On one-relator groups having elements of finite order

Abstract

Let <italic>G</italic> be a one-relator group having torsion. It is easy to show that there exists a normal subgroup <italic>N</italic> which is of finite index and torsion-free. We prove that <italic>N</italic> is free iff <italic>G</italic> is the free product of a free group and a finite cyclic group; <italic>N</italic> is a proper free product iff <italic>G</italic> is a proper free product of a free group and a one-relator group. In the proof, use is made of the following result: the elements of finite order in <italic>G</italic> generate a group which is the free product of conjugates of a finite cyclic group.