# Solvable group

## 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. |

The length of the derived series, and the smallest possible length of a series for any of the other equivalent definitions, is termed the derived length or solvable length of the group.

This definition is presented using a tabular format. |View all pages with definitions in tabular format

### Equivalence of definitions

`Further information: Equivalence of definitions of solvable group, equivalence of definitions of derived length`

## Examples

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

### Extreme examples

- The trivial group is solvable.
- Symmetric group:S3 is the smallest solvable non-abelian group.

### Groups satisfying the property

Here are some basic/important groups satisfying the property:

GAP ID | |
---|---|

Cyclic group:Z2 | 2 (1) |

Cyclic group:Z3 | 3 (1) |

Cyclic group:Z4 | 4 (1) |

Group of integers | |

Klein four-group | 4 (2) |

Symmetric group:S3 | 6 (1) |

Trivial group | 1 (1) |

Here are some relatively less basic/important groups satisfying the property:

GAP ID | |
---|---|

Alternating group:A4 | 12 (3) |

Dihedral group:D8 | 8 (3) |

Direct product of Z4 and Z2 | 8 (2) |

Quaternion group | 8 (4) |

Special linear group:SL(2,3) | 24 (3) |

Symmetric group:S4 | 24 (12) |

Here are some even more complicated/less basic groups satisfying the property:

GAP ID | |
---|---|

Binary octahedral group | 48 (28) |

Central product of D8 and Z4 | 16 (13) |

Dihedral group:D16 | 16 (7) |

Direct product of A4 and Z2 | 24 (13) |

Direct product of D8 and Z2 | 16 (11) |

General linear group:GL(2,3) | 48 (29) |

Generalized quaternion group:Q16 | 16 (9) |

M16 | 16 (6) |

Mathieu group:M9 | 72 (41) |

Nontrivial semidirect product of Z4 and Z4 | 16 (4) |

Semidihedral group:SD16 | 16 (8) |

### Groups dissatisfying the property

Here are some basic/important groups that do not satisfy the property:

Here are some relatively less basic/important groups that do not satisfy the property:

GAP ID | |
---|---|

Alternating group:A5 | 60 (5) |

Alternating group:A6 | 360 (118) |

Free group:F2 | |

Projective special linear group:PSL(3,2) | 168 (42) |

Special linear group:SL(2,5) | 120 (5) |

Symmetric group:S5 | 120 (34) |

Symmetric group:S6 | 720 (763) |

Here are some even more complicated/less basic groups that do not satisfy the property:

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 definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated 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

## Relation with other properties

### Stronger properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions | Comparison |
---|---|---|---|---|---|

hypoabelian group | transfinite derived series reaches identity; equivalent to solvable in the finite case |
solvable implies hypoabelian | hypoabelian not implies solvable | Residually solvable group|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 equivalent to solvable in the finite case |
||||

residually solvable group | every non-identity element has a non-identity image in some solvable quotient equivalent to solvable in the finite case |

### 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:

- By applying the poly operator to the group property of being abelian
- By applying the finite normal series operator to the group property of being abelian
- By applying the finite characteristic series operator to the group property of being abelian

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 ofGroups, 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

Book | Page number | Chapter and section | Contextual information | View |
---|---|---|---|---|

Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347^{More info} |
105 | formal definition | ||

Topics in Algebra by I. N. Herstein^{More info} |
116 | formal definition, introduced between exercises | ||

Algebra by Serge Lang, ISBN 038795385X^{More info} |
18 | definition in paragraph | ||

A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info} |
121 | formal definition | ||

Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261^{More info} |
95 | definition in paragraph | ||

An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444^{More info} |
171 | definition in paragraph | ||

A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907^{More info} |
194 | Definition 3.4.16 | formal definition | |

Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189^{More info} |
102 | Definition 7.9 | formal definition | |

Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716^{More info} |
563 | |||

Topics in Algebra by I. N. Herstein^{More info} |
116 | formal definition, introduced between exercises |