LAUSR

All groups in the family of Baumslag-Solitar groups (i.e., groups of the form G ( m,n ) = 〈 a, b ; a −1 b m a = b n 〉, where m and n are nonzero integers) for which the residual nilpotence condition holds if and only if the residual p -finiteness condition holds for some prime number p are described. It has turned out, in particular, that the group G ( p r , − p r ), where p is an odd prime and r ≥ 1, is residually nilpotent, but it is residually q -finite for no prime q . Thus, an answer to the existence problem for noncyclic one-relator groups possessing such a property (formulated by McCarron in his 1996 paper) is obtained. A simple proof of the statement that an arbitrary residually nilpotent noncyclic one-relator group which has elements of finite order is residual p -finite for some prime p , which was announced in the same paper of McCarron, is also given.