Title:

Some applications of set theory to algebra

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 nonfree X_{alpha+n }free algebra of cardinality X_{alpha+n} from a suitable nonfree X_{alpha}free algebra, when is regular. The algebras constructed turn out to be elementarily equivalent in the language L_{infinityXalpha+n } to free algebras in the variety. As applications of the technique, it is shown that for any positive integer n there are 2^{Xn} X_{n}free algebras which are generated X_{n} elements, cannot be generated by fewer than this number and are L_{infinityXn}equvalent to free algebras in each of the following varieties: any torsionfree variety of groups, all rings with a 1, all commutative rings with a 1, all Kalgebras (with K a notnecessarily 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 lambdafree group of uncountable cardinality lambda is free (respectively, equivalent in L_{infinitylambda} 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 Xfree 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 lambdafree group of cardinality lambda is parafree, while if lambda is strongly compact, then a lambdafree group of any cardinality is parafree.
