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Math Help - Torsion

  1. #1
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    Torsion

    Consider the (additive) factor group Q/Z.
    (a) Show that every coset of Z in Q contains exactly one representative q ∈ Q in the range0 ≤ q < 1.
    (b) Show that every element of Q/Z has finite order, but there are elements of arbitrarily large order.
    (c) Show that Q/Z is the torsion subgroup of R/Z.
    (d) Show that Q/Z is isomorphic to the multiplicative group U∗ consisting of all roots of unityin C. (That is, U∗ = {z ∈ C|z^n = 1 for some n ∈ Z+}.)

    I got a and b, but I can't figure out c or d, any help?
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  2. #2
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    Quote Originally Posted by DanielThrice View Post
    Consider the (additive) factor group Q/Z.
    (a) Show that every coset of Z in Q contains exactly one representative q ∈ Q in the range0 ≤ q < 1.
    (b) Show that every element of Q/Z has finite order, but there are elements of arbitrarily large order.
    (c) Show that Q/Z is the torsion subgroup of R/Z.
    (d) Show that Q/Z is isomorphic to the multiplicative group U∗ consisting of all roots of unityin C. (That is, U∗ = {z ∈ C|z^n = 1 for some n ∈ Z+}.)

    I got a and b, but I can't figure out c or d, any help?

    (c) We have that if r+\mathbb{Z}\in \mathbb{R}/\mathbb{Z}\,,\,\,ord(r+\mathbb{Z}) = n , then nr\in\mathbb{Z}\Longleftrightarrow r\in \mathbb{Q}

    (d) Check the map \phi: Q\rightarrow U^*\,,\,\, \phi(q):=e^{2\pi iq}

    Tonio
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