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Math Help - when a subgroup is in the center of other subgroup

  1. #1
    MHF Contributor Amer's Avatar
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    Jordan
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    when a subgroup is in the center of other subgroup

    The question is
    Let G be a group, N \lhd G such that \[G:N\] = r
    let H \lhd G with finite order show that
    if
    H \cap N = e then nh = hn \;\; \forall \; h\in H \; , n\in N
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  2. #2
    MHF Contributor

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    Re: when a subgroup is in the center of other subgroup

    i am a little unsure as to how [G:N] = r is relevant to your question, other than saying N is of finite index. i also don't know what you mean by "one subgroup is in the center of the other" in terms of your subsequent question.

    if H is in Z(N), of course nh = hn, but then it cannot be the case that H∩N = {e}, unless Z(N) is likewise trivial.

    but if H,N are both normal in G, with H∩N = {e}, then for any h in H, and n in N, we have:

    hnh^{-1}n^{-1} \in N since hnh^{-1}, n^{-1} \in N, because N is normal, while

    hnh^{-1}n^{-1} \in H, since h, nh^{-1}n^{-1} \in H because H is normal.

    since H∩N = {e}, we see that h and n commute.
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  3. #3
    MHF Contributor Amer's Avatar
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    Re: when a subgroup is in the center of other subgroup

    thanks
    the problem in Abstract algebra when you see the answer you say oooh it is easy how i did not figure that
    the hardness is the idea of the solution
    I think if I had thought in it in reverse nh = hn and see what will happen
    anyway thanks very much
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