Generating groups of nilpotent varieties

Abstract

If is a variety of groups which are nilpotent of class c then is generated by its free group of rank c. It is proved that under certain general conditions cannot be generated by its free group of rank c - 2, and that under certain other conditions is generated by its free group of rank c - 1. It follows from these results that if is the variety of all groups which are nilpotent of class c, then the least value of k such that the free group of of rank k generates is c - 1. This extends known results of L.G. Kovács, M.F. Newman, P.P. Pentony (1969) and F. Levin (1970).