# Supersolvable group

From Groupprops

## Definition

### Symbol-free definition

A group is said to be **supersolvable** if it has a normal series (wherein all the members are normal in the whole group) of finite length, starting from the trivial group and ending at the whole group, such that all the successive quotients are cyclic.

### Definition with symbols

A group is said to be **supersolvable** if there exists a normal series:

where each and further, each is cyclic.

## Examples

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

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

pseudovarietal group property | Yes | supersolvability is pseudovarietal | Supersolvability is closed under taking subgroups, quotients, and finite direct products (see below). |

subgroup-closed group property | Yes | supersolvability is subgroup-closed | If is supersolvable, and is a subgroup, is supersolvable. |

quotient-closed group property | Yes | supersolvability is quotient-closed | If is supersolvable, and is a normal subgroup of , the quotient group is supersolvable. |

finite direct product-closed group property | Yes | supersolvability is finite direct product-closed | If are all supersolvable, the external direct product is also supersolvable. |

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 Supersolvable group, all facts related to Supersolvable 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]

This is a variation of solvability|Find other variations of solvability |

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

## Relation with other properties

### Conjunction with other properties

Conjunction | Other component of conjunction |
---|---|

finite supersolvable group | finite group |

### Stronger properties

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

finite abelian group | Finitely generated abelian group|FULL LIST, MORE INFO | |||

finitely generated abelian group | Finitely generated nilpotent group|FULL LIST, MORE INFO | |||

finite nilpotent group | Finitely generated nilpotent group|FULL LIST, MORE INFO | |||

finitely generated nilpotent group | |FULL LIST, MORE INFO | |||

finite supersolvable group | |FULL LIST, MORE INFO |

### Weaker properties

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

polycyclic group | has a subnormal series with cyclic quotients | |FULL LIST, MORE INFO | ||

solvable group | Polycyclic group|FULL LIST, MORE INFO | |||

finitely generated solvable group | Polycyclic group|FULL LIST, MORE INFO | |||

group with nilpotent derived subgroup | Supersolvable implies nilpotent commutator subgroup |

## Study of the notion

### Mathematical subject classification

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