Francesco de Giovanni

On groups with many almost normal subgroups

SummaryAn anti-FC-group is a group in which every subgroup either is finitely generated or has only a finite number of coniugates. In this article a classification is given of (generalized) soluble anti-FC-groups which neither are central-by-finite nor satisfy the maximal condition on subgroups. Moreover, groups in which every non-cyclic subgroup has only a finite number of coniugates are characterized.

Groups in which every finite subnormal subgroup is normal

A group G is called a T-group if normality in G is a transitive relation, i.e. all subnormal subgroups of G are normal. The structure of soluble T-groups is well-known, and several authors have investigated groups in which the normality condition is imposed only to a relevant system of subnormal subgroups. We consider here (generalized) soluble groups in which all finite subnormal subgroups are normal.

GROUPS WITH FINITELY MANY DERIVED SUBGROUPS

A study is made of groups with finitely many derived groups of subgroups or of infinite subgroups. These groups are classified completely in the locally graded case. In the general case, detailed structural information about groups in each class is found.

Groups with a supersoluble triple factorization

In the investigation of factorized groups very often one has to study groups with a triple factorization G=AB=AK=BK, where A and B are subgroups and K is a normal subgroup of G (see, for instance, [2,4, 7, 15, 221). In [3] it was shown that under certain liniteness conditions the triple factorized group G satisfies some nilpotency requirement if A, B, and K satisfy the same nilpotency requirement. In the following, similar statements are proved for some supersolubility conditions.

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.

On locally finite groups factorized by locally nilpotent subgroups

Abstract The structure of a periodic radical group G = AB , factorized by two locally nilpotent subgroups A and B , is investigated. In particular, many theorems already known for finite produts of nilpotent groups are extended to the case of periodic radical groups.

Strongly Inertial Groups

A subgroup X of a group G is strongly inert if the index | ⟨ X, X g ⟩: X| is finite for all elements g ∈ G, and a group is strongly inertial if all its subgroups are strongly inert. This article investigates the structure of strongly inertial groups. In particular, strongly inertial groups which are either finitely generated or minimax are completely classified. Moreover, groups in which many subgroups are strongly inert are studied.

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.

Isomorfismi tra reticoli di sottogruppi normali di gruppi nilpotenti senza torsione

RiassuntoIn questo lavoro sono studiati gli isomorfismi tra reticoli di sottogruppi normali di gruppi nilpotenti senza torsione. Si prova che ogni gruppo iperciclico con la struttura normale di gruppo nilpotente senza torsione è un gruppo nilpotente sanza torsione.SummaryLattice isomorphisms between the normal structures of torsion-free nilpotent groups are studied in this paper. We prove that every hypercyclic group with the normal structure of a torsion-free nilpotent group is a torsion-free nilpotent group.

Groups Whose Proper Subgroups of Infinite Rank Have Polycyclic Conjugacy Classes

A group G is called a PC-group if the factor group G/CG(〈x〉G) is polycyclic for each element x of G. It is proved here that if G is a group of infinite rank whose proper subgroups of infinite rank have the property PC, then G itself is a PC-group, provided that G has an abelian non-trivial homomorphic image. Moreover, under the same assumption, a complete classification of minimal non-PC groups is obtained.