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Normal Subgroup

  1. Nov 16, 2007 #1
    Hi, I need some help with this problem:

    Let H be a subgroup of an arbitary group G. Prove H is normal iff it has the following property: For all a,b, in G, ab is in H iff ab is in H.
     
  2. jcsd
  3. Nov 16, 2007 #2
    Are you sure this is the exact question? Did you notice that this is actually true for all subgroups of G, and is equivalent to saying c is in H if and only if c is in H?
     
  4. Nov 16, 2007 #3
    Re

    oops, I meant to write for all a,b in G, ab is in H iff ba is in H.
     
  5. Nov 16, 2007 #4

    matt grime

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    It is a prerequisite of the forums that you say what you're trying to do to solve the problem. Start with the definition of normal.
     
  6. Nov 16, 2007 #5
    I'll do half of the proof the other is the same, but with the letters reversed.

    if ab is an element of H and H is a normal group then

    abab = h for some h element of H

    ba = (a^-1)h(b^-1)

    gbag^-1 = g(a^-1)h(b^-1)g^-1 for any element g of G


    gbag^-1 = g(a^-1)(abb^-1a^-1)h(b^-1)g^-1

    but since ab is an element of H so is b^-1a^-1 since H is a group

    so
    gbag^-1 = (gb)h'(gb)^-1 for h' = b^-1a^-1h which is also an element of H

    now by definition of a normal group (gb)h'(gb)^-1 = h'' for some element of H

    Therefore gbag^-1 = h'' some element of H

    That completes one direction the other direction can be done by the same method.
     
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