup.

Motivation

Historically, the word "solvable" arose fromGalois theoryand theproofof the general unsolvability ofquinticequations. Specifically, apolynomial equationis solvable inradicalsif and only ifthe correspondingGalois groupis solvable(note this theorem holds only incharacteristic0). This means associated to a polynomial$f\in F[x]$there is a tower of field extensions

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

such that

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

Example

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

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

gives a solvable group. It has associated field extensions

$\mathbb {Q} \subseteq \mathbb {Q} ({\sqrt {2}})\subseteq \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\subseteq \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\left(e^{2i\pi /5}\right)\subseteq \mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\left(e^{2i\pi /5},a\right)$

giving a solvable group of Galois extensions containing the followingcomposition factors:

• $\mathrm {Aut} \left(\mathbb {Q({\sqrt {2}})} \right/\mathbb {Q} )\cong \mathbb {Z} /2$with group action$f\left(\pm {\sqrt {2}}\right)=\mp {\sqrt {2}},\ f^{2}=1$,andminimal polynomial$x^{2}-2$.
• $\mathrm {Aut} \left(\mathbb {Q({\sqrt {2}},{\sqrt {3}})} \right/\mathbb {Q({\sqrt {2}})} )\cong \mathbb {Z} /2$with group action$g\left(\pm {\sqrt {3}}\right)=\mp {\sqrt {3}},\ g^{2}=1$,and minimal polynomial$x^{2}-3$.
• $\mathrm {Aut} \left(\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\left(e^{2i\pi /5}\right)/\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\right)\cong \mathbb {Z} /4$with group action$h^{n}\left(e^{2im\pi /5}\right)=e^{2(n+1)mi\pi /5},\ 0\leq n\leq 3,\ h^{4}=1$,and minimal polynomial$x^{4}+x^{3}+x^{2}+x+1=(x^{5}-1)/(x-1)$containing the 5^throots of unity excluding$1$.
• $\mathrm {Aut} \left(\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\left(e^{2i\pi /5},a\right)/\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})\left(e^{2i\pi /5}\right)\right)\cong \mathbb {Z} /5$with group action$j^{l}(a)=e^{2li\pi /5}a,\ j^{5}=1$,and minimal polynomial$x^{5}-\left({\sqrt {2}}+{\sqrt {3}}\right)$.

,where$1$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$fgh^{3}j^{4}$.A general element in the group can be written as$f^{a}g^{b}h^{n}j^{l},\ 0\leq a,b\leq 1,\ 0\leq n\leq 3,\ 0\leq l\leq 4$for a total of 80 elements.

It is worthwhile to note that this group is notabelianitself. For example:

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

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

In fact, in this group,$jh=hj^{3}$.The solvable group is isometric to$(\mathbb {C} _{5}\rtimes _{\varphi }\mathbb {C} _{4})\times (\mathbb {C} _{2}\times \mathbb {C} _{2}),\ \mathrm {where} \ \varphi _{h}(j)=hjh^{-1}=j^{2}$,defined using thesemidirect productanddirect productof thecyclic groups.In the solvable group,$\mathbb {C} _{4}$is not a normal subgroup.

Definition

A groupGis calledsolvableif it has asubnormal serieswhosefactor groups(quotient groups) are allabelian,that is, if there aresubgroups

$1=G_{0}\triangleleft G_{1}\triangleleft \cdots \triangleleft G_{k}=G$

meaning thatG_j−1isnormalinG_j,such thatG_j /G_j−1is an abelian group, forj= 1, 2,...,k.

Or equivalently, if itsderived series,the descending normal series

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

where every subgroup is thecommutator subgroupof the previous one, eventually reaches the trivial subgroup ofG.These two definitions are equivalent, since for every groupHand everynormal subgroupNofH,the quotientH/Nis abelianif and only ifNincludes the commutator subgroup ofH.The leastnsuch thatG⁽ⁿ⁾= 1 is called thederived lengthof the solvable groupG.

For finite groups, an equivalent definition is that a solvable group is a group with acomposition seriesall of whose factors arecyclic groupsofprimeorder.This is equivalent because a finite group has finite composition length, and everysimpleabelian group is cyclic of prime order. TheJordan–Hölder theoremguarantees 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 tonth roots (radicals) over somefield.The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the groupZofintegersunder addition isisomorphictoZitself, it has no composition series, but the normal series {0,Z}, with its only factor group isomorphic toZ,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, allnilpotent groupsare solvable. In particular, finitep-groupsare solvable, as all finitep-groupsare nilpotent.

Quaternion groups

In particular, thequaternion groupis a solvable group given by the group extension

$1\to \mathbb {Z} /2\to Q\to \mathbb {Z} /2\times \mathbb {Z} /2\to 1$

where the kernel$\mathbb {Z} /2$is the subgroup generated by$-1$.

Group extensions

Group extensionsform 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.

Non-abelian group which is non-nilpotent

A small example of a solvable, non-nilpotent group is thesymmetric groupS₃.In fact, as the smallest simple non-abelian group isA₅,(thealternating groupof degree 5) it follows thateverygroup with order less than 60 is solvable.

Finite groups of odd order

TheFeit–Thompson theoremstates 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 groupS₅is not solvable — it has a composition series {E,A₅,S₅} (and theJordan–Hölder theoremstates that every other composition series is equivalent to that one), giving factor groups isomorphic toA₅andC₂;andA₅is not abelian. Generalizing this argument, coupled with the fact thatA_nis a normal, maximal, non-abelian simple subgroup ofS_nforn> 4, we see thatS_nis not solvable forn> 4. This is a key step in the proof that for everyn> 4 there arepolynomialsof degreenwhich are not solvable by radicals (Abel–Ruffini theorem). This property is also used in complexity theory in the proof ofBarrington's theorem.

Subgroups of GL₂

Consider the subgroups

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

for some field$\mathbb {F}$.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

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

Note the determinant condition on$GL_{2}$implies$ac\neq 0$,hence$\mathbb {F} ^{\times }\times \mathbb {F} ^{\times }\subset B$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$\mathbb {F}$since$b\in \mathbb {F}$.Since we can take any matrix in$B$and multiply it by the matrix

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

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

Remark

Notice that this description gives the decomposition of$B$as$\mathbb {F} \rtimes (\mathbb {F} ^{\times }\times \mathbb {F} ^{\times })$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

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

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

Borel subgroups

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

Borel subgroup in GL₃

In$GL_{3}$there are the subgroups

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

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

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

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

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

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 whosep-Sylow subgroupsare cyclic is asemidirect productof two cyclic groups, in particular solvable. Such groups are calledZ-groups.

OEIS values

Numbers of solvable groups with ordernare (start withn= 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,... (sequenceA201733in theOEIS)

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,... (sequenceA056866in theOEIS)

Properties

Solvability is closed under a number of operations.

• IfGis solvable, andHis a subgroup ofG,thenHis solvable.
• IfGis solvable, and there is ahomomorphismfromGontoH,thenHis solvable; equivalently (by thefirst isomorphism theorem), ifGis solvable, andNis a normal subgroup ofG,thenG/Nis solvable.
• The previous properties can be expanded into the following "three for the price of two" property:Gis solvable if and only if bothNandG/Nare solvable.
• In particular, ifGandHare solvable, thedirect productG×His solvable.

Solvability is closed undergroup extension:

• IfHandG/Hare solvable, then so isG;in particular, ifNandHare solvable, theirsemidirect productis also solvable.

It is also closed under wreath product:

• IfGandHare solvable, andXis aG-set, then thewreath productofGandHwith respect toXis also solvable.

For any positive integerN,the solvable groups ofderived lengthat mostNform asubvarietyof the variety of groups, as they are closed under the taking ofhomomorphicimages,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

Main article:Burnside's theorem

Burnside's theorem states that ifGis afinite groupoforderp^aq^bwherepandqareprime numbers,andaandbarenon-negativeintegers,thenGis solvable.

Related concepts

Supersolvable groups

Main article:supersolvable group

As a strengthening of solvability, a groupGis calledsupersolvable(orsupersoluble) if it has aninvariantnormal series whose factors are all cyclic. Since a normal series has finite length by definition,uncountablegroups are not supersolvable. In fact, all supersolvable groups arefinitely generated,and an abelian group is supersolvable if and only if it is finitely generated. The alternating groupA₄is 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

A groupGis calledvirtually solvableif it has a solvable subgroup of finite index. This is similar tovirtually 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 afinitestage. 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 ahypoabelian group,and every solvable group is a hypoabelian group. The first ordinalαsuch thatG^(α)=G^(α+1)is called the (transfinite) derived length of the groupG,and it has been shown that every ordinal is the derived length of some group (Malcev 1949).

p-solvable

A finite group is p-solvable for some prime p if every factor in the composition series is ap-groupor has order prime to p. A finite group is solvable iff it is p-solvable for every p.

See also

• Prosolvable group

References

• Malcev, A. I.(1949), "Generalized nilpotent algebras and their associated groups",Mat. Sbornik,New Series,25(67): 347–366,MR0032644
• Rotman, Joseph J.(1995),An Introduction to the Theory of Groups,Graduate Texts in Mathematics, vol. 148 (4 ed.), Springer,ISBN978-0-387-94285-8

External links

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