Polycyclic Groups

Finite soluble groups and infinite polycyclic groups have special generating systems chosen in accordance with a subnormal series with cyclic factors. These allow highly efficient computation in such groups using collection of words in these generators. This general idea can be used both with such polycyclic generating systems consisting e. g. of permutations and with polycyclic presentations for such polycyclic generating systems.

The main GAP library contains a wide variety of functions utilizing these methods and being able to compute efficiently

• conjugacy classes,
• centralizers,
• normalizers,
• intersections and complements of subgroups as well as
• cohomology groups.

The two packages FORMAT and CRISP give access to the highly developed theory of the subgroup structure of finite soluble groups related to notions such as formations, Schunck classes, and homomorphs.

The package Polycyclic extends the possibility of detailed structure analysis to infinite polycyclic groups allowing to compute e. g. Hirsch length and torsion subgroup. It utilizes an interface Alnuth to the KANT / KASH system for algebraic number theory.

The package Polenta allows to find polycyclic presentations for matrix groups.