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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 might assume $A$ and $B$ both abelian.

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.

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 $B$?

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 might assume $A$ and $B$ both abelian.