A famous theorem of B. H. Neumann states that a group G is central-by-finite if and only if all normalizers of subgroups of G have finite index, and Y. D. Polovickii proved that this is also equivalent to the property that the group G has only finitely many normalizers of subgroups. Here the structure of groups in which all but finitely many normalizers of (in-finite) subgroups have finite index is investigated, and the above results are extended to this more general situation.

normalizer subgroup, almost normal subgroup