Let Gbe an infinite group with one conjugate class of non-subnormal subgroups. It is shown that the following are equivalent: Gis locally nilpotent; Gsatisfies the normalizer condition; Gis a Baer group. If G is a group with the given property, then Gneeds not be locally finite. If Gis an inner finite group, then Gis an extension of the center by a Tarski p-group. Moreover, Gis a simple inner finite group with the given property if and only if Gis a Tarski p-group.

locally nilpotent, non-subnormal subgroup, conjugate, locally finite, inner-finite.