INTRODUCTION

A group G is a PC-group (power commutative) if for any elements a,b in G of infinite orders, proper powers of a,b permute only if a,b permute, i.e. [am,bn] = 1, m,n ≠ 0, implies [a,b] = 1. G is a CT-group (commutationtransitive) if for any a,b,c in G, b noncentral, [a,b] = [b,c] = 1 implies [a,c] = 1. In [1], B. Fine notes that free groups and Fuchsian groups are both PC and CT groups and poses the problem of finding further classes of PC and of CT groups (the latter problem is attributed to L. Greenberg). In this note we construct various examples of PC and CT groups and consider generalized free products and HNN-extensions of such groups.

GL(2,K)

The example of a free nilpotent group of class at least 4 shows that not all PC-groups are CT-groups. Below (Theorem 2) we give an example of a CT-group which is not a PC-group. On the other hand, a group with presentation (a,b,t : tat-1 = a-1), which is an HNN-extension of a free group is neither a CT-group nor a PC-group.

In this section we consider examples arising from linear groups. Let K be any field and GL(2,K) the general linear group of 2 x 2 matrices over K. If K contains elements of infinite orders GL(2,K) contains elements of infinite order which are non-central but have finite powers which are central, and, hence, GL(2,K) will not be a PC-group.