mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a polycyclic group is a

solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates ...

that satisfies the maximal condition on

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, which makes them interesting from a computational point of view.

Terminology

Equivalently, a group ''G'' is polycyclic if and only if it admits a

subnormal series In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups: :1 = A_0 \leq A_1 \leq \cdots \leq A_n = G where 1 is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simple ...

with cyclic factors, that is a finite set of subgroups, let's say ''G''₀, ..., ''G''_''n'' such that * ''G''_''n'' coincides with ''G'' * ''G''₀ is the trivial subgroup * ''G''_''i'' is a normal subgroup of ''G''_''i''+1 (for every ''i'' between 0 and ''n'' - 1) * and the quotient group ''G''_''i''+1 / ''G''_''i'' is a

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

(for every ''i'' between 0 and ''n'' - 1) A

metacyclic group In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group ''G'' for which there is a short exact sequence :1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1,\, where ''H'' and ''K'' ar ...

is a polycyclic group with ''n'' ≤ 2, or in other words an

extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...

of a cyclic group by a cyclic group.

Examples

Examples of polycyclic groups include finitely generated abelian groups, finitely generated

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the class ...

groups, and finite solvable groups.

Anatoly Maltsev Anatoly Ivanovich Maltsev (also: Malcev, Mal'cev; Russian: Анато́лий Ива́нович Ма́льцев; 27 November N.S./14 November O.S. 1909, Moscow Governorate – 7 June 1967, Novosibirsk) was born in Misheronsky, near Moscow, and ...

proved that solvable subgroups of the integer

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

are polycyclic; and later

Louis Auslander Louis Auslander (July 12, 1928 – February 25, 1997) was a Jewish American mathematician. He had wide-ranging interests both in pure and applied mathematics and worked on Finsler geometry, geometry of solvmanifolds and nilmanifolds, locally aff ...

(1967) and Swan proved the converse, that any polycyclic group is up to isomorphism a group of integer matrices. The holomorph of a polycyclic group is also such a group of integer matrices.

Strongly polycyclic groups

A polycyclic group ''G'' is said to be strongly polycyclic if each quotient ''G''_''i''+1 / ''G''_''i'' is infinite. Any subgroup of a strongly polycyclic group is strongly polycyclic.

Polycyclic-by-finite groups

A virtually polycyclic group is a group that has a polycyclic subgroup of finite

index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...

, an example of a virtual property. Such a group necessarily has a ''normal'' polycyclic subgroup of finite index, and therefore such groups are also called polycyclic-by-finite groups. Although polycyclic-by-finite groups need not be solvable, they still have many of the finiteness properties of polycyclic groups; for example, they satisfy the maximal condition, and they are finitely presented and

residually finite {{unsourced, date=September 2022 In the mathematical field of group theory, a group ''G'' is residually finite or finitely approximable if for every element ''g'' that is not the identity in ''G'' there is a homomorphism ''h'' from ''G'' to a fini ...

. In the textbook and some papers, an M-group refers to what is now called a polycyclic- by-

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

, which by Hirsch's theorem can also be expressed as a group which has a finite length subnormal series with each factor a finite group or an infinite

. These groups are particularly interesting because they are the only known examples of

Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...

group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...

s , or group rings of finite injective dimension.

Hirsch length

The Hirsch length or Hirsch number of a polycyclic group ''G'' is the number of infinite factors in its subnormal series. If ''G'' is a polycyclic-by-finite group, then the Hirsch length of ''G'' is the Hirsch length of a polycyclic

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...

''H'' of ''G'', where ''H'' has finite

in ''G''. This is independent of choice of subgroup, as all such subgroups will have the same Hirsch length.

References

* *

Notes

{{reflist Properties of groups Solvable groups