The weak minimal condition on subgroups which fail to be close to normal subgroups

A subgroup H of a group G is commensurable (or close) to a normal subgroup if there is a normal subgroup N of G such that the index |HN:(H∩N)| is finite; if further the subgroup N can be chosen to be contained in H, i.e. if H/HG is finite, then H is called core-finite. We describe the structure of (generalized) soluble groups satisfying the weak minimal condition on subgroups that are not commensurable with a normal subgroup. Our results describe also (generalized) soluble groups satisfying the weak minimal condition on non-(core-finite) subgroups.