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Math Help - Monoids and Groups

  1. #1
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    Monoids and Groups

    Can I get help on the following problem:
    Is the additive group of rationals isomorphic to the multiplicative group of non-zero rationals.
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  2. #2
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    Quote Originally Posted by thomas_donald View Post
    Can I get help on the following problem:
    Is the additive group of rationals isomorphic to the multiplicative group of non-zero rationals.
    Assume that such an isomorphism does exist.

    Let

    \theta : (\mathbb{Q}, +) \rightarrow (\mathbb{Q} ^ \times,\times)

    Then for all q \in \mathbb{Q} we have:

    \theta(q + q) = \theta (q) \theta(q)

    as \theta is an isomorphism.

    Can you see where a contradiction could lie in this arguement?

    Spoiler:
    As  \theta is an isomorphism, there exists a q such that \theta(q) = 2, so...


    Hope this helps.
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  3. #3
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    Smile

    Thanks a lot. I get it.
    If we take 2=(2q)=(q)*(q) we get: (q)=square root of 2
    which does not belong to Q.
    Great.
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  4. #4
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    nice one.
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  5. #5
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    Quote Originally Posted by thomas_donald View Post
    Can I get help on the following problem:
    Is the additive group of rationals isomorphic to the multiplicative group of non-zero rationals.
    There is another way to see the non-isomorphism. In \mathbb{Q}^{\times} there exists an element, different from the identity, which when multiplied on itself produces the identity. This is -1 since (-1)^2=1. However, in \mathbb{Q} under addition you cannot find an element which when added to itself will produce the identity other than the identity. Thus, the two cannot be isomorphic.
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