Is $(\mathbb{Z},+)$ a cyclic group?

Question

$(\Bbb Z,+)$ is cyclic since it is generated by $\pm1$, e.g. $n=n\cdot1$ for an $n\in\Bbb Z$.

I think that $(\mathbb{Z},+)$ is not cyclic cause it is not finite group.

Answer 1

A group $(G,*)$ is said to be cyclic, if it is generated by one group element. Concretely this means there exists some $g\in G$ such that for every element h there is a natural number $n \in \mathbb{Z}$ with $h=g^{*n} = g * ... * g$ ($\vert n \vert$ times). Here $g^{*(-k)} = (g^{-1})^{*k}$.

$(\mathbb Z, +)$ is cyclic, as (somewhat tautologically) every element $n$ can be written as $n=1^{+n}$. Up to group isomorphism it is the only infinite cyclic group though.

Indeed, if $G$ is another cyclic group generated by some element $g$ the assignment $$\begin{array}{rcl} \mathbb Z & \rightarrow & G\\ 1^{+n} & \mapsto & g^{*n} \end{array}$$ yields an isomorphism of groups (it is surjective as $G$ is cyclic, injective as $G$ is infinite and clearly a homomorphism).