If there monomorphisms $f : A \rightarrow B$ and $g : B \rightarrow A$ then there is an isomorphism $h : A \rightarrow B$

Question

Consider the following set theoretical result of Schröder-Bernstein-Cantor:

Let $A$, $B$ be sets. Assume we have injections $f : A \rightarrow B$ and $g : B \rightarrow A$. Then there exists a bijection $h : A \rightarrow B$.

Although this fact seems obvious, its proof is not completely trivial (see Wikipedia ) But what about the following statements?

• Let $A$ and $B$ be finite-dimensional vector spaces. Assume we have monomorphisms $f : A \rightarrow B$ and $g : B \rightarrow A$. Then there exists an isomorphism $h : A \rightarrow B$. (true)

• Let $A$ and $B$ be vector spaces. Assume we have monomorphisms $f : A \rightarrow B$ and $g : B \rightarrow A$. Then there exists an isomorphism $h : A \rightarrow B$. (true?)

• Let $A$ and $B$ be Hilbert spaces. Assume we have monomorphisms $f : A \rightarrow B$ and $g : B \rightarrow A$. Then there exists an isomorphism $h : A \rightarrow B$. (true)

• Let $A$ and $B$ be groups. Assume we have monomorphisms $f : A \rightarrow B$ and $g : B \rightarrow A$. Then there exists an isomorphism $h : A \rightarrow B$. (true?)

How far can you generalize the original result of Schröder-Bernstein-Cantor, at least in categories where monomorphisms are injective and isomorphisms are bijective?

Answer 1

For groups this is not true. Let $F_n$ be the free group with $n$ generators, then $F_2$ has a subgroup isomorphic to $F_3$ and obviously $F_3$ has a subgroup isomorphic to $F_2$.