Assuming the axiom DC($\omega\_1$), is there a definition of the rank of a group ?


Another related question: assuming DC($\omega\_1$), if we have two groups $A$ and $B$ of the same infinite rank, is there necessarily a surjective homomorphism from $A$ onto $B$? 


Edit: in the second question , we assume $A$ and $B$ both free abelian.



Axiom of dependent choice (up to $\omega_1$) and group rank
Without sitting to verify the details in full, here is a sketch of a proof:


Consider Lauchli's construction of a vector space with two bases of different cardinality, as outlined in Jech **The Axiom of Choice** in problem 10.5 (p. 149). The construction is to create two infinite sets which span isomorphic vector spaces, but have different cardinalities.


Replace "countable" by $\aleph\_2$ and "finite" by $\aleph\_1$. The result will satisfy $\sf DC\_{\aleph\_1}$, and the conclusion *should* still hold. Moreover, you can take the free abelian group generated by those sets instead of a vector space, and it would solve the problem.


If one prefers to do that by forcing instead of atoms, then it is also possible, although slightly more technical.


