TITLE: Infinite cyclic group generated by every single element?
QUESTION [9 upvotes]: This probably is a stupid question, but is there an infinite cyclic group generated by every single one of its nonidentity elements?

REPLY [13 votes]: The infinite cyclic group is unique up to isomorphism.  If $C$ is infinite cyclic and is generated by $x$ and by $y$, then $x=ky$ and $y=hx$ for some $k,h\in\mathbb Z$.  But then $x=khx$, so $hk=1$ and either $h=k=1$ or $h=k=-1$.  So either $x=y$ or $x=-y$.

REPLY [11 votes]: Up to isomorphism there is only one infinite cyclic group, the integers under addition; it has only two generators, $1$ and $-1$.

REPLY [4 votes]: No. The only infinite cyclic group up to isomorphism is $\mathbb{Z}$, which has only two generators.

REPLY [4 votes]: The only infinite cyclic group is $\mathbb{Z}$. So, no.