Infinite groups satisfying the normalizer condition for nonprimary subgroups

In this paper we study infinite groups satisfying the normalizer condition for nonprimary subgroups. We show, in particular, that a nonprimary periodic group satisfying this condition is locally finite if the intersection of all its nonprimary subgroups is finite. We establish the local nilpotency of a nonperiodic group satisfying the normalizer condition for nonprimary subgroups. This implies the theorem of S. N. Chernikov which states that a nonperiodic group in which each infinite proper subgroup is different from its normalizer satisfies the normalizer condition.