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Math Help - It's a pity Herstein didn't explain.

  1. #1
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    It's a pity Herstein didn't explain.

    Hi:
    Herstein, in his book Topics in Algebra, 2nd edition,
    states as theorem 2.7.1: Let phi be a homomorphism of G
    onto G' with kernel K. Then G/K isomorphic with G'.
    [G, G' are groups and phi is group homomorphism.]

    Then he comments: Theorem 2.7.1 is important, for it
    tells us precisely what groups can be expected to arise
    as homomorphic images of a given group. These must be
    expressible in the form G/K, where K is normal in G [G
    normal because it's the kernel of phi]. But by lemma
    2.7.1, for any normal subgroup N of G, G/N is a
    homomorphic image of G [Of course]. Thus there is a
    one-to-one correspondence between homomorphic images of
    G and normal subgroups of G.

    O.K. He does not bother to give a formal proof of the
    existence of the one-to-one correspondence. I've tried hard to find one, and have even recoursed to the Schroder-Berstein theorem but in vain.
    If the above one-to-one correspondence exists then there must be a bijection from the set of normal subgroups of G to the set of homomorphic images of G. And therefore either one must produce one such bijection or otherwise prove the existence by some other means.

    Because of the fundamental importance of theorem 2.7.1, I have always been frustrated at my not being able to understand Herstain's remark. Any hint will be welcome. Regards.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Well, more precisely, there is a bijection between the normal subgroups of G and the set of classes of homomorphic images of G, where two homomorphic images of G are in the same class iff they are isomorphic.

    The bijection is given by

    N \mapsto \{H : \phi(G)=H, \ker \phi = N\}.

    Try to show it and report back if you still have trouble.
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  3. #3
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    Thanks for you kind reply.
    Last edited by ENRIQUESTEFANINI; December 7th 2009 at 09:17 PM.
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