This paper proposes to represent subgroups of finitely presented groups by their image in a quotient. It gives algorithms for basic operations in this representation and investigates how iteration of this approach can be used to extend known quotients groups with a solvable normal subgroup. Traditionally, algorithms in group theory have represented a subgroup U of a finitely presented group G either by a set of generators (as words in the generators of the full group) or via a coset table. Both representations have distinctive disadvantages: A coset table for U has [G:U] rows, thus the space requirement for storage can be substantial. Practically this restricts this representation to index a few million at most. On the other hand, the representation via a generating set is not particularly suited to calculations and many algorithms compute a coset table as a first step and then effectively work with the coset table, discarding the generating set. Furthermore, for subgroups which are obtained as output of other algorithms, generating sets often are not what a user wants: The standard way to obtain generators is as Schreier generators. As this is a process which “lives” in a free group the number of generators produced grows linearly with the index, and any reduction of this number (if possible at all) would require further work. The first aim of this note is to introduce a different representation, namely as a subgroup of a quotient: Given a group G and a homomorphism ϕ, we represent a subgroup U which is a full preimage under ϕ by the pair (ϕ;Uϕ). We call U a quotient subgroupand the pair (ϕ;Uϕ) the quotient representationof U. Such representation comes naturally to systems such as Magnus[The New York Group Theory Cooperative], quotpic[Holt and Rees 1993] or XGAP[Celler and Neunh¨ offer 1999] with a graphical user interface in which subgroups are objects whose generators or coset table are (usually) never displayed. We will show how such representations can be created, and how to calculate with subgroups in this representation. We will then examine how this approach can be iterated in an attempt to extend known quotients. The approach has been implemented by the author in the system GAP [The GAP Group 2000] and is used there as a default method for many operations for subgroups of finitely presented groups.
['Alexander Hulpke']
Representing Subgroups of Finitely Presented Groups by Quotient Subgroups