The nilpotency of some groups with all subgroups subnormal.
Let G be a group with all subgroups subnormal. A normal subgroup N of G is said to be G-minimax if it has a finite G-invariant series whose factors are abelian and satisfy either max-G or min- G. It is proved that if the normal closure of every element of G is G-minimax then G is nilpotent and the normal closure of every element is minimax. Further results of this type are also obtained.