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Give an example where H is not a subgroup.

  1. Feb 25, 2009 #1
    1. The problem statement, all variables and given/known data
    If G is an abelian group, show that H = { a in G | a^2=1} is a subgroup of G.
    Give an example where H is not a subgroup.


    3. The attempt at a solution

    For showing H is a subgroup of G, hh' in G and h^-1 in G.
    (a^2)(a^2) in G also a = a^-1 in G so H is a subgroup of G.. right?

    counterexample is Z+_(4) = {0,1,2,3,4}.. right?

    Thanks
     
  2. jcsd
  3. Feb 25, 2009 #2

    MathematicalPhysicist

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    Gold Member

    Re: subgroup

    To show that H is a subgroup you need to show that it's closed under multiplication under H, take h,h' in H then you need to show that (hh')^2=1 which is game in the pond.
    The same goes for h^-1 and 1.

    For an example where Z+_(4) well if it's Z/4Z then no, cause it doesn't satisfy a+a=0.
     
  4. Feb 25, 2009 #3
    Re: subgroup

    so.. If a = hh', then hh' in H
    and (hh')^2 = a^2 = 1 in H... ?
     
  5. Feb 25, 2009 #4

    MathematicalPhysicist

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    Re: subgroup

    Not really, (hh')^2=hh'hh'=h^2h'^2 for the last equality I used G being abelian.
     
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