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Every subgroup of index 2 is normal?

  1. Jul 18, 2011 #1
    I have to prove that every subgroup H of a group G with index(number of distinct cosets of the subgroup) 2 is normal.

    I dont know how to start :'(

    Please help.
     
  2. jcsd
  3. Jul 18, 2011 #2

    micromass

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    Hi Oster! :smile:

    You must prove that gN=Ng for all g. But what exactly are the cosets of N??
     
  4. Jul 19, 2011 #3
    If the element g comes from H then both gH and Hg are equal to H.
    If g comes from H complement then i know it must represent the "other" coset. Since cosets partition G, i know both these cosets must be equal to H complement.
    More or less correct?
     
  5. Jul 19, 2011 #4

    micromass

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    Yes, that's good!
     
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