Refined solvable presentations for polycyclic groups

TL;DR

This paper introduces refined solvable presentations for polycyclic groups, improving computational efficiency by refining subgroup series and providing effective criteria for their consistency in algebra systems.

Contribution

It presents a new class of polycyclic presentations called refined solvable presentations, with criteria for their consistency and improved computational performance.

Findings

Refined solvable presentations can be effectively described using presentation maps.

Consistency criteria for these presentations are established.

Implementation shows faster performance than existing methods.

Abstract

We describe a new type of polycyclic presentations, that we will call refined solvable presentations, for polycyclic groups. These presentations are obtained by refining a series of normal subgroups with abelian sections. These presentations can be described effectively by presentation maps which yield the basis data structure to define a polycyclic group in computer-algebra-systems like {\scshape Gap} or {\scshape Magma}. We study refined solvable presentations and, in particular, we obtain consistency criteria for them. This consistency implementation demonstrates that it is often faster than the existing methods for polycyclic groups.