Finitely Presented Groups
Magma implements Todd-Coxeter coset enumeration, Reidemeister-Schreier rewriting, and low-index subgroup algorithms for finitely presented groups. Abelian, polycyclic, and profinite quotients can all be computed.
Contributors
Group Theory: Finitely-Presented Groups
See also the subsection Group Theory: Soluble Groups.
A package for testing whether a finitely presented group is large, developed by Jack Button (Cambridge), was installed in Magma V2.20 (2013).
A new algorithm for computing all normal subgroups of a finitely-presented group up to a specified index has been designed and implemented by David Firth and Derek Holt (both Warwick).
The function for determining whether a given finite permutation group is a homomorphic image of a finitely presented group has been implemented in C by Volker Gebhardt from a Magma language prototype developed by Derek Holt (Warwick). A variant developed by Derek, allows one to determine whether a small soluble group is a homomorphic image.
Versions of Magma from V2.8 onwards employ the Advanced Coset Enumerator designed by George Havas (Queensland) and implemented by Colin Ramsay (also of Queensland). George has also contributed to the design of the machinery for finitely presented groups.
Derek Holt (Warwick) developed a modified version of his program, kbmag, for inclusion within Magma. The Magma facilities for groups and monoids defined by confluent rewrite systems, as well as automatic groups, are supported by this code.
Derek Holt (Warwick) has provided a Magma implementation of his algorithm for testing whether two finitely presented groups are isomorphic.
A small package for working with subgroups of free groups has been developed by Derek Holt (Warwick). He has also provided code for computing the automorphism group of a free group.
An improved version of the Plesken–Fabianska algorithm for finding L2-quotients of a finitely presented group was designed and implemented by Sebastian Jambor (Aachen). Among other features this version removes the restriction to two generators.
The algorithms designed and implemented by Sebastian Jambor (Aachen) for finding L3- and U3-quotients of a finitely-presented group have been incorporated into Magma.
The function for finding all subgroups having index less than a specified finite bound was implemented by Catherine Playoust (Magma). The implementation was based on a Pascal program written by Charlie Sims (Rutgers).