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.

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 $\displaystyle \mathbb{Q}$