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Math Help - Prove that Z and Q are not isomorphic

  1. #1
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    Prove that Z and Q are not isomorphic

    Prove that additive groups \mathbb{Z} and \mathbb{Q} are not isomorphic
    So I looked through the isomorphic properties and they seem to satisfy all of them...
    - The order of \mathbb{Z} equals the order of \mathbb{Q}
    - Both are abelian
    - The order of any element in the groups are all infinite (except the identity)

    The only thing I can think of is that \mathbb{Z} has a generator and \mathbb{Q} doesn't, but does that have anything to do with them being isomorphisms?
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    Re: Prove that Z and Q are not isomorphic

    If one group is cyclic and the other isn't, can they be isomorphic?
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    Re: Prove that Z and Q are not isomorphic

    As noted above, we need to prove $\Bbb Q$ is NOT cyclic, which would settle the question (since an isomorph of the integers would be cyclic).

    So, suppose $\Bbb Q$ had a generator, which we will write as $q = \dfrac{a}{b}$ with $\text{gcd}(a,b) = 1$ (a fraction "in simplest terms"), and of course $b \neq 0$. We can further insist that $b > 0$, for if not, replace $\dfrac{a}{b}$ with $\dfrac{-a}{-b}$.

    Consider $r = \dfrac{1}{b+1}$, which is clearly an element of $\Bbb Q$. Since we are supposing $\Bbb Q$ cyclic, there is some INTEGER $k$ with:

    $k\left(\dfrac{a}{b}\right) = \dfrac{1}{b+1}$.

    This leads to the equation of integers:

    $ka(b+1) = b$, which after some re-arranging, we can write as:

    $k = b(1 - ka)$.

    It is clear from this equation that $b$ divides $k$, so we have $k = nb$ for some integer $n$. This means that $k\left(\dfrac{a}{b}\right) = \dfrac{ka}{b} = \dfrac{nab}{b} = na \in \Bbb Z$.

    Thus:

    $na = \dfrac{1}{b+1}$, so that $na(b+1) = 1$. Thus $b+1$ divides 1, and since $b+1 > 0$, we must have $b+1 = 1$ so that $b = 0$, contradiction.

    In short, for any element $\dfrac{a}{b}$, we have found an element $r$ NOT in $\left\langle \dfrac{a}{b} \right\rangle$, namely $r = \dfrac{1}{b+1}$.
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