di Grazia, Giovanna (2016) Some properties of large subgroups of groups. [Tesi di dottorato]

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Starting from the nineteen twenties until now, a relevant part of investigation in infinite groups was based on the fact that in many case the imposition of finiteness conditions on an infinite group forces the group to be “close” to a finite group. Recently, a new point of view has been adopted, focusing the attention on groups which are far from finiteness. The subject of this thesis is the investigation of groups which are “large” in some sense. A subgroup property P is an embedding property if in any group G all images under automorphism of G of P-subgroups likewise have the property P. Let X be a class of groups. Then X is said to be a class of large groups if it satisfies the following condition: - If a group G contains an X-subgroup, the G belongs to X; - If G is any X-group and N is a normal subgroup of G, then at least one of the groups N and G/N belongs to X; - No finite cyclic group belongs X. An obvious example of class of large groups is the class of groups of infinite rank. A group G is said to have finite (Prüfer) rank if there exists a positive integer r such that all finitely generated subgroups of G can be generated by at most r elements and r is the least positive integer with such property. In recent years, many authors have been published a series of relevant papers which shows that the subgroups of infinite rank of a group of infinite rank have the power to influence the structure of the whole group; in fact, it has been proved that, for some choices of group theoretical properties X, if G is a group in which all subgroups of infinite rank satisfy the property X, then the same happens also to the subgroups of finite rank (in other words, the class of groups of infinite rank controls X). The doctoral thesis will illustrate some results obtained in this area by different authors, then focusing on results, obtained by Giovanna di Grazia joint with the doctor A.V. De Luca, which show that the class of groups of infinite rank controls, in a suitable universe of generalized soluble groups, the following properties: the property of being either normal or self-normalizing; the property of being either normal or contranormal (i.e. the normal closure of the subgroups is the whole group); the property of being either subnormal or contranormal. Furthermore, groups in which every infinite subsets of subgroups contains a pair such that at least one of them (or both) is modular in the subgroup generated by them are examined, and it illustrates the result obtained by Giovanna di Grazia and A.V. De Luca , that shows that these groups are central-by-finite. Finally, we focus on some results contained in a work in progress on groups of infinite rank which are isomorphic to their non-abelian subgroups of infinite rank. It is proved that if G is a periodic soluble group with such property, then G is abelian-by-finite and in the torsion-free nilpotent case it is even abelian.

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