Groups in which every non-abelian subgroup is permutable

A subgroupH of a groupG is said to bepermutable ifHX=XH for every subgroupX ofG. In this paper the structure of groups in which every subgroup either is abelian or permutable is investigated. This work was done while the last author was visiting the University of Napoli Federico II. He thanks the “Dipartimento di Matematica e Applicazioni” for its financial support.     

Electrochemical Approach for the Production of Layered Double Hydroxides with a Well-Defined Co/MeIII Ratio

Layered double hydroxides (LDHs) have been widely studied for their plethora of fascinating features and applications. The potentiostatic electrodeposition of LDHs has been extensively applied in the literature as a fast and direct method to substitute classical chemical routes. However, the electrochemical approach does not usually allow for a fine control of the M^II/M^III ratio in the synthesized material. By employing a recently proposed potentiodynamic method, LDH films of controlled composition are herein prepared with good reproducibility, using different ratios of the trivalent (Fe or Al) to bivalent (Co) cations in the electrolytic solution. All the obtained materials are shown to be effective oxygen evolution reaction (OER) catalysts, and are thoroughly characterized by a multi-technique approach, including FE-SEM, XRD, Raman, AES and a wide range of electrochemical procedures.     

Isomorphisms between lattices of almost normal subgroups

A subgroupH of a groupG is said to bealmost normal inG if it has only finitely many conjugates inG. The setan(G) of almost normal subgroups ofG is a sublattice of the lattice of all subgroups ofG. Isomorphisms between lattices of almost normal subgroups ofFC-soluble groups are considered in this paper. In particular, properties of images of normal subgroups under such an isomorphism are investigated.     

A note on commutator subgroups in groups of large cardinality

Monatshefte für Mathematik - If G is an uncountable group of regular cardinality $$\aleph $$, we shall denote by $${\mathfrak {L}L}_\aleph (G)$$ the set of all subgroups of G of cardinality...     

Groups whose cyclic subgroups have bounded permutability properties

It follows from classical results of Neumann and Macdonald that a group G has finite commuator subgroup if and only if either the normalizers of cyclic subgroups of G have boundedly finite indices or cyclic subgroups of G have bounded indices in their normal closures. In this paper, groups with a similar condition are considered, when normality is replaced by permutability.      

On a class of metahamiltonian groups

A group is metahamiltonian if all its non-abelian subgroups are normal. It is known that any infinite (generalized) soluble group whose proper subgroups are metahamiltonian is itself metahamiltonian. Moreover, it turns out that the study of soluble groups whose infinite proper subgroups are metahamiltonian can be reduced to the case of a finite extension of a central subgroup of type $p^\infty $ for some prime $p$ . A classification of metahamiltonian groups in this latter class is given.     

The Schur property for subgroup lattices of groups

A classical theorem of Schur states that if the centre of a group G has finite index, then the commutator subgroup G′ of G is finite. A lattice analogue of this result is proved in this paper: if a group G contains a modularly embedded subgroup of finite index, then there exists a finite normal subgroup N of G such that G/N has modular subgroup lattice. Here a subgroup M of a group G is said to be modularly embedded in G if the lattice is modular for each element x of G. Some consequences of this theorem are also obtained; in particular, the behaviour of groups covered by finitely many subgroups with modular subgroup lattice is described. Received: 16 October 2007, Final version received: 22 February 2008     

Detecting the index of a subgroup in the subgroup lattice

A theorem by Zacher and Rips states that the finiteness of the index of a subgroup can be described in terms of purely lattice-theoretic concepts. On the other hand, it is clear that if  is a group and is a subgroup of finite index of , the index cannot be recognized in the lattice of all subgroups of , as for instance all groups of prime order have isomorphic subgroup lattices. The aim of this paper is to give a lattice-theoretic characterization of the number of prime factors (with multiplicity) of .

     

Polycyclic groups with modular finite homomorphic images

A group G is said to be a modular group if it has modular subgroup lattice. We will prove in this paper that a polycyclic group G is modular if and only if all its finite homomorphic images are modular groups. Similar results will also be obtained for other conditions of modular type.     

Groups Whose Finite Homomorphic Images are Metahamiltonian

A group G is metahamiltonian if all its non-abelian subgroups are normal. It is proved here that a finitely generated soluble group is metahamiltonian if and only if all its finite homomorphic images are metahamiltonian; the behaviour of soluble minimax groups with metahamiltonian finite homomorphic images is also investigated. Moreover, groups satisfying the minimal condition on non-metahamiltonian subgroups are described.