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Math Help - Direct Sum and Isomorphism

  1. #1
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    Direct Sum and Isomorphism

    If, A,B, C are finite abelian groups then prove

    (i) A direct sum B isomorphic to A direct sum C implies that B is isomorphic to C
    (ii) A direct sum A isomorphic to B direct sum B implies A isomorphic to B.
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  2. #2
    MHF Contributor
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    Re: Direct Sum and Isomorphism

    Use commutative diagrams:

    \begin{matrix}A\bigoplus B & \to & A\bigoplus C \\ \downarrow & \searrow & \downarrow \\ B & \to & C\end{matrix}

    You may need to make the diagrams bigger (include groups like (A\oplus B) / (A \oplus \{e_B\}), depending on how much detail you need).
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