Trivial extensions by torsion-free groups

Let $A$ be an abelian group. Recall that $A$ is

($\bullet$) a Whitehead group if $\text{Ext}(A,\mathbb Z)=0$,

($\bullet$) a free abelian group if $\text{Ext}(A,D)=0$ for every abelian group $D$.

Now fix a cardinal $\mathfrak k$ and consider the following property of $A$:

($\star$) $\text{Ext}(A,D)=0$ for every torsion-free abelian group $D$ with cardinality at most $\mathfrak k$.

What are the groups $A$ satisfying condition ($\star$)?

Observe: If the cardinality of $A$ itself is at most $\mathfrak k$ then ($\star$) implies that $A$ is free abelian. (In such case $A$ has a free resolution $0\rightarrow D\rightarrow F\rightarrow A\rightarrow 0$ with $D$ as in ($\star$) and this sequence splits by the assumptions.) What happens if the cardinality of $A$ is greater than $\mathfrak k$?

• Every of your groups is $\mathfrak k^+$-free (that is each subgroup of cardinality at most $\mathfrak k$ is free). I am not sure if the converse holds. $\kappa$-free groups have an extensive literature - search for "almost free groups". You may start with Eklof and Mekler "Almost free modules" chapter VII. – Adam Przeździecki May 29 '14 at 16:31
• What do you mean by "what are the groups with ($\star$)"? an answer is "these are groups with ($\star$)". What's the question? Is there a characterization? in terms of what? Or are you just asking whether there are non-free examples? – YCor May 29 '14 at 16:50
• I did not express myself sufficiently clearly, sorry. Consider the second condition I mentioned. You can define a free abelian group by means of this condition but the definition of a free abelian group one encounters in the literature is in most cases different. What I meant is the following: Does the class of groups with ($\star$) have a special name and is it of an iterest? The $\mathfrak k$−free groups mentioned by Adam came across my mind but they do not lie between free abelian and Whitehead. For instance an $\aleph_1$−free group need not be Whitehead ($A=\mathbb Z^{\mathbb N}$). – William of Baskerville May 29 '14 at 20:30
• Do you have a non-free example satisfying $(\star)$ for $\mathfrak{l}=\aleph_0$? – YCor May 29 '14 at 22:17
• @Matthew I guess there's no possible answer to your question since, depending on your set theory, all Whitehead groups may be free. – Fernando Muro May 30 '14 at 7:56