The Two - Generator Subgroups of One - Relator Groups with Torsion

The main aim of this paper is to show that every two-generator subgroup of any one-relator group with torsion is either a free product of cyclic groups or is a one-relator group with torsion. This result is proved by using techniques for reducing pairs of elements in certain HNN groups. These techniques not only apply to one-relator groups with torsion but also to a large number of other groups, and some additional applications of the techniques are included in the paper. In particular, examples are given to show that the following result of K. Honda is no longer true for infinite groups: if g is a commutator in a finite group G then every generator of sgp{ g) is a commutator in G. This confirms a conjecture of B. H. Neumann.