D. I. Zajtsev [Ukr. Math. J. 20 (1968), 408-416 (1969); translation from Ukr. Mat. Zh. 20, 472-482 (1968; Zbl 0212.04801)] characterized the locally nilpotent groups satisfying either the weak minimal or weak maximal condition to be nilpotent minimax. Being inspired by this result the authors of the article under review are able to prove that in Baer groups the weak minimal and the weak maximal conditions for normal subgroups are equivalent. Moreover, these groups are nilpotent. It follows from L. A. Kurdachenko [Sib. Mat. Zh. 20, 1068-1076 (1979; Zbl 0425.20025); translation in Sib. Math. J. 20, 755-761 (1980)] that in the wider class of Gruenberg groups this assertion is false.