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Math Help - problems on order??

  1. #1
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    problems on order??

    Q1: let G be a group.For any x belongs to G,prove that |x|<=|G| ?
    Q2:Prove that (Q,+) is not isomorphic to (Q,*)?
    Q3:Prove that a cyclic group is isomorphic either to (Zn,+) for some positive integer n or to (Z,+) ?
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  2. #2
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    Quote Originally Posted by Mathventure View Post
    Q1: let G be a group.For any x belongs to G,prove that |x|<=|G| ?
    .

    Assume G is finite. For x\in G construct \left < x \right>. Then, |x| = |\left< x \right>|\leq |G|.
    Q2:Prove that (Q,+) is not isomorphic to (Q,*)?
    There are two solutions to x^2 = 1 in the multiplicative group but only one solution to 2x=0 in the additive one.

    Q3:Prove that a cyclic group is isomorphic either to (Zn,+) for some positive integer n or to (Z,+) ?
    If G is infinite and G = \left< a\right> then define \phi : G\to \mathbb{Z} by \phi(a^k) = k as the isomorphism. Be sure to argue this is well-defined.

    If G is finite and G = \left< a\right> with |G|=n then define \phi : G\to \mathbb{Z}_n by \phi (a^k) = [k]_n. Be sure to argue this is well-defined.
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