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Math Help - Quotient group question

  1. #1
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    Quotient group question

    Suppose  N \unlhd G and  N \unlhd GN .

    I'm having trouble understanding why  GN/N \neq G/N . Here's my reasoning:

    Let  H = GN , then  H/N = \{hN \mid h \in H \} = \{gnN \mid g \in G, n \in N \} = \{gN \mid g \in G \} = G/N .

    Where I used  nN = N since  n \in N . What is wrong with this 'proof'?
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  2. #2
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    Re: Quotient group question

    what is the context in which this takes place (what group is GN a subset of)?

    note that if N is a subgroup of G, |GN| = |G||N|/|G∩N| = |G||N|/|N| = |G|, so that in fact, GN = G.

    usually, one considers two subgroups S,N normal in G. and the reason SN/N ≠ S/N is that there is no reason to suppose S/N even makes sense

    (N may be "too big" to fit inside S, or may only intersect S trivially).

    if one isn't considering G,N to both be subgroups of a larger group, then GN makes no sense, because it is not clear what the product should be.

    on the other hand, GxN/({e}xN) is isomorphic to G, not G/N.
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  3. #3
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    Re: Quotient group question

    Ok, let me modify this a bit then. Let's take  H to be a group and let's pick a normal group, say  Z its center.

    Say  G \leq H , then  GZ is a group and  Z \unlhd GZ . Shouldn't  GZ/Z = G/Z from essentially my argument above?
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  4. #4
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    Re: Quotient group question

    no. Z is not a subgroup of G, it's a subgroup of H. you need to find "something like Z that fits in G". that's exactly what Z∩G is.
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