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**joestevens** Suppose $\displaystyle A$ and $\displaystyle B$ are Abelian groups and $\displaystyle \varphi :A \rightarrow B$ is a group homomorphism. Suppose that there exists another group homomorphism $\displaystyle \psi : B \rightarrow A$ such that $\displaystyle \psi \circ \varphi = id_A$. Prove that $\displaystyle B$ is isomorphic to $\displaystyle A \oplus M$ for some other group $\displaystyle M$.

Hint: Set $\displaystyle M=B/A$ (note that $\displaystyle A \rightarrow B$ is injective, so viewing $\displaystyle A$ as a subgroup of $\displaystyle B$ is essentially harmless).

This was an extra credit question on my last exam. I just want to know what the proof is.