A height sequence s is a function on primes p with values \(s_ p\) natural numbers or \(\infty\). The height sequence \(| x|\) of an element x in a torsion-free abelian group G is defined by \(| x|_ p=height\) of x at p. For a height sequence s, \(G(s)=\{x\in G:| x| \geq s\}\), \(G(ps)=\{x\in G(s):| x|_ p\geq s_ p+1\}\), \(G(s^*)=\{x\in G(s):\sum_{p}(| x|_ p-s_ p)\) is unbounded) and \(G(s^*,p)=G(s^*)+G(ps)\). A subgroup H of G is *-pure if for all s and p \(H(s^*,p)=H\cap G(s^*,p)\). The subgroup H is knice in G if for each finite subset X of G there is a finite rank completely decomposable subgroup K such that \(X\subset H+K=H\oplus K\) and this sum is *-pure in G. Further, G is a k-group if \(\{\) \(0\}\) is knice in G. These concepts are due to \textit{P. Hill} and \textit{C. Megibben} [Trans. Am. Math. Soc. 295, 735-751 (1986; Zbl 0597.20047)] except that the authors strengthen the meaning of knice (equivalent to pure and knice in the old sense). Among the results are the following. Proposition 4: G is a k-group if and only if G is the quotient of a completely decomposable group modulo a knice subgroup if and only if every countable subset of G is contained in a *-pure completely decomposable subgroup of G. Theorem 7: A knice subgroup of a completely decomposable group is \(\aleph_ 1\)-separable. Theorem 9: A knice subgroup of cardinality \(\leq \aleph_ 1\) of a completely decomposable group is completely decomposable. Theorem 12: A k-group (in particular a separable group) of cardinality \(\leq \aleph_ n\) has balanced projective dimension \(\leq n\).