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This probably is a stupid question, but is there an infinite cyclic group generated by every single one of its nonidentity elements?

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    A cyclic group is generated by any of its non-identity elements if and only if it is cyclic of prime order. This thread covers the infinite case, while if your group is of order $n=ab$ and generated by an element $x$ then $x^a$ has order $b$ in your group so cannot generate the whole group.2012-05-29

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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 $x=-y$.

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

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No. The only infinite cyclic group up to isomorphism is $\mathbb{Z}$, which has only two generators.

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The only infinite cyclic group is $\mathbb{Z}$. So, no.