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# 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 G1, ..., Gn+1 such that

• G1 coincides with G
• Gn+1 is the trivial subgroup
• Gi+1 is a normal subgroup of Gi (for every i between 1 and n)
• and the quotient group Gi / Gi+1 is a cyclic group (for every i between 1 and n)

A metacyclic group is, according to the current standard definition , 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 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 , or group rings of finite injective dimension.