Some congruences of subnormal subgroups

Si discutono congruenze e semicongruenze e del posetR(G), costituito dai sottogruppi normali di un gruppoG. Più dettagliata è l'analisi quandoR(G)/ρ è una catena a quandoR(G) è un reticolo.

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Conferenza tenuta il 16 maggio 1995     

Inductive sources and subnormal subgroups

A character pair (H, ) in a group G is a subgroup H and a character Irr(H). Following Dade, we say that a character pair (H, ) is an inductive source in G if induction to G defines an injective map from the irreducible characters of the stabilizer of (H, ) that lie over into Irr(G). (By the Clifford correspondence, this necessarily happens if H is normal in G.) A character pair is said to be conjugate stable if its conjugates satisfy a certain technical condition. We show that an inductive source must be conjugate stable and we present an example of a character pair that is conjugate stable but is not an inductive source. Finally, if (H, ) is conjugate stable and H is a subnormal subgroup of G, we show that (H, ) must be an inductive source in G.     

Permutable subnormal subgroups of finite groups

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called -groups. In particular, it is shown that the finite solvable -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not -groups but all subnormal subgroups of defect two are permutable. Received: 13 August 2008     

Conjugation of two finite subnormal subgroups

A necessary and sufficient condition for two finite subnormal subgroups to be conjugate is presented in this paper.     

On subnormal subgroups of linear groups

We describe the subnormal subgroups of 2-dimensional linear groups over local and full rings in which 2 is invertible, as well as the subnormal subgroups of symplectic groups over local rings in which 2 is invertible.     

Torsion-free groups with all subgroups subnormal

It is proved that a torsion-free group with all subgroups subnormal is nilpotent.     

Finite groups with formation subnormal primary subgroups

Given a hereditary saturated formation F containing all nilpotent groups, we describe finite soluble groups in which every primary subgroup is self-normalizing or F-subnormal.     

Finite groups with subgroups supersoluble or subnormal

The aim of this paper is to study the structure of finite groups whose non-subnormal subgroups lie in some subclasses of the class of finite supersoluble groups.     

Groups with restrictions on infinite subnormal subgroups

In this article groups are investigated in which every infinite subnormal subgroup has finitely many conjugates or has finite index in its normal closure.