Finitely Presented Groups [HB 20, 21]

 

Changes:

• Coercion of elements of a finitely presented group into a subgroup defined by a set of generating words is now possible to some limited extent. Words in the supergroup which are freely equivalent to one of the defining generators of the subgroup or the inverse of such a generator can be coerced.
• Coercing elements of a finitely presented group into a subgroup defined in terms of Schreier generators is now possible for both unsimplified and simplified presentations.
• Known information about normality of subgroups is stored more consistently, yielding a better performance of functions like IsNormal, Normaliser, NormalClosure and Core. The caching of coset tables has been improved in general.
• The functions computing the normal closure of a subgroup were revised and now are able to produce answers in more cases.
• Several functions can now handle finitely presented groups which do not have a presentation assigned, e.g. subgroups defined by a coset table. Among them are GModule, GModulePrimes, subgroup constructor and normal closure constructor.
• The functions for the representation theory of fp-groups have been revised completely. The new functions GModulePrimes and GModule are much faster than the old versions and are less restrictive with regard to the maximal size of possible computations. The function GModule now returns -- like the corresponding functions for other categories of groups -- an epimorphism onto the constructed module.
• Some bugs and memory leaks have been fixed. In particular, the system is more tolerant against invalid user input.
• A new version of Éamonn O'Brien's p-quotient algorithm has been installed. In particular, the interactive computation of p-quotients has been improved considerably in this version.

• A new version of George Havas' coset enumerator ACE has been installed. Controlled by a wide range of parameters, it enables the user to exert more precise control over the Todd-Coxeter procedure than the old version did. This set of parameters is accepted by various functions which indirectly invoke a coset enumeration, e.g. Order, Index etc. All functions can handle the old parameters, ensuring backwards compatibility. A global set of parameters, controlling coset enumerations invoked indirectly, e.g. by computing intersections of subgroups, has been introduced. This provides improved performance for many standard operations on finitely presented groups.

• The functions for simplifying a presentation using Tietze transformations have been revised completely. The new version uses an improved simplification strategy and allows more control about the simplification procedure by providing an extended set of parameters. Functions SetOptions and ShowOptions have been introduced to control the parameter settings of a simplification process. The new function SimplifyLength allows partial simplification of a presentation. The elimination of generators is stopped, if further eliminations start to increase the total length of the presentation. This not only may save time, but the resulting presentations in general are more suitable for coset enumeration.

• The functions Simplify and Rewrite now return an isomorphism from the original group onto the constructed presentation. The group returned is created as a subgroup of the original group, which allows coercing elements from the original group into the new group.

• The function SchreierGenerators now by default applies a heuristic method for removing redundant generators. (This feature can be turned off.) Since this reduction is also heavily used in functions internally constructing a generating set of a group, the performance of several functions of the module is thereby improved.

• The function Order has been revised completely. New strategies for computing the order of a group or proving its infiniteness are applied. The new function more frequently completes successfully and is in general faster than the old version.

New features:

• Interactive coset enumeration based on the ACE program of Havas.
• For homomorphisms with a domain of type GrpFP, an attempt is made to compute the kernel, when the kernel is accessed directly or indirectly as a consequence of another function call. This is done by trying to construct a regular permutation representation of the image and, if successful, defining a subgroup of the domain using the obtained coset table. If the kernel can be constructed, computing preimages of subgroups of the codomain is possible.
• The new function BraidGroup returns the braid group on a given number of generators as a finitely presented group.
• The new functions AbelianQuotient and ElementaryAbelianQuotient return the abelian quotient and the p-elementary abelian quotient for a given prime p, respectively, of a finitely presented group.
• The new function DihedralGroup can now also construct the infinite dihedral group.
• The new function PresentationLength returns the total length of the relators of a presentation.
• The new function ReduceGenerators tries to obtain a presentation of a group on fewer generators. It is particularly useful for eliminating redundant generators in subgroups of finitely presented groups obtained as preimages under a homomorphism.

Bug fixes:

• A bug in Darstellungsgruppe has been fixed, which was responsible for incorrect answers for groups having an infinite abelian quotient.