
SUMARY
A group is said to be locally finite if every finite subset of G generates a finite subgroup. The class of locally finite groups is placednear the crossroads 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 finitesoluble groups with minp for all primes p then the Sylow ¼subgroups are very well behaved
if ¼ or its complementary in the set of all primes is finite. The conjugacy of Sylow psubgroups 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 minp for all primes p. Moreover, using an extension of the Frattini subgroup introduced by Tomkinson, it has been proved a Gasch¨utzLubeseder 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 minp 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 Bsubgroup of
a c¯Lgroup is the Fitting subgroup. This fact allows us to prove some results
1
concerning the Fitting subgroup of a c¯Lgroup 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 Binjectors 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 pnilpotent groups. We extend
some results of finite groups to the above universe using a local version of
a Frattinilike subgroup. In particular, some properties appear relating the
Frattini and Fitting subgroups. The injectors associated to this class of
generalized pnilpotent groups are also characterized.
Finally, Chapter 3 is concerned with the structure of a radical locally
finite group with minp for all p, G = AB, factorized by two subgroups A
and B in the class B. We extend the wellknown 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 wellknown results that will be used
throughout this thesis.
