The Engel elements in generalized FC-groups

We generalize to FC*, the class of generalized FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J. 28 (2002), 241-254], a result of Baer on Engel elements. More precisely, we prove that the sets of left Engel elements and bounded left Engel elements of an FC*-group G coincide with the Fitting subgroup; whereas the sets of right Engel elements and bounded right Engel elements of G are subgroups and the former coincides with the hypercentre. We also give an example of an FC*-group for which the set of right Engel elements contains properly the set of bounded right Engel elements.Comment: to appear in "Illinois Journal of Mathematics

Some characterisations of groups in which normality is a transitive relation by means of subgroup embedding properties

In this survey we highlight the relations between some subgroup embedding properties that characterise groups in which normality is a transitive relation in certain universes of groups with some finiteness properties

Some Results on Subnormal-like Subgroups

In questo mio lavoro studio diverse ragionevoli generalizzazioni della subnormalità, in relazione a diversi insiemi di sottogruppi di un dato gruppo: provo, ad esempio, che talune sono riconoscibili mediante l'ausilio dei sottogruppi numerabili di un gruppo e che altre lo sono mediante quello dei sottogruppi di rango infinito

On generalised FC-groups in which normality is a transitive relation

We extend to soluble FC∗-groups, the class of generalised FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, `Groups with restricted conjugacy classes', Serdica Math. J. 28(3) (2002), 241-254], the characterisation of finite soluble T-groups obtained recently in [G. Kaplan, `On T-groups, supersolvable groups and maximal subgroups', Arch. Math. 96 (2011), 19-25]

Generalized norms of groups

In this survey paper the authors specify all the known findings related to the norms of the group and their generalizations. Special attention is paid to the analysis of their own study of different generalized norms, particularly the norm of non-cyclic subgroups, the norm of Abelian non-cyclic subgroups, the norm of infinite subgroups, the norm of infinite Abelian subgroups and the norm of other systems of Abelian subgroups

Some topics in the theory of generalized fc-groups

2009 - 2010A finiteness condition is a group-theoretical property which is possessed by all finite groups: thus it is a generalization of finiteness. This embraces an immensely wide collection of properties like, for example, finiteness, finitely generated, the maximal condition and so on. There are also numerous finiteness conditions which restrict, in some way, a set of conjugates or a set of commutators in a group. Sometimes these restrictions are strong enough to impose a recognizable structure on the group. R. Baer and B.H. Neumann were the first authors to discuss groups in which there is a limitation on the number of conjugates which an element may have. An element x of a group G is called FC-element of G if x has only a finite number of conjugates in G, that is to say, if |G : CG(x)| is finite or, equivalently, if the factor group G/CG(⟨x⟩G) is finite. It is a basic fact that the FC-elements always form a characteristic subgroup. An FC-element may be thought as a generalization of an element of the center of the group, because the elements of the latter type have just one conjugate. For this reason the subgroup of all FC-elements is called the FC-center and, clearly, always contains the center. A group G is called an FC-group if it equals its FC-center, in other words, every conjugacy class of G is finite. Prominent among the FC-groups are groups with center of finite index: in such a group each centralizer must be of finite index, because it contains the center. Of course in particular all abelian groups and all finite groups are FC-groups. Further examples of FC-groups can be obtained by noting that the class of FC-groups is closed with respect to forming subgroups, images and direct products. The theory of FC-groups had a strong development in the second half of the last century and relevant contributions have been given by several important authors including R. Baer, B.H. Neumann, Y.M. Gorcakov, Chernikov,L.A. Kurdachenko, and many others. We shall use the monographs , as a general reference for results on FC-groups. The study of FC-groups can be considered as a natural investigation on the properties common to both finite groups and abelian groups. A particular interest has been devoted to groups having many FC-subgroups or many FC-elements. [edited by the author]IX n.s

Groups with restriction on non-normal subgroups

This thesis contains a study of groups with restrictions on non-normal subgroups and of groups whose subgroups not satisfying a property X have finitely many normalizers

Product of Polycyclic-By-Finite Groups (PPFG)

In this paper we show that If the soluble-by-finite group G=AB is the product of two polycyclic-by-finite subgroups A and B, then G is polycyclic-by-finit