# Introduction

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`.

This chapter describes how to use polycyclic presentations to compute with $p$-groups and other finite soluble groups in Magma. While most functions apply to any soluble group, a small number of functions specific to $p$-groups are identified in the text.

Over the past two decades 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.

## Power-Conjugate Presentations

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

$$
< a_1, \ldots,a_n\ |\ a_j ^{p_j}= w_{jj}, \quad 1\leq j \leq n,
\quad a_j ^{a_i}= w_{ij},\quad  1 \leq i < j\leq n >
$$

where

**(i)**
$p_j$ is the least prime such that $a_j^{p_j} \in < a_{j+1}, \ldots,  a_n>$ for $j < n$, and $a_j^{p_j}$ is the identity for $j = n$, and

**(ii)**
$w_{ij}$ is a word in the generators $a_{i+1}, \ldots, a_n$,

will be called a *power-conjugate presentation (pc-presentation)* for $G$. The generators of $G$ corresponding to $a_1, \ldots, a_n$ in this presentation are known as a *power-conjugate generating sequence (pc-generators)* for $G$.

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*

$$
a_1^{\alpha_1}\ldots a_n^{\alpha_n}, 0 \leq \alpha_i < p_i
\quad\hbox{\rm for }
i = 1, \ldots, n.
$$

Given such a pc-presentation for $G$ there exists an algorithm (the *collection algorithm*), which given an arbitrary word in the pc-generators $a_1, \ldots, a_n$, 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.