Transitivity-Systems of Certain One-Relator Groups

Abstract

For different non-negative integers n the pairs $\left( {{a^{{2^n}}},b} \right)$, which are generating pairs of G = gp(a, b; b ⁻¹ a ² b = a ³), are shown to lie in different T-systems of G. The presentation $\left( {x,y;{x^{ - 1}}{{\left[ {x,y} \right]}^2},\left[ {x,{x^{{y^n}}}} \right]} \right)$ is associated with the generating pair $\left( {{a^{{2^n}}},b} \right)$ of G for each positive integer n. It is shown that a two generator group which is an extension of an abelian group by an infinite cyclic group has only one T-system of generating pairs. This implies that the quotient group of G by its second derived group has only one T-system of generating pairs.