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Title: Some applications of set theory to algebra
Author: Pope, Alun Lloyd
Awarding Body: University of London
Current Institution: Royal Holloway, University of London
Date of Award: 1982
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This thesis deals with two topics. In Part I it is shown that if ZFC is consistent, then so is ZF + the order extension principle + there is an abelian group without a divisible hull. The proof is by forcing. In Part II a technique is developed which, in many varieties of algebras, enables the construction for each positive integer not a non-free Xalpha+n -free algebra of cardinality Xalpha+n from a suitable non-free Xalpha-free algebra, when is regular. The algebras constructed turn out to be elementarily equivalent in the language LinfinityXalpha+n to free algebras in the variety. As applications of the technique, it is shown that for any positive integer n there are 2Xn Xn-free algebras which are generated Xn elements, cannot be generated by fewer than this number and are LinfinityXn-equvalent to free algebras in each of the following varieties: any torsion-free variety of groups, all rings with a 1, all commutative rings with a 1, all K-algebras (with K a not-necessarily commutative integral domain), all Lie algebras over a given field. By a different analysis it is shown too that in any variety of nilpotent groups, a lambda-free group of uncountable cardinality lambda is free (respectively, equivalent in Linfinitylambda to a free group) if and only if its abelianisation is, in the abelian part of the variety. Finally, sufficient conditions are given for a X-free group in a variety of groups to be also para free in the variety. The results imply that in the varieties of all groups soluble of length at most k and of all groups polynil potent of given class, if lambda is singular or weakly compact, then a lambda-free group of cardinality lambda is parafree, while if lambda is strongly compact, then a lambda-free group of any cardinality is parafree.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics