Cardinal invariants of infinite groups

LetG be a group. CallG akC-group if every element ofG has less thank conjugates. Denote byP(G) the least cardinalk such that any subset ofG of sizek contains two elements which commute.It is shown that the existence of groupsG such thatP(G) is a singular cardinal is consistent withZFC. So is the existence of groupsG which are notkC but haveP(G) wherek is a limit cardinal. On the other hand, ifk is a singular strong limit cardinal, andG is akC-group, thenP(G)k. This partially answers questions, and improves results, of Faber, Laver and McKenzie.The present paper has non-trivial intersection with the author's Diplomarbeit written under the direction of Prof. Ulrich Felgner at the University of Tübingen, W. Germany, 1988