# Two abelian groups, each being direct factor of the other

Let $M$ and $N$ be two abelian groups. Suppose that $M$ is a direct factor of $N$ (i.e. there are homomorphisms $i:M\rightarrow N$ and $p:N\rightarrow M$ such that $p\circ i=id_M$) and $N$ is also a direct factor of $M$. Is $M$ isomorphic to $N$?

No. A classic result of Corner (On a conjecture of Pierce concerning direct decompositions of Abelian groups. 1964 Proc. Colloq. Abelian Groups (Tihany, 1963) pp. 43–48, MR0169905 (30 #148)) shows that for any positive integer $r$, there exist a countable torsion free abelian group $G$ such that the direct sum of $m$ copies of $G$ is isomorphic to the direct sum of $n$ copies of $G$ if and only if $n\equiv m\pmod{r}$.
In particular, take $r=2$; there is an abelian group $G$ such that $G\cong G\oplus G\oplus G$, but $G\not\cong G\oplus G$. Take $M=G\oplus G$ and $N=G\cong G\oplus G\oplus G$. Then $M$ is clearly a direct factor of $N$, and $N$ is a direct factor of $N\oplus G= (G\oplus G\oplus G)\oplus G \cong G\oplus G\cong M$, but $N\not\cong M$.