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Thread: Finding an inifnite group that is not cyclic

  1. #1
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    Finding an inifnite group that is not cyclic

    Problem: Find an infinite group that is not cyclic.

    I was looking at the set $\displaystyle \mathbb{Q}$ for this.

    I said that if $\displaystyle b=\frac{x}{y}$ so that $\displaystyle b \in \mathbb{Q}$, then the generator $\displaystyle <a> = b^n$ cannot generate all of $\displaystyle \mathbb{Q}$, and therefore, since there is no generator for $\displaystyle \mathbb{Q}$, it is not cyclic.

    For all $\displaystyle n$ with $\displaystyle b^n$, $\displaystyle b$ must be zero in order to generate the zero element that is present in $\displaystyle \mathbb{Q}$. However, if $\displaystyle b=0$, then ONLY the zero element is generated, the rest of $\displaystyle \mathbb{Q}$ is not generated, and so $\displaystyle \mathbb{Q}$ is not cyclic.

    In the other case, if $\displaystyle b \neq 0$, $\displaystyle b$ could potentially generate $\displaystyle \mathbb{Q}$ but without the zero element since b is not zero.

    I have the nagging suspicion that I am incorrect.

    Any input is appreciated.
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  2. #2
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    Re: Finding an inifnite group that is not cyclic

    How about something like... if $\displaystyle |b|<1$ then for all $\displaystyle n$, $\displaystyle b^n$ ...?
    and hence $\displaystyle b$ didn't generate which rationals ?

    Similarly,
    if $\displaystyle |b|=1$, then

    and if $\displaystyle |b|>1$ then for all $\displaystyle n$, $\displaystyle b^n$ ...?

    Hence b cannot generate all of $\displaystyle \mathbb{Q}$
    Last edited by MonroeYoder; Oct 3rd 2011 at 08:18 PM. Reason: make more clear
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