# Article

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Keywords:
torsion-free abelian groups; pure subgroup; $P$-pure subgroup
Summary:
Let $\lambda$ be an infinite cardinal. Set $\lambda _0=\lambda$, define $\lambda _{i+1}=2^{\lambda _i}$ for every $i=0,1,\dots$, take $\mu$ as the first cardinal with $\lambda _i<\mu$, $i=0,1,\dots$ and put $\kappa = (\mu ^{\aleph _0})^+$. If $F$ is a torsion-free group of cardinality at least $\kappa$ and $K$ is its subgroup such that $F/K$ is torsion and $|F/K|\le \lambda$, then $K$ contains a non-zero subgroup pure in $F$. This generalizes the result from a previous paper dealing with $F/K$ $p$-primary.
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