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
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.
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.