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Thread: Proove an isomorphism

  1. #1
    Junior Member
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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:
    $\displaystyle (N_1+N_2)/N_2 \simeq N_1/(N_1 \cap N_2)$

    Well, I make an exact sequence:

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

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

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

    Showing that the kernel of $\displaystyle \kappa \circ \iota$ is equal to $\displaystyle N_1\cap N_2$ 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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    Ok, I think I got it now, thanks.
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