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Thread: Group homomorphisms and short exact sequences

  1. #1
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    Group homomorphisms and short exact sequences

    I have no idea how to do this problem so I was hoping someone could show me how to do it? Or even at least how to do part (a). If anyone can help me out I would appreciate so much!

    Let $N\underline{\triangleleft}G$ be a normal subgroup, and assume that there exists a group homomorphism $p:G\rightarrow N$ that is left inverse to the inclusion of $N$ in $G.$

    (a) Show that there exists a group homomorphism
    $$
    f:G\rightarrow N\times(G/N)
    $$
    satisfying
    $$
    (\forall n\in N)\ f(n)=(n, N)
    $$
    and
    $$
    (\forall g\in G)\ (pr_{2}\circ f)(g)=gN.
    $$
    Here $pr_{2}$ denotes the projection to the second factor.

    (b) Prove that the homomorphism $f$ you constructed in part (a) is in fact an isomorphism.

    (c) Reformulate the questions and answers above in terms of short exact sequences.
    Last edited by lucarioimpersius; Apr 7th 2017 at 11:05 PM.
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  2. #2
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    Re: Group homomorphisms and short exact sequences

    Define

    f(x)=(p( x),x N) for x\in G
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