In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated).
Polycyclic groups are finitely presented, which makes them interesting from a computational point of view.
Anatoly Maltsev proved that solvable subgroups of the integer general linear group are polycyclic; and later Louis Auslander (1967) and Swan proved the converse, that any polycyclic group is up to isomorphism a group of integer matrices.
A virtually polycyclic group is a group that has a polycyclic subgroup of finite index, an example of a virtual property.
[citation needed] The Hirsch length or Hirsch number of a polycyclic group G is the number of infinite factors in its subnormal series.