Infinite groups with short balanced chains of subgroups

A subgroup H of a group G is said to be abnormal in G if g∈〈H,Hg〉 for each element g∈G. A balanced chain connecting a subgroup H to the group G is a chain of subgroups H=H0⩽H1⩽⋯⩽Hn−1⩽Hn=G such that for each j, 0⩽j⩽n−1, either Hj is normal in Hj+1, or Hj is abnormal in Hj+1. In the current paper we study the groups whose subgroups connected to the group by balanced chains of length n⩽2. More precisely, the subjects of investigation are the groups whose every subgroup is abnormal in its normal closure and the groups in which all subgroups have abnormal normalizers. In passing, some interesting new characterizations of the groups with transitivity of normality have been obtained.