Groups with H/core(H) satisfying max or min for all subgroups H

Let G be a group in which satisfies min, the minimal condition on subgroups, for all subgroups H of G, where HG denotes the normal core of H in G. We show that, with the additional hypothesis that G has all of its periodic images locally finite, G has an abelian normal subgroup A such that has min; further consequences are then established. With the maximal condition replacing the minimal condition, a similar conclusion does not hold: we give an example of a (torsion-free) nilpotent group G such that satisfies max for all subgroups H but G is not abelian-by-max.