  For certain uncountable cardinals $\kappa$ we produce a group of cardinality
$\kappa$ which is freely indecomposable, strongly $\kappa$-free, and whose
abelianization is free abelian of rank $\kappa$. The construction takes place
in G\"odel's constructible universe $L$. This strengthens an earlier result of
Eklof and Mekler.
