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.