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Math Help - normal subgroup and cosets

  1. #1
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    normal subgroup and cosets

    If H is a subgroup of a finite group G of index [G: H] two, then H is normal in G.
    I need to show that the left cosets equals to the right cosets, but what are the cosets here?
    Last edited by mr fantastic; November 21st 2009 at 10:12 PM. Reason: Changed posts title
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  2. #2
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    Quote Originally Posted by apple2009 View Post
    If H is a subgroup of a finite group G of index [G: H] two, then H is normal in G.
    I need to show that the left cosets equals to the right cosets, but what are the cosets here?
    there are more than one way to prove this. one way is to write G=H \cup xH, where x \notin H. let g \in G. if g \in H, then clearly g H g^{-1} = H. so we may assume that g = xh, for some h \in H.

    then gHg^{-1}=xhHh^{-1}x^{-1}=xHx^{-1}. now if xh_1x^{-1} \in xH, for some h_1 \in H, then x \in H, which is a false result. so xHx^{-1} \subseteq H and we're done.

    by the way, i just realized that you've already posted this question in here. next time don't post your question twice.
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