Polycyclic group

Polycyclic group

In mathematics, especially in the area of abstract algebra known as group theory, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated).

Equivalently, a group "G" is polycyclic if and only if it admits a subnormal series with cyclic factors, that is a finite set of subgroups, let's say "G""1", ..., "G""n+1" such that
* "G""1" coincides with "G"
* "G""n+1" is the trivial subgroup
* "G""i+1" is a normal subgroup of "G""i" (for every "i" between "1" and "n")
* and the quotient group "G""i" / "G""i+1" is a cyclic group (for every "i" between "1" and "n")

A metacyclic group is, according to the current standard definition [http://mathworld.wolfram.com/MetacyclicGroup.html] , a polycyclic group with "n" ≤ 2, or in other words an extension of a cyclic group by a cyclic group.

Polycyclic groups are finitely presented, and this makes them very interesting from a computational point of view.

Examples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent groups, and finite solvable groups.

Polycyclic-by-finite groups

A virtually polycyclic group is a group that has a polycyclic subgroup of finite index, 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.

In the textbook harv|Scott|1964|loc=Ch 7.1 and some papers , an M-group refers to what is now called a polycyclic-by-finite group, which by Hirsch's theorem can also be expressed as a group which has a finite length normal series with each factor a finite group or an infinite cyclic group.

These groups are particularly interesting because they are the only known examples of noetherian group rings harv|Ivanov|1989, or group rings of finite injective dimension.Fact|date=February 2008

ee also

*supersolvable group

References

*Citation | last1=Ivanov | first1=S. V. | title=Group rings of Noetherian groups | id=MathSciNet | id = 1051052 | year=1989 | journal=Akademiya Nauk SSSR. Matematicheskie Zametki | issn=0025-567X | volume=46 | issue=6 | pages=61–66
*


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