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Math Help - Abelian and isomorphic groups, homomorphisms

  1. #1
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    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.
    Last edited by Ackbeet; March 1st 2011 at 02:30 AM. Reason: Fixed grammar in title.
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  2. #2
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    Quote Originally Posted by joestevens View Post
    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
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  3. #3
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    Quote Originally Posted by tonio View Post
    If you know a little about short exact sequences then the hint is huge:
    I don't know anything about short exact aequences.
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