Finding an inifnite group that is not cyclic

**tangibleLime** · October 3rd 2011, 06:27 PM

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.

**MonroeYoder** · October 3rd 2011, 07:05 PM

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 $\mathbb{Q}$

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