Resumen de Sobre grupos radicales localmente finitos con min-p para todo primo p.
SUMARY 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 about finite groups can be phrased in such a way that their statements still make sense for locally finite groups. However, in general, Sylows Theorems do not hold in the class of locally finite groups and there are a number 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 1/4-subgroups are very well behaved if 1/4 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 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.
It is therefore appropriate to study the class c¯L of all radical locally finite groups with min-p for all primes p in more detail. In this thesis we have obtained results which help to understand better the groups in this class.
Consequently, the unspoken rule is that all groups considered in the three chapters of this thesis belong to the class c¯L. The work is organized as follows.
In Chapter 1, we explore the class B of generalized nilpotent groups in the universe c¯L. We obtain that this class behaves in the universe c¯L as the nilpotent groups in the finite universe and we determine the structure of B- groups explicitly. Moreover, we show that the largest normal B-subgroup of a c¯L-group is the Fitting subgroup. This fact allows us to prove some results 1 concerning the Fitting subgroup of a c¯L-group which are extensions of the finite ones. The aim of the last section is to study the injectors associated to the class B. In fact, we obtain a description of the B-injectors similar to the characterization of nilpotent injectors of a finite soluble group.
Chapter 2 is devoted to study the local version of the class B. This is a natural generalization of the class of finite p-nilpotent groups. We extend some results of finite groups to the above universe using a local version of a Frattini-like subgroup. In particular, some properties appear relating the Frattini and Fitting subgroups. The injectors associated to this class of generalized p-nilpotent groups are also characterized.
Finally, Chapter 3 is concerned with the structure of a radical locally finite group with min-p for all p, G = AB, factorized by two subgroups A and B in the class B. We extend the well-known results of finite products of nilpotent groups to the above universe.
We have introduced a Chapter 0 establishing the notation and terminology.
It also presents many of the well-known results that will be used throughout this thesis.
