Forcing a Basis into ${\mathrm{\aleph }}_1$-Free Groups

In this paper, we address the question of when a non-free ${\mathrm{\aleph }}_1$-free group $H$ can be be free in a transitive cardinality-preserving model extension. Using the $\mathrm{\Gamma }$-invariant, denoted $\mathrm{\Gamma }(H)$, we present a necessary and sufficient condition resolving this question for ${\mathrm{\aleph }}_1$-free groups of cardinality ${\mathrm{\aleph }}_1$. Specifically, if $\mathrm{\Gamma }(H)=[{\mathrm{\aleph }}_1]$, then $H$ will be free in a transitive model extension if and only if ${\mathrm{\aleph }}_1$ collapses, while for $\mathrm{\Gamma }(H)\mathrm{\ne }[{\mathrm{\aleph }}_1]$ there exist cardinality-preserving forcings that will add a basis to $H$. In particular, for $\mathrm{\Gamma }(H)\mathrm{\ne }[{\mathrm{\aleph }}_1]$, we provide a poset $(\mathcal P_{\rm pb}, \leq)$ofpartialbasesforaddingabasisto$H$withoutcollapsing$\aleph_1$.