Order of generators in infinite cyclic group.

Question

I read that order of generators in an infinite cyclic group can be 0 or $\infty$. I understand how it is infinity but I don't get how can the order of a generator be 0.

Can someone please help me understand the above statement ?

Answer 1

An infinite cyclic group $G=\langle a \rangle $ has exactly two generators, namely $a$ and $a^{-1}$. By definition, the order of an element is the smallest positive exponent $n\ge 1$ with $a^n=e$, or infinity.

References:

Orders of Elements in a Group

An Infinite Cyclic Group has Exactly Two Generators: Is My Proof Correct?