Sobre grupos radicales localmente finitos con min-p para todo primo p.

Abstract

SUMARY<br/>A group is said to be locally finite if every finite subset of G generates a fi-nite subgroup. The class of locally finite groups is placednear the cross-roads of finite group theory and the general theory of infinite groups. Many theorems<br/>about finite groups can be phrased in such a way that their statements still make sense for locally finite groups. However, in general, Sylow's Theorems do not hold in the class of locally finite groups and there are a number<br/>of generic examples which show that locally finite groups can be very varied and complex. If we restrict our attention to locally finite-soluble groups with min-p for all primes p then the Sylow ¼-subgroups are very well behaved<br/>if ¼ or its complementary in the set of all primes is finite. The conjugacy of Sylow p-subgroups in these groups is a very strong condition which have guaranteed the successful development of formation theory and interesting<br/>results on Fitting classes in the universe c¯L of all radical locally finite groups with min-p for all primes p. Moreover, using an extension of the Frattini subgroup introduced by Tomkinson, it has been proved a Gasch¨utz-Lubeseder type theorem characterizing saturated formations in this universe.<br/>It is therefore appropriate to study the class c¯L of all radical locally finite<br/>groups with min-p for all primes p in more detail. In this thesis we have<br/>obtained results which help to understand better the groups in this class.<br/>Consequently, the unspoken rule is that all groups considered in the three<br/>chapters of this thesis belong to the class c¯L. The work is organized as follows.<br/>In Chapter 1, we explore the class B of generalized nilpotent groups in<br/>the universe c¯L. We obtain that this class behaves in the universe c¯L as the<br/>nilpotent groups in the finite universe and we determine the structure of B-<br/>groups explicitly. Moreover, we show that the largest normal B-subgroup of<br/>a c¯L-group is the Fitting subgroup. This fact allows us to prove some results<br/>1<br/>concerning the Fitting subgroup of a c¯L-group which are extensions of the<br/>finite ones. The aim of the last section is to study the injectors associated<br/>to the class B. In fact, we obtain a description of the B-injectors similar to<br/>the characterization of nilpotent injectors of a finite soluble group.<br/>Chapter 2 is devoted to study the local version of the class B. This is<br/>a natural generalization of the class of finite p-nilpotent groups. We extend<br/>some results of finite groups to the above universe using a local version of<br/>a Frattini-like subgroup. In particular, some properties appear relating the<br/>Frattini and Fitting subgroups. The injectors associated to this class of<br/>generalized p-nilpotent groups are also characterized.<br/>Finally, Chapter 3 is concerned with the structure of a radical locally<br/>finite group with min-p for all p, G = AB, factorized by two subgroups A<br/>and B in the class B. We extend the well-known results of finite products<br/>of nilpotent groups to the above universe.<br/>We have introduced a Chapter 0 establishing the notation and terminology.<br/>It also presents many of the well-known results that will be used<br/>throughout this thesis.