Maria De Falco

On the upper central series of infinite groups

Two relevant theorems by R. Baer and P. Hall show that a group is finite over a term with finite ordinal type of its upper central series if and only if it is finite-by-nilpotent. Extending these results, we prove here that if is any group, the hypercentre factor group is finite if and only if contains a finite normal subgroup such that is hypercentral (where the hypercentre of is defined as the last term of its upper central series).

Groups with normality conditions for subgroups of infinite rank

A well-known theorem of B. H. Neumann states that a group has finite conjugacy classes of subgroups if and only if it is central-by-finite. It is proved here that if G is a generalized radical group of infinite rank in which the conjugacy classes of subgroups of infinite rank are finite, then every subgroup of G has finitely many conjugates, and so G=Z(G) is finite. Corresponding results are proved for groups in which every subgroup of infinite rank has fiznite index in its normal closure, and for those in which every subgroup of infinite rank is finite over its core.

Groups with many subgroups having a transitive normality relation

A group is said to be aT-group if all its subnormal subgroups are normal. The structure of groups satisfying the minimal condition on subgroups that do not have the propertyT is investigated. Moreover, locally soluble groups with finitely many conjugacy classes of subgroups which are notT-groups are characterized.

Groups with Finitely Many Normalizers of Non-polycyclic Subgroups

The structure of locally graded groups with flnitely many normalizers of non- polycyclic subgroups is investigated. In particular, it is proved that such groups either are polycyclic or have • Cernikov commutator subgroups. 2000 Mathematics Subject Classiflcation: 20F24

Groups with Finiteness Conditions on Commutators

The structure of groups for which certain sets of commutator subgroups are finite is investigated. In particular, the relation between such groups and groups with finite conjugacy classes of elements is discussed.

Locally Finite Products of Totally Permutable Nilpotent Groups

A group G = AB is said to be totally factorized by its subgroups A and B if XY = Y X for all subgroups X of A and Y of B. It is known that any flnite group totally factorized by supersoluble subgroups is supersoluble, and that a flnite group totally factorized by nilpotent subgroups is abelian-by-nilpotent. This latter result is extended here to certain classes of inflnite groups. 2000 Mathematics Subject Classiflcation: 20F19

Groups Satisfying the Maximal Condition on Non-modular Subgroups

In this paper, (generalized) soluble groups for which the set of non-modular subgroups verifies the maximal condition and groups for which the set of non-permutable subgroups satisfies the same property are classified.

The metanorm, a characteristic subgroup: Embedding properties

Abstract The norm of a group was introduced by R. Baer as the intersection of all normalizers of subgroups, and it was later proved that the norm is always contained in the second term of the upper central series of the group. The aim of this paper is to study embedding properties of the metanorm of a group, defined as the intersection of all normalizers of non-abelian subgroups. The metanorm is related to the so-called metahamiltonian groups, i.e. groups in which all non-abelian subgroups are normal, and it is known that every locally graded metahamiltonian group is finite over its second centre. Among other results, it is proved here that if G is a locally graded group whose metanorm M is not nilpotent, then M ′ / M ′′ {M^{\prime}/M^{\prime\prime}} is a small eccentric chief factor and it is the only obstruction to a strong hypercentral embedding of M in G.

Groups with Supersoluble Non-normal Subgroups

The structure of groups in which many subgroups have a certain property χ has been investigated for several choices of the property χ. In particular, groups whose non-normal subgroups are supersoluble are studied in this paper. Moreover, groups with only finitely many normalizers of non-supersoluble groups are considered.

Groups with few conjugacy classes of non-normal subgroups

The structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.