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Introduction

This section presents some more advanced techniques available for computing with finitely-presented groups (fp-groups for short) within Magma. The features considered here are regarded as more advanced, either because they are technically or theoretically more complex and they are therefore expected to be used mainly by specialists, or because their efficient (and in a few cases even their merely correct) use requires some more detailed knowledge on the user's part.

Trying to summarise the expected main purpose of the functions described in this section, one could think of two main situations: On the one hand, user written functions, which may benefit from low-level tools for manipulating presentations or words, or which make use of interruptible process versions of some standard Magma functions for fp-groups. On the other hand, the solution of very hard problems, which require careful fine-tuning of the strategy employed or for which some iterative approach, using feedback of information obtained during the computation, is necessary.

The following topics are discussed in detail. First, some rather low-level operations on presentations and elements of fp-groups (words) are described. Then, the features for interactive coset enumeration in Magma are presented. This section also contains the complete description of all the parameters available for controlling the execution of the Todd-Coxeter procedure, which also applies to the appropriate standard functions documented in Chapter FINITELY PRESENTED GROUPS. After that, we describe the process version of the p-quotient algorithm. Note that some care has to be taken when interpreting results obtained with this interactive p-quotient computation; incorrect use of the existing functions may result in incomplete or wrong answers. The chapter ends with a treatise of the soluble quotient algorithm available in Magma. This final section contains a brief review of the theory underlying the soluble quotient algorithm, a description of the parameters available for functions computing soluble quotients, and the documentation of the interactive soluble quotient facilities.

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012