# Thread: Finding an inifnite group that is not cyclic

1. ## 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 $ = 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.

2. ## 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 $\mathbb{Q}$