# Polycyclic group

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## Definition

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

1. 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.
2. It is a solvable group and all the factor groups between successive members of its derived series are finitely generated abelian groups.
3. 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 $G$ is said to be polycyclic if there exists a series of subgroups:

$1 = H_0 \triangleleft H_1 \triangleleft H_2 \triangleleft \ldots \triangleleft H_n = G$

where each $H_{i+1}/H_i$ is cyclic.

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW 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
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This is a variation of solvable group|Find other variations of solvable group |
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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
Finitely generated abelian group Finitely generated nilpotent group, Supersolvable group|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
solvable group (any infinitely generated solvable group will do) Finitely generated solvable group, Finitely presented solvable group|FULL LIST, MORE INFO
finitely generated group has a generating set that is finite (any finitely generated non-solvable group will do) Finitely generated solvable group, Finitely presented solvable group, Group in which every subgroup is finitely presented, Group with polynomial-time solvable word problem, Group with solvable word problem, Noetherian group|FULL LIST, MORE INFO
Noetherian group every subgroup is finitely generated Group in which every subgroup is finitely presented|FULL LIST, MORE INFO
finitely generated solvable group finitely generated and solvable not implies polycyclic Finitely presented solvable group|FULL LIST, MORE INFO
finitely presented group has a presentation that uses a finite number of generators and a finite number of relations polycyclic implies finitely presented (any finitely presented non-solvable group will do) Finitely presented solvable group, Group in which every subgroup is finitely presented|FULL LIST, MORE INFO
finitely presented solvable group both finitely presented and solvable (via separate implications for finitely presented and solvable) finitely presented and solvable not implies polycyclic |FULL LIST, MORE INFO

## 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.