Application of computational tools for finitely presented groups

Abstract

Computer based techniques for recognizing finitely presented groups are quite powerful. Tools available for this purpose are outlined. They are available both in stand-alone programs and in more comprehensive systems.A general computational approach for investigating finitely presented groups by way of quotients and subgroups is described and examples are presented. The techniques can provide detailed information about group structure. Under suitable circumstances a finitely presented group can be shown to be soluble and its complete derived series can be determined, using what is in effect a soluble quotient algorithm.

A Polycyclic Quotient Algorithm

“…Effective algorithms for determining p-quotients were originated by the work of Macdonald (1973Macdonald ( , 1974. His techniques were later extended by Wamsley (1974), Bayes et al (1974), Newman (1976) and Havas and Newman (1980). Nickel (1994) developed and implemented an algorithm to compute the nilpotent quotients for finitely presented groups.…”

“…Computing nilpotent quotients is also covered in depth in Chapter 11 of Sims (1994). Research on determining finite solvable quotients can be found in Plesken (1987), Niemeyer (1994), Wegner (1992) and Havas and Robertson (1994). As for determining polycyclic quotients, in 1981, Baumslag, et al described a polycyclic quotient algorithm in Baumslag et al (1981a) and Baumslag et al (1981b).…”

“…a, a, a], [b, a, a, b], [b, a, b, b] The elements [b, a, a, a], [b, a, a, b] and [b, a, b, b] which appear as relators in the presentation are precisely the basic commutators of weight four defined in (Hall, 1959). The fact that G is the free nilpotent group of rank two, class three is proved by Groves and the result appears in Havas and Nicholson (1976). We use pcqa to compute the derived series of G. However, direct computation is not successful.…”

A Reidemeister-Schreier theorem for finitely $L$-presented groups

“…The Reidemeister rewriting allows us to rewrite the relations of the group to relations of the subgroup [16,31,21]. Note that a method to compute a finite presentation for a finite index subgroup can be applied in the investigation of the structure of a group by its finite index subgroups; see [17].…”

“…Our proof of Theorem 1.1 is constructive and it yields a finite L-presentation for the subgroup. These finite L-presentations can be applied in the investigation of the underlying groups as the methods in [17] suggest for finitely presented groups. Notice that Theorem 1.1 was already posed in Proposition 2.9 of [1].…”