Contributions to the theory of factorized groups

Abstract

In chapter 1 we begin by describing certain group theoretical concepts which appear during the course of this thesis. We also supply a brief survey of results concerning factorized groups, relating them to our investigations. In chapter 2, section 2.2, we consider groups which possess a triple factorization. We show that if a Cernikov group is factorized by three nilpotent subgroups it is itself nilpotent. It is then possible to generalize this result to a wider class of infinite groups, denoted by Ǽ In section 2.3 we continue this theme by examining groups which have a triple factorization by three abelian subgroups. If such a group has finite abelian total rank then it must be nilpotent. In section 2.4 we investigate the circumstances under which a subgroup inherits the factorization of the group. We show that if a Cernikov group is factorized by two abelian subgroups, then its Fitting subgroup factorizes. Once again this result holds for the class Ǽ, furthermore we are able to show that the Hirsch-Plotkiu radical also factorizes. Chapter 3 examines this question in relation to the formation subgroups of a group. Let § denote a formation of finite soluble groups as defined in section 3.1. We begin by reviewing the existence and behaviour of the L§ -covering subgroups and L§ -normalizers of a periodic (LŊ)Ø-group. Then, by taking § to be the formation of finite nilpotent groups, we prove that, if such a group is factorized by two nilpotent subgroups, then there is an L§ -covering subgroup which also factorizes. By specializing to Cernikov groups we are able to show that the above holds for an arbitrary saturated formation §. In the final chapter of this thesis we consider the situation where the product of two abelian subgroups of a group G is not itself a group. We then examine a subgroup M of G which lies in the product set. By imposing extra conditions we are able to produce some bounds on the exponent of M in terms of those of the factors. Lastly we show that if the torsion-free nilpotent group G is generated by two infinite cyclic subgroups then a subgroup which lies in their product is abelian