Thread: Abelian and isomorphic groups, homomorphisms

1. Abelian and isomorphic groups, homomorphisms

Suppose $A$ and $B$ are Abelian groups and $\varphi :A \rightarrow B$ is a group homomorphism. Suppose that there exists another group homomorphism $\psi : B \rightarrow A$ such that $\psi \circ \varphi = id_A$. Prove that $B$ is isomorphic to $A \oplus M$ for some other group $M$.
Hint: Set $M=B/A$ (note that $A \rightarrow B$ is injective, so viewing $A$ as a subgroup of $B$ 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 $A$ and $B$ are Abelian groups and $\varphi :A \rightarrow B$ is a group homomorphism. Suppose that there exists another group homomorphism $\psi : B \rightarrow A$ such that $\psi \circ \varphi = id_A$. Prove that $B$ is isomorphic to $A \oplus M$ for some other group $M$.
Hint: Set $M=B/A$ (note that $A \rightarrow B$ is injective, so viewing $A$ as a subgroup of $B$ 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 $\psi\circ\phi=1_a$ , then $\psi$ is onto and $\phi$ is 1-1 , and mentioned there, so that

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

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

there's a homom. $f:B\rightarrow A$ s.t. $f\circ\phi=1_A$ ...well, put now $\phi=f$ 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.