# Hirsch length of a polycyclic group

From Groupprops

*This article defines an arithmetic function on a restricted class of groups, namely:* polycyclic groups

## Definition

The **Hirsch length** or **Hirsch number** of a polycyclic group is defined as the number of infinite factors in a polycyclic series for the group. By the Schreier refinement theorem, this number is independent of the choice of the polycyclic series.

The concept of Hirsch length can be generalized to a virtually polycyclic group, for which it is the Hirsch length of any polycyclic normal subgroup of finite index. It has also been generalized to the Hirsch length of an amenable group.