On a graph related to permutability in finite groups

For a finite group G we define the graph $\Gamma(G)$ to be the graph whose vertices are the conjugacy classes of cyclic subgroups of G and two conjugacy classes $\{\mathcal {A}, \mathcal {B}\}$ are joined by an edge if for some $\{A \in \mathcal {A},\, B \in \mathcal {B}\, A\}$ and B permute. We characterise those groups G for which $\Gamma(G)$ is complete.

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 PT -groups. In particular, it is shown that the finite solvable PT -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 conjugatepermutability 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, equivalentl…

On the exponent of mutually permutable products of two abelian groups

In this paper we obtain some bounds for the exponent of a finite group, and its derived subgroup, which is a mutually permutable product of two abelian subgroups. They improve the ones known for products of finite abelian groups, and they are used to derive some interesting structural properties of such products.

Graphs and classes of finite groups

[EN] There are different ways to associate to a finite group a certain graph. An interesting question is to analyse the relations between the structure of the group, given in group-theoretical terms, and the structure of the graph, given in the language of graph theory. This survey paper presents some contributions to this research line.

Finite groups with subgroups supersoluble or subnormal

Abstract 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.

On irreducible representations ofp-soluble groups in characteristicp

On the abnormal structure of finite groups

We study finite groups in which every maximal subgroup is supersoluble or normal. Our results answer some questions arising from papers of Asaad and Rose.

Permutable products of supersoluble groups

We investigate the structure of finite groups that are the mutually permutable product of two supersoluble groups. We show that the supersoluble residual is nilpotent and the Fitting quotient group is metabelian. These results are consequences of our main theorem, which states that such a product is supersoluble when the intersection of the two factors is core-free in the group.

On two questions from the Kourovka Notebook

Abstract The aim of this paper is to give answers to some questions concerning intersections of system normalisers and prefrattini subgroups of finite soluble groups raised by the third author, Shemetkov and Vasil'ev in the Kourovka Notebook [10] . Our approach depends on results on regular orbits and it can be also used to extend a result of Mann [9] concerning intersections of injectors associated to Fitting classes.

On totally permutable products of finite groups

[EN] The behaviour of totally permutable products of finite groups with respect to certain classes of groups is studied in the paper. The results are applied to obtain information about totally permutable products of T, PT, and PST-groups.

Generalised norms in finite soluble groups

Abstract We give a framework for a number of generalisations of Baerʼs norm that have appeared recently. For a class C of finite nilpotent groups we define the C -norm κ C ( G ) of a finite group G to be the intersection of the normalisers of the subgroups of G that are not in C . We show that those groups for which the C -norm is not hypercentral have a very restricted structure. The non-nilpotent groups G for which G = κ C ( G ) have been classified for some classes. We give a classification for nilpotent classes closed under subgroups, quotients and direct products of groups of coprime order and show the known classifications can be deduced from our classification.

THE STRUCTURE OF MUTUALLY PERMUTABLE PRODUCTS OF FINITE NILPOTENT GROUPS

We consider mutually permutable products G = AB of two nilpotent groups. The structure of the Sylow p-subgroups of its nilpotent residual is described.

On Formations of Finite Groups with the Wielandt Property for Residuals

Abstract Given two subgroups U, V of a finite group which are subnormal subgroups of their join 〈U, V〉 and a formation F , in general it is not true that 〈U, V〉 F  = 〈U F , V F 〉. A formation is said to have the Wielandt property if this equality holds universally. A formation with the Wielandt property must be a Fitting class. Wielandt proved that the most usual Fitting formations (e.g., nilpotent groups and π-groups) have the Wielandt property. At present, neither a general satisfactory result on the universal validity of the Wielandt property nor a counterexample is known. In this paper a criterion for a Fitting formation to have the Wielandt property is given. As an application, it is p…

On finite groups generated by strongly cosubnormal subgroups

[EN] Two subgroups A and B of a group G are cosubnormal if A and B are subnormal in their join and are strongly cosubnormal if every subgroup of A is cosubnormal with every subgroup of B. We find necessary and sufficient conditions for A and B to be strongly cosubnormal in and, if Z is the hypercentre of G=, we show that A and B are strongly cosubnormal if and only if G/Z is the direct product of AZ/Z and BZ/Z. We also show that projectors and residuals for certain formations can easily be constructed in such a group. Two subgroups A and B of a group G are N-connected if every cyclic subgroup of A is cosubnormal with every cyclic subgroup of B (N denotes the class of nilpotent groups). Thou…