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Math Help - Proove an isomorphism

  1. #1
    Junior Member
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    Oct 2008
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    Proove an isomorphism

    Hi there

    I'm a little stuck in this one and could use a hand:
    I need to show this isomorphism:
    (N_1+N_2)/N_2 \simeq N_1/(N_1 \cap N_2)

    Well, I make an exact sequence:

    0\rightarrow N_1 \underrightarrow{\iota} (N_1 +N_2) \underrightarrow{\kappa}  (N_1+N_2)/N_2 \rightarrow 0

    where \iota is the inclusion and \kappa the canonical mapping.
    And then I find the Kernel of \kappa \circ \iota but how do I get on from here? As far as I have understood I must find that the Kernel is exactly (N_1 \cap N_2).
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  2. #2
    Member
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    Nov 2010
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    Right idea, but you need to include one more detail:

    We know the natural map \kappa is surjective. But to use the First Isomorphism Theorm, we need to know that the map \kappa \circ \iota is surjective. Why is this? (Consider what an arbitrary element of N_1+N_2/N_2 would look like.)

    Showing that the kernel of \kappa \circ \iota is equal to N_1\cap N_2 should be fairly easy. What would it mean for an element to be in the kernel of this map?
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  3. #3
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    Oct 2008
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    Ok, I think I got it now, thanks.
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