New criteria for freeness in abelian groups

Hill, Paul

doi:10.1090/S0002-9947-1973-0325805-5

New criteria for freeness in abelian groups
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Trans. Amer. Math. Soc. 182 (1973), 201-209 Request permission

Abstract:

A new criterion is established for an abelian group to be free. The criterion applies to an ascending chain of free subgroups. The result is used to construct groups that are almost free but not free. In particular, we construct examples that show that the class of free abelian groups is not definable in the logical language ${L_{\infty \kappa }}$ if $\kappa \leq {\aleph _2}$. In doing so, we take advantage of a recent theorem of P. Eklof.

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