SUMARYA 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 theoremsabout 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 numberof 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 behavedif 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 interestingresults 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 finitegroups with min-p for all primes p in more detail. In this thesis we haveobtained results which help to understand better the groups in this class.Consequently, the unspoken rule is that all groups considered in the threechapters 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 inthe universe c¯L. We obtain that this class behaves in the universe c¯L as thenilpotent groups in the finite universe and we determine the structure of B-groups explicitly. Moreover, we show that the largest normal B-subgroup ofa c¯L-group is the Fitting subgroup. This fact allows us to prove some results1concerning the Fitting subgroup of a c¯L-group which are extensions of thefinite ones. The aim of the last section is to study the injectors associatedto the class B. In fact, we obtain a description of the B-injectors similar tothe characterization of nilpotent injectors of a finite soluble group.Chapter 2 is devoted to study the local version of the class B. This isa natural generalization of the class of finite p-nilpotent groups. We extendsome results of finite groups to the above universe using a local version ofa Frattini-like subgroup. In particular, some properties appear relating theFrattini and Fitting subgroups. The injectors associated to this class ofgeneralized p-nilpotent groups are also characterized.Finally, Chapter 3 is concerned with the structure of a radical locallyfinite group with min-p for all p, G = AB, factorized by two subgroups Aand B in the class B. We extend the well-known results of finite productsof 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 usedthroughout this thesis.
| Identifer | oai:union.ndltd.org:TDX_UV/oai:www.tdx.cat:10803/9462 |
| Date | 27 March 2003 |
| Creators | Pedraza Aguilera, Tatiana |
| Contributors | Ballester Bolinches, Adolfo, Universitat de València. Departament d'Àlgebra |
| Publisher | Universitat de València |
| Source Sets | Universitat de València |
| Language | Spanish |
| Detected Language | English |
| Type | info:eu-repo/semantics/doctoralThesis, info:eu-repo/semantics/publishedVersion |
| Format | application/pdf |
| Source | TDX (Tesis Doctorals en Xarxa) |
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