Almost free groups in varieties

Abstract

A group G in a variety V is almost free in V if it has the property that every subgroup which can be generated by fewer elements than the cardinality of G is contained in a V-free subgroup of G. Clearly, a V-free group is almost free in V. In this paper it is shown that in any torsion-free variety V of groups there is a non-free almost free group in V of cardinality K,, for each positive integer n. This is done by a genera1 construction which works in any variety of algebras in which factoring out by a subalgebra makes sense. Applications to rings and Lie algebras are considcrcd. In [I], Eklof shows that if K is a regular intinite cardinal and there is an almost free non-free abelian group of cardinality K, then there is an almost free non-free abelian group of cardinality K*, the next largest cardinal. It follows that there are almost free non-free ahelian groups of cardinality K,! for each positive integer n. In 14 1, Mekler extends the methods and the result to the variety of all groups. In both these papers, properties of free groups specific to the variety concerned are employed, the most notable being the Schreier property: every subgroup of a free group is free. The construction given below enables the results of [I 1 and [ 4 1 to be extended to non-Schrcicr varieties. The paper is organised as follows. In Section I, the set-theoretic background is given and in Section 2 the fundamental algebraic concepts are reviewed and terminology established. In the next section, the inductive step of the construction is given, and in Section 4 it is shown that the induction can be started. In the final section, extensions in various directions are considered, including applications to varieties of rings and Lie algebras. The material presented here was part of the author’s Ph. D. thesis 161, written under the supervision of Wilfrid Hodges, whose help is gratefully acknowledged.