# Polycyclic group

## Contents

## Definition

A group is said to be **polycyclic** if it satisfies the following equivalent conditions:

- It has a subnormal series (wherein each member is normal in its successor) of finite length, starting from the trivial group and ending at the whole group, such that all the successive quotients are cyclic.
- It is a solvable group and all the factor groups between successive members of its derived series are finitely generated abelian groups.
- It is both a Noetherian group (also called a slender group, i.e., every subgroup is finitely generated) and a solvable group.

### Definition with symbols

A group is said to be **polycyclic** if there exists a series of subgroups:

where each is cyclic.

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: Noetherian group and solvable group

View other group property conjunctions OR view all group properties

This is a variation of solvable group|Find other variations of solvable group |

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 Polycyclic group, all facts related to Polycyclic group) |Survey articles about this | Survey articles about definitions built on thisVIEW RELATED: Analogues of this | Variations of this | Opposites of this |

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## History

### Origin of the concept

Hirsch, in the years 1938-1954, obtained results on polycylic groups, but he used the term S-group for them.

### Origin of the term

The term **polycyclic group** was first used by Hall in 1954. It is now the commonly accepted term.

## Relation with other properties

### Stronger properties

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

Cyclic group | Characteristically metacyclic group, Characteristically polycyclic group, Finitely generated abelian group, Metacyclic group|FULL LIST, MORE INFO | |||

Metacyclic group | |FULL LIST, MORE INFO | |||

Supersolvable group | |FULL LIST, MORE INFO | |||

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

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

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

### Weaker properties

## Metaproperties

### Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property

View a complete list of subgroup-closed group properties

Any subgroup of a polycyclic group is polycyclic. The subnormal serise for the subgroup is obtained by intersecting with it the subnormal series of the whole group.

### Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property

View a complete list of quotient-closed group properties

Any quotient of a polycyclic group is polycyclic. The subnormal series for the quotient is obtained by taking the image of the subnormal series for the whole group, via the quotient map.

### Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property

View other finite direct product-closed group properties

A finite direct product of polycyclic groups is polycyclic.