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
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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 |
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
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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
|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|
This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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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.
This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
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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.
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
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A finite direct product of polycyclic groups is polycyclic.