How to prove a group is cyclic

Question

What are some methods for proving that a group is cyclic?

For example, I found this link https://www.math3ma.com/blog/ways-to-show-a-group-is-abelian for different ways to prove a group is abelian. Could anyone make a list like this for different ways to show a group is cyclic?

Answer 1

Here is a (short) list for recognizing a cyclic group:

1. If your group is finite of order $n$, try to give a group isomorphism to $C_n$.

2. If your group is infinite, try to give a group isomorphism to $\Bbb Z=C_{\infty}$.

3. In general, try to find a generator $g$. If you succedd then $G$ is cyclic and consists of the integral powers of $g$.

For example, take $G$ as the subgroup of $GL_2(\Bbb Z)$ consisting of the matrices $$ \begin{pmatrix} 1 & k \cr 0 & 1 \end{pmatrix} $$ for all $k\in \Bbb Z$. Can you decide whether or not $G$ is cyclic using 1.,2. or 3.?