Polycyclic Groups

Polycyclic groups (both finite and infinite) have special generating sets which are compatible with the polycyclic structure of the underlying group. These allow highly efficient computation in polycyclic groups. For example, there are effective algorithms available to compute:

• centralizers of elements
• normalizers of subgroups
• intersections of subgroups
• complements to normal subgroups
• cohomology groups

Methods for this purpose are contained in the main GAP library for finite polycyclic groups and in the Polycyclic package for arbitrary polycyclic groups.

The main GAP library also contains methods for special structure investigations in finite soluble groups such as the computation of Hall subgroups, Sylow systems and maximal subgroups. The packages FORMAT and CRISP extend this to the computation of all kinds of formation and Schunk classes related structures.

The Polycyclic package also contains methods for special structure investigations in infinite polycyclic groups such as the computation of the Hirsch length, the torsion subgroup (if it exists) or a torsion free normal subgroup of finite index. It utilizes an interface to the PARI/GP system from the Alnuth package.

For finite polycyclic groups it is also possible to determine the automorphism group or a minimal generating set. The AutPGrp package contains a particularly effective method to determine the automorphism group of a finite p-group.

The package Polenta allows to determine a polycyclic presentation for polycyclic matrix groups over finite fields or the rational numbers. The main GAP library contains a method to determine a polycyclic presentation for permutation groups. The nq and ANUPQ packages contain methods to determine polycyclic presentations for nilpotent or p-group quotients of finitely presented groups, respectively.