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Title: Parasolubility in Lie rings and Lie algebras
Author: Brazier, Stephen George
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1974
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Parasoluble groups have been defined by Wehrfritz as a generalization of nilpotent and supersoluble groups, and this thesis is concerned with the analogous Lie rings and Lie algebras. Most results are proved for Lie rings only, but in several cases the proofs for general Lie algebras are identical. Chapter 1 sets up notation and terminology and introduces the concepts of power derivations and quasi-centralizers. Chapter 2 defines the pparalyzer and stabilizer of a series as terms similar to those used in group theory. Also defined are quasicentral series, and parasoluble Lie rings as those with a finite quasicentral series. In chapter 3 it is shewn that under certain conditions the paralyzer of a finite series is parasoluble. This is always true for torsion-free Lie rings, but an example is given to shew that it is not true in general. Chapter 4 is concerned with paralyzers of ascending series and results are obtained which are generalizations of Lie ring analogues of some results of Hall and Hartley. Chapter 5 looks at the join problem for hypercyclic, parasoluble and supersoluble Lie rings. Chapter 6 is concerned with the class of soluble Lie rings in which all subideals are ideals. These are the Lie ring analogue of similarly defined groups of Robinson. Chapter 7 deals with local parasolubility and we shew that Lie rings which locally have a quasicentral series of bounded length are parasoluble. Chapter 8 employs some of the methods of the preceding chapters to obtain group-theoretic results, the main one being an improvement of a theorem of Hill.
Supervisor: Not available Sponsor: Science Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics