Finding an inifnite group that is not cyclic

**Problem:** Find an infinite group that is not cyclic.

I was looking at the set for this.

I said that if so that , then the generator cannot generate all of , and therefore, since there is no generator for , it is not cyclic.

For all with , must be zero in order to generate the zero element that is present in . However, if , then ONLY the zero element is generated, the rest of is not generated, and so is not cyclic.

In the other case, if , could potentially generate 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 then for all , ...?

and hence didn't generate which rationals ?

Similarly,

if , then

and if then for all , ...?

Hence b cannot generate all of