Subsections

• Introduction
• Power-conjugate Presentations

Computation in an arbitrary finitely presented group offers considerable difficulties because of the unsolvability of the word problem for groups. However, the class of groups defined by means of a polycyclic presentation has a soluble word problem. In particular, every soluble group possesses a polycyclic presentation. We consider the class of finite soluble groups where the groups are defined by polycyclic presentations. In this situation we will follow existing convention and use the term power-conjugate presentation in preference to the term polycyclic presentation.

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.

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 pcp representation of a soluble group be used whenever intensive calculation with soluble groups is necessary.

Magma will compute an internal pc-presentation which will be used for internal computation, but the user's presentation will be used for all input and output. The recommended way to access the internal presentation is via the intrinsic ConditionedGroup.

Since soluble groups are often given in a form other than by a pc-presentation, it is usually necessary to explicitly construct such a presentation. If G is a finitely presented group known to be a p-group, then the p-quotient algorithm may be used to construct a pc-presentation for G. Unfortunately, if G is a finitely presented soluble group, not a p-group, there is no good algorithm available at the time of writing for constructing a pc-presentation for G from the given presentation. It is hoped that such an algorithm will be developed in the near future. If G is given as a permutation group, then Magma contains a standard function, PCGroup, which will construct a pc-presentation.