'Normally' the following is a definition of a cyclic group:
A group $G$ is called cyclic if there exists a $g \in G$ such that $G=\{g^k : k\in \mathbb{Z}\}.$
Aluffi's Algebra books defines a cyclic group as follows:
A group G is cyclic if it is isomorphic to $\mathbb{Z}$, or to $\mathbb{Z}/n\mathbb{Z}$ for some n.
I can't prove the two definition are equivalent because I don't know how to prove that any group that is not isomorphic to either $\mathbb{Z}$, or to $\mathbb{Z}/n\mathbb{Z}$ is cannot be written as $G=\{g^k : k\in \mathbb{Z}\}.$ Please help!
Best Answer
To prove equivalence of the definitions, we do the following:
$(1)$ If $G=\{g^k:k\in\Bbb Z\}$ then we need to show it is isomorphic to $\Bbb Z/n\Bbb Z$ for some $n$ or $\Bbb Z$. This can be done by showing $g^k\mapsto k$ is an isomorphism (if $|G|<\infty$ then we map to $\Bbb Z/|G|\Bbb Z$).
$(2)$ If $G$ is isomorphic to $\Bbb Z/n\Bbb Z$ for some $n$ or $\Bbb Z$, we need to show it can be written as $\{g^k:k\in\Bbb Z\}$. Under this isomorphism, you map the identity in $\Bbb Z$ or $\Bbb Z/n\Bbb Z$ to some $g\in G$ then show that $G=\{g^k:k\in\Bbb Z\}$. Take an arbitrary $x\in G$, then $x=\phi(k)$ for some $k$ where $\phi$ is the isomorphism with range $G$. Then, $\phi(k)=\phi(1)^k=g^k$.