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Multiplicative groups of nonzero reals and pos. reals

  1. Dec 3, 2011 #1
    WTS is that [itex]\mathbb R^*/N \ \cong \ \mathbb R^{**}[/itex] where [itex]N = (-1, 1)[/itex]

    then prove that [itex]\mathbb R^*/\mathbb R^{**} \ is \ \cong \ to \mathbb Z/2\mathbb Z[/itex]

    So the best answer in my opinion is to construct a surjection and use the first iso thm.

    [itex]f:\mathbb R^*\rightarrow\mathbb R^{**}[/itex]

    [tex]f(x)=|x|,[/tex] is onto by construction. clearly a homomorphism

    [itex]Ker \ (f) = N[/itex], hence [itex]\mathbb R^*/N \ \cong \ \mathbb R^{**}[/itex]

    part 2

    [itex]ψ:\mathbb R^*\rightarrow\ \ N[/itex]

    [tex]ψ(x)=1 \ if \ x>0 \ and \ ψ(x)=-1 \ if \ x<0[/tex]

    by same thm, [itex]\mathbb R^*/\mathbb R^{**} \cong \mathbb Z/2\mathbb Z[/itex]

    because it has 2 elements one of each is the identity.
  2. jcsd
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