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Math Help - cosets

  1. #1
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    cosets

    Let G be a group and let N be a subgroup of G. Using the definitions of the sets gN, Ng, and gNg^{-1}, prove that Ng = gN for every g in G if and only if G = N_G(N).

    N_G(N) = { g in G | gNg^{-1} = N}
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  2. #2
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    Quote Originally Posted by ll2008 View Post
    Let G be a group and let N be a subgroup of G. Using the definitions of the sets gN, Ng, and gNg^{-1}, prove that Ng = gN for every g in G if and only if G = N_G(N).

    N_G(N) = { g in G | gNg^{-1} = N}
    gN = Ng if and only if N is normal subgroup if and only if the normalizer of N is G.
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  3. #3
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    I thought about that, but I don't think I can use those theorems to prove it. I would have to prove it using the definitions.
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  4. #4
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    Hacker's way is valid. It's using the definitions, not theorems. Here it is essentially broken down into baby steps.

    If G=N_G(G), then g\in G implies g\in N_G(G), whence gN=Ng.

    If gN=Ng for all g\in G, it follows that G \subseteq N_G(G) (and so equality too of course).
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