Introduction

This chapter describes how to use polycyclic presentations to compute with p-groups and other finite soluble groups in Magma{}. Several additional functions and features specific to p-groups are described in Chapter p-GROUPS. Any finite soluble group has a subnormal series with cyclic factors. Such a series gives rise to various polycyclic presentations. These polycyclic presentations are useful because the word problem in such presentations can be solved in an algorithmic fashion. In Magma{}, we use the specific form called a power-conjugate presentation (pc-presentation), which is described below. The Magma{} category of groups represented by a power-conjugate presentation (pc-groups for short) is called GrpPC.

Over the past decade a considerable body of efficient algorithms has been developed for computing with soluble groups defined in terms of pc-presentations. It is recommended that the GrpPC representation of a soluble group be used whenever intensive calculation with that group is necessary.

Subsections

• Power-Conjugate Presentations

Power-Conjugate Presentations

Let G be a finite soluble group. A presentation for G of the form

< a_1, ..., a_n | a_j ^(p_j)= w_(jj), 1 <= j <= n, a_j ^(a_i)= w_(ij), 1 <= i < j <= n > where

• p_j is the least prime such that a_j^(p_j) in < a_(j + 1), ..., a_n> for j < n, and a_j^(p_j) is the identity for j = n, and
• w_(ij) is a word in the generators a_(i + 1), ..., a_n,

It is easy to show that every finite soluble group possesses a pc-presentation. If such a presentation satisfies a certain additional condition (the consistency condition) then every element a of G can be written uniquely in the normal form a_1^(alpha_1) ... a_n^(alpha_n), 0 <= alpha_i < p_i for i = 1, ..., n. Given such a pc-presentation for G there exists an algorithm (the collection algorithm), which given an arbitrary word in the pc-generators a_1, ..., a_n, will determine the corresponding normal word. In particular, collection can be used to compute the normal word which is equal to the product of two given normal words, thus implementing the group multiplication.