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Math Help - 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 \mathbb{Q} for this.

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

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

    In the other case, if b \neq 0, b could potentially generate \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
    Junior Member
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    Sep 2011
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    Re: Finding an inifnite group that is not cyclic

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

    Similarly,
    if |b|=1, then

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

    Hence b cannot generate all of \mathbb{Q}
    Last edited by MonroeYoder; October 3rd 2011 at 09:18 PM. Reason: make more clear
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