A class of deficiency zero soluble groups of derived length 4

Abstract

SynopsisIn this paper we study a class of 2-generator 2-relator groups G(m) and show that they are all finite. Moreover, two infinite subclasses are soluble of derived length 4.

On the commutator lengths of certain classes of finitelypresented groups

“…(3.6) By an inductive method we may show that g n = (1 + f n−2 ) 2 − f n−1 (−1 + f n−3 ) and then x gn = 1. A similar proof exists for y gn = 1 and since G 1 is of order 2g n (see [5]), g n is the order of y. Now consider the relations [x 2 , y] = [y 2 ,x] = 1 and conclude that the words xy −1+gn (or y −1+gn x) are the words of largest length; that is, c(G…”

“…The groups G 1 and G 2 are finite and nonmetabelian soluble groups for many values of n (see [4,5]). The following propositions are our main results on the commutator lengths of these groups.…”

“…The groups G i (i = 1,2,3,4) are soluble and have been studied for their structures and orders in [3][4][5]. These groups are generalizations of the well-known Coxeter groups.…”

A Class of 3-Generated Groups

“…Detailed information on the deficiency of a presentation of a finitely presented group may be found [1][2][3][4][5][6]. In this paper, the Modified Todd-Coxeter enumeration algorithm is used as given in [7][8] to get a presentation for all subgroups of G(n).…”

“…In this paper, the Modified Todd-Coxeter enumeration algorithm is used as given in [7][8] to get a presentation for all subgroups of G(n). Further an application of this algorithm may be found in [4][5]9]. Our notation is standard and follows [6]; in our calculations to certain results of [4,6,7,10] is referred.…”