If I have two groups, $G$ and $H$ and two injective homomorphisms $\phi:G \to H$ and $\psi: H \to G$, then by the first isomorphism theorem applied to $\phi$, we have that $G \cong \mathrm{Im} (\phi)$, a subgroup of $H$. Similarly, $H$ is isomorphic to a subgroup of $G$. For finite groups, this guarantees that $G \cong H$ but does this hold for infinite groups?
Weird things can happen for infinite groups, e.g $n\mathbb Z \subsetneq \mathbb Z$ but $n \mathbb Z \cong \mathbb Z$. I'm wondering if this kind of thing stops an isomorphism from occurring.
I think this question generalises to rings, modules, fields and so on. Can this be answered for all of these structures?
Best Answer
For groups this property - sometimes called the Cantor-Schröder-Bernstein property after the corresponding theorem for plain sets - is wrong. Let $G = F_2$ the free group on two generators $\{a,b\}$ and $H = F_3$ the free group on three generators $x,y,z$. Then there are monomorphisms $f \colon G \to H$, given by $f(a) = x$, $f(b) = y$ and $g \colon H \to G$ given by $g(x) = a^2$, $g(y) = b^2$, $g(z) = ab$, there is no isomorphism.