Solvable group: Difference between revisions

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{{semibasicdef}}
{{pivotal group property}}
{{finite-at|finite solvable group}}
==History==
{{term introduced by|Galois}}
The notion of ''solvable group'' arose from the attempt to characterize the Galois groups of those field extensions which could be solved by means of radicals.
In fact, the term ''solvable'' arose precisely because a normal field extension is contained in a radical extension if and only if the Galois group is solvable.
==Definition==
==Definition==


===Symbol-free definition===
'''Solvable''' is also called '''soluble''' by some people.


A [[group]] is said to be '''solvable''' (or '''soluble''') if any of the following equivalent conditions holds:
===Equivalent definitions in tabular format===


* There is a [[normal series]] of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient being an [[abelian group]].
{| class="sortable" border="1"
* There is a [[subnormal series]] of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient being an [[abelian group]].
! No. !! Shorthand !! A group is termed solvable if ... !! A group <math>G</math> is termed solvable if ...
* The [[derived series]] reaches the identity in finitely many steps. The number of steps needed is termed the [[derived length]] (sometimes also called solvable length) of the solvable group.
|-
* There is a [[characteristic series]] of finite length with each successive quotient being an [[abelian group]].
| 1 || normal series, abelian quotients || there is a [[defining ingredient::normal series]] of finite length starting from the trivial subgroup and ending at the whole group with each successive [[quotient group]] being an [[defining ingredient::abelian group]]. || there exists a series of subgroups <math>\{ e \} = H_0 \le H_1 \le \ldots \le H_n = G</math> such that each <math>H_i</math> is [[normal subgroup|normal]] in <math>G</math> and each <math>H_{i+1}/H_i</math> is [[abelian group|abelian]].
* There is a [[fully invariant series]] of finite length with each successive quotient being an [[abelian group]].
|-
 
| 2 || subnormal series, abelian quotients || there is a [[defining ingredient::subnormal series]] of finite length starting from the trivial subgroup and ending at the whole group with each successive [[quotient group]] being an [[abelian group]]. || there exists a series of subgroups:<math>\{ e \}= H_0 \triangleleft H_1 \triangleleft \ldots \triangleleft H_n = G</math> such that each <math>H_i</math> is normal in <math>H_{i+1}</math> and each <math>H_{i+1}/H_i</math> is [[abelian group|abelian]].
===Definition with symbols===
|-
 
| 3 || derived series finite length || the [[defining ingredient::derived series]] reaches the identity in finitely many steps || the [[derived series]] of <math>G</math>, i.e., the series <math>G^{(n)}</math> where <math>G^{(0)} = G</math> and <math>G^{(i+1)} = [G^{(i)}, G^{(i)}]</math> is the [[defining ingredient::derived subgroup]] of its predecessor, reaches the trivial subgroup in finitely many steps.
A [[group]] <math>G</math> is said to be '''solvable''' if it satisfies any of the following equivalent conditions;
|-
 
| 4 || characteristic series, abelian quotients || there is a [[defining ingredient::characteristic series]] of finite length starting from the trivial subgroup and ending at the whole group with each successive [[quotient group]] being an [[defining ingredient::abelian group]]. || there exists a series of subgroups <math>\{ e  \} = H_0 \le H_1 \le \ldots \le H_n = G</math> such that each <math>H_i</math> is [[characteristic subgroup|characteristic]] in <math>G</math> and each <math>H_{i+1}/H_i</math> is [[abelian group|abelian]].
* There exists a series of subgroups:
|-
 
| 5 || fully invariant series, abelian quotients || there is a [[defining ingredient::fully invariant series]] of finite length starting from the trivial subgroup and ending at the whole group with each successive [[quotient group]] being an [[defining ingredient::abelian group]]. || there exists a series of subgroups <math>\{ e  \} = H_0 \le H_1 \le \ldots \le H_n = G</math> such that each <math>H_i</math> is [[fully invariant subgroup|fully invariant]] in <math>G</math> and each <math>H_{i+1}/H_i</math> is [[abelian group|abelian]].
<math>e = H_0 \le H_1 \le \ldots \le H_n = G</math>
|}
 
such that each <math>H_i</math> is [[normal subgroup|normal]] in <math>G</math> and each <math>H_i/H_{i-1}</math> is [[Abelian group|Abelian]].
 
* There exists a series of subgroups:
 
<math>e = H_0 \triangleleft H_1 \triangleleft \ldots \triangleleft H_n = G</math>
 
such that each <math>H_{i-1}</math> is normal in <math>H_i</math> and each <math>H_i/H_{i-1}</math> is [[Abelian group|Abelian]].
 
* The [[derived series]] of <math>G</math>, viz the series <math>G^{(n)}</math> where <math>G^{(0)} = G</math> and <math>G^{(i+1)} = [G^{(i)}, G^{(i)}]</math>, reaches the trivial subgroup in finitely many steps.


===Equivalence of definitions===
===Equivalence of definitions===


{{further|[[Equivalence of definitions of solvable group]]}}
{{further|[[Equivalence of definitions of solvable group]]}}
==Formalisms==
{{group extension operator}}
The group property of being solvable can be obtained in either of these equivalent ways:
* By applying the [[poly operator]] to the group property of being [[Abelian group|Abelian]]
* By applying the [[finite normal series operator]] to the group property of being [[Abelian group|Abelian]]
* By applying the [[finite characteristic series operator]] to the group property of being [[Abelian group|Abelian]]
Note that all these three operators have the same effect in the case of Abelian groups, though in general they may not have.


==Examples==
==Examples==
Line 65: Line 32:
* [[Symmetric group:S4|The symmetric group of degree four]] is also solvable.
* [[Symmetric group:S4|The symmetric group of degree four]] is also solvable.
* The [[dihedral group]] of any order is solvable, Further it is nilpotent only when the order is a power of 2.
* The [[dihedral group]] of any order is solvable, Further it is nilpotent only when the order is a power of 2.
{{semibasicdef}}
{{pivotal group property}}
{{finite-at|finite solvable group}}
==Metaproperties==
{| class="sortable" border="1"
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
| [[satisfies metaproperty::extension-closed group property]] || Yes || [[solvability is extension-closed]] || Suppose <math>H</math> is a [[normal subgroup]] of <math>G</math> such that both <math>H</math> and the [[quotient group]] <math>G/H</math> are [[solvable group]]s. Then <math>G</math> is a [[solvable group]].
|-
| [[satisfies metaproperty::quasivarietal group property]] || Yes || [[solvability is quasivarietal]] || subgroup-closed, quotient-closed, and closed under finite direct products (see items  below)
|-
| [[satisfies metaproperty::subgroup-closed group property]] || Yes || [[solvability is subgroup-closed]] || If <math>G</math> is solvable, and <math>H \le G</math> is a [[subgroup]], then <math>H</math> is solvable.
|-
| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[solvability is quotient-closed]] || If <math>G</math> is solvable, and <math>H</math> is a [[normal subgroup]] of <math>G</math>, the [[quotient group]] <math>G/H</math> is solvable.
|-
| [[satisfies metaproperty::finite direct product-closed group property]] || Yes || [[solvability is finite direct product-closed]] || If <math>G_1, G_2, \times, G_n</math> are solvable, the [[external direct product]] <math>G_1 \times G_2 \times \dots \times G_n</math> is also solvable.
|-
| [[satisfies metaproperty::finite normal join-closed group property]] || Yes || [[solvability is finite normal join-closed]] || If <math>G</math> is a group and <math>N_1,N_2,\dots,N_r</math> are all [[solvable normal subgroup]]s of <math>G</math>, the [[join of subgroups]] (in this case also the [[product of subgroups]]) <math>N_1N_2\dots N_r</math> is also solvable.
|}


==Relation with other properties==
==Relation with other properties==
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|}
|}


==Metaproperties==
==Formalisms==
 
{{group extension operator}}
The group property of being solvable can be obtained in either of these equivalent ways:
 
* By applying the [[poly operator]] to the group property of being [[abelian group|abelian]]
* By applying the [[finite normal series operator]] to the group property of being [[abelian group|abelian]]
* By applying the [[finite characteristic series operator]] to the group property of being [[abelian group|abelian]]


{| class="sortable" border="1"
Note that all these three operators have the same effect in the case of abelian groups, though in general they may not have.
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
| [[satisfies metaproperty::extension-closed group property]] || Yes || [[solvability is extension-closed]] || Suppose <math>H</math> is a [[normal subgroup]] of <math>G</math> such that both <math>H</math> and the [[quotient group]] <math>G/H</math> are [[solvable group]]s. Then <math>G</math> is a [[solvable group]].
|-
| [[satisfies metaproperty::quasivarietal group property]] || Yes || [[solvability is quasivarietal]] || subgroup-closed, quotient-closed, and closed under finite direct products (see items  below)
|-
| [[satisfies metaproperty::subgroup-closed group property]] || Yes || [[solvability is subgroup-closed]] || If <math>G</math> is solvable, and <math>H \le G</math> is a [[subgroup]], then <math>H</math> is solvable.
|-
| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[solvability is quotient-closed]] || If <math>G</math> is solvable, and <math>H</math> is a [[normal subgroup]] of <math>G</math>, the [[quotient group]] <math>G/H</math> is solvable.
|-
| [[satisfies metaproperty::finite direct product-closed group property]] || Yes || [[solvability is finite direct product-closed]] || If <math>G_1, G_2, \times, G_n</math> are solvable, the [[external direct product]] <math>G_1 \times G_2 \times \dots \times G_n</math> is also solvable.
|-
| [[satisfies metaproperty::finite normal join-closed group property]] || Yes || [[solvability is finite normal join-closed]] || If <math>G</math> is a group and <math>N_1,N_2,\dots,N_r</math> are all [[solvable normal subgroup]]s of <math>G</math>, the [[join of subgroups]] (in this case also the [[product of subgroups]]) <math>N_1N_2\dots N_r</math> is also solvable.
|}


==Testing==
==Testing==

Revision as of 01:07, 19 June 2011

Definition

Solvable is also called soluble by some people.

Equivalent definitions in tabular format

No. Shorthand A group is termed solvable if ... A group is termed solvable if ...
1 normal series, abelian quotients there is a normal series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group. there exists a series of subgroups such that each is normal in and each is abelian.
2 subnormal series, abelian quotients there is a subnormal series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group. there exists a series of subgroups: such that each is normal in and each is abelian.
3 derived series finite length the derived series reaches the identity in finitely many steps the derived series of , i.e., the series where and is the derived subgroup of its predecessor, reaches the trivial subgroup in finitely many steps.
4 characteristic series, abelian quotients there is a characteristic series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group. there exists a series of subgroups such that each is characteristic in and each is abelian.
5 fully invariant series, abelian quotients there is a fully invariant series of finite length starting from the trivial subgroup and ending at the whole group with each successive quotient group being an abelian group. there exists a series of subgroups such that each is fully invariant in and each is abelian.

Equivalence of definitions

Further information: Equivalence of definitions of solvable group

Examples

VIEW: groups satisfying this property | groups dissatisfying this property
VIEW: Related group property satisfactions | Related group property dissatisfactions

Particular note-worthy examples are given below:


This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this: (facts closely related to Solvable group, all facts related to Solvable group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki

This article defines a group property that is pivotal (i.e., important) among existing group properties
View a list of pivotal group properties | View a complete list of group properties [SHOW MORE]

The version of this for finite groups is at: finite solvable group


Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
extension-closed group property Yes solvability is extension-closed Suppose is a normal subgroup of such that both and the quotient group are solvable groups. Then is a solvable group.
quasivarietal group property Yes solvability is quasivarietal subgroup-closed, quotient-closed, and closed under finite direct products (see items below)
subgroup-closed group property Yes solvability is subgroup-closed If is solvable, and is a subgroup, then is solvable.
quotient-closed group property Yes solvability is quotient-closed If is solvable, and is a normal subgroup of , the quotient group is solvable.
finite direct product-closed group property Yes solvability is finite direct product-closed If are solvable, the external direct product is also solvable.
finite normal join-closed group property Yes solvability is finite normal join-closed If is a group and are all solvable normal subgroups of , the join of subgroups (in this case also the product of subgroups) is also solvable.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions Comparison
abelian group commutator subgroup is trivial abelian implies solvable solvable not implies abelian (see also list of examples)
cyclic group (see also list of examples)
nilpotent group lower central series reaches the identity nilpotent implies solvable solvable not implies nilpotent (see also list of examples) For intermediate notions between solvable group and nilpotent group, click here. nilpotent versus solvable
metabelian group abelian normal subgroup with abelian quotient; derived length two (see also list of examples) For intermediate notions between metabelian group and solvable group, click here.
supersolvable group normal series with cyclic factor groups supersolvable implies solvable solvable not implies supersolvable (see also list of examples) For intermediate notions between solvable group and supersolvable group, click here.
polycyclic group subnormal series with cyclic factor groups polycyclic implies solvable solvable not implies polycyclic (see also list of examples) For intermediate notions between solvable group and polycyclic group, click here.
metacyclic group cyclic normal subgroup with cyclic quotient group (see also list of examples) For intermediate notions between solvable group and metacyclic group, click here.

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions Comparison
hypoabelian group transfinite derived series reaches identity solvable implies hypoabelian hypoabelian not implies solvable |FULL LIST, MORE INFO
imperfect group no nontrivial perfect quotient group solvable implies imperfect imperfect not implies solvable |FULL LIST, MORE INFO
locally solvable group every finitely generated subgroup is solvable
residually solvable group every non-identity element has a non-identity image in some solvable quotient

Conjunction with other properties

Conjunction Other component of conjunction Additional comments
finite solvable group finite group For finite groups, being solvable is equivalent to being polycyclic, and has many other alternative characterizations.
solvable T-group T-group
solvable HN-group HN-group

Formalisms

In terms of the group extension operator

This group property can be expressed in terms of the group extension operator and/or group property modifiers that arise from this operator The group property of being solvable can be obtained in either of these equivalent ways:

Note that all these three operators have the same effect in the case of abelian groups, though in general they may not have.

Testing

The testing problem

Further information: Solvability testing problem

The problem of testing whether a group is solvable or not reduces to the problem of computing its derived series. This can be done when the group is described by means of a generating set, if the normal closure algorithm can be implemented.

GAP command

This group property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for this group property is:IsSolvableGroup
View GAP-testable group properties

To determine whether a group is solvable or not, we cna use the following GAP command: 

IsSolvableGroup(group);

where group may be a definition of the group or a name for a group previously defined.

Study of this notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20F16

The class 20F16 is used for the general theory of solvable groups, while the class 20D10 (coming under 20D which is for finite groups) focusses on finite solvable groups.

Also closely related is 20F19: Generalizations of nilpotent and solvable groups.

References

Textbook references

  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 105 (formal definition)
  • Topics in Algebra by I. N. Herstein, More info, Page 116 (formal definition, introduced between exercises)
  • Algebra by Serge Lang, ISBN 038795385X, More info, Page 18 (definition in paragraph)
  • A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, More info, Page 121 (formal definition)
  • Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, More info, Page 95 (definition in paragraph)
  • An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444, More info, Page 171 (definition introduced in paragraph)
  • A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, More info, Page 194, Definition 3.4.16 (formal definition)
  • Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, More info, Page 102, Definition 7.9 (formal definition)
  • Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716, More info, Page 563
  • Topics in Algebra by I. N. Herstein, More info, Page 116 (formal definition, introduced between exercises)

External links

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