# Thread: Abelian and isomorphic groups, homomorphisms

1. ## Abelian and isomorphic groups, homomorphisms

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.

2. Originally Posted by 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.

If you know a little about short exact sequences then the hint is huge:

as $\displaystyle \psi\circ\phi=1_a$ , then $\displaystyle \psi$ is onto and $\displaystyle \phi$ is 1-1 , and mentioned there, so that

$\displaystyle 0\rightarrow A\,\,\xrightarrow{\phi}\,\,B\,\,\xrightarrow {\pi}\,\,B/A\,\,\rightarrow 0\,,\,\,\pi=$ the canonical projection,

is an exact sequence of abelian groups, and we know this sequence splits iff

there's a homom. $\displaystyle f:B\rightarrow A$ s.t. $\displaystyle f\circ\phi=1_A$ ...well, put now $\displaystyle \phi=f$ and we're done!

Tonio

3. Originally Posted by tonio
If you know a little about short exact sequences then the hint is huge:
I don't know anything about short exact aequences.