When can a pair of groups be embedded in each other?

Solution 1:

Let $F$ be a free group of finite rank $r > 1$. Then the commutator subgroup $[F,F]$ of $F$ is a free group of (countably!) infinite rank. Similarly but more easily, a free group of countably infinite rank contains as subgroups free groups of all finite ranks.

From this it follows that for any $r_1, r_2$ with $2 \leq r_1, r_2 \leq \aleph_0$, $r_1 \neq r_2$, the free group of rank $r_1$ and the free group of rank $r_2$ can be embedded in each other.

Comment: It is a lot easier to find examples of groups which are isomorphic to proper subgroups of themselves (or, in fancier terminology, non co-Hopfian groups). For instance an infinite cyclic group has this property, as does any nontrivial free abelian group or any infinite-dimensional vector space over $\mathbb{F}_p$ or $\mathbb{Q}$. (Added after seeing Arturo's answer: or, more generally, an infinite direct sum of copies of any nontrivial group!)