Looking for structure in the group of units of integral group rings.

摘要

The Thesis is framed in the context of non commutative algebra and more concretely in the generalization of the concept of integer extension associated to a number field. In this generalization the role of the number field is played by the rational group algebra Q G over a finite group G and the integer extensions are the integral group ring Z G.; The aim of this Thesis is to study the group of unit of an integral group ring, pointing out the analysis of the virtual structure, that is to say, the structure of a subgroup of finite index. Next we give a brief description of the chapters of the thesis.; In Chapter 2 we study the structure of the group generated by two bicyclic units in the particular case of the dihedral group, proving in this case that the group generated by two bicyclic units of the same type is abelian free or free of rank 2.; In Chapter 3 we construct explicitly a subgroup of finite minimal index and minimal rank in the group of unit of Z G which is a direct products of free groups for each finite group G for which this is possible.; Finally in Chapter 4 we characterize the finite nilpotent group G that we call of Kleinian type (which are discrete subgroups of SL2 ( C )), in terms of the Wedderburn decomposition of Q G and describe the structure of those that the commutator subgroup is central.