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Proving subgroup of group

  1. Sep 20, 2011 #1
    1. The problem statement, all variables and given/known data
    G is an abelian group
    Let [itex]H = {x \in G : x = x^{-1}[/itex]

    Prove H is a subgroup of G.

    I have two methods in my arsenal to do this (and Im writing them out additively just for ease):
    1. Let a,b be in H. If a + b is in H AND -a is in H then H<G.
    or
    2.Let a,b be in H. if a-b is in H then H<G.

    Solution:
    If I use method one the 2nd part is given practically (if a is in H then a^-1 = a is certainly in H).

    Then I need to show ab is in H. this is what Im struggling with... I feel (since it is given) I should use the fact that G is abelian but not sure where/how to do that!
     
  2. jcsd
  3. Sep 20, 2011 #2

    micromass

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    So you need to show that

    [tex]ab=(ab)^{-1}[/tex]

    First, write out what [itex](ab)^{-1}[/itex] is. Then us that a and b are in H.
     
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