Is $\Bbb C\setminus \{0\}$ a cyclic group?

Question

Is the group $\Bbb C^\times$ (where $\Bbb C$ is the complex numbers and the $\times$ means not including zero) a cyclic group thus meaning that a subgroup of $\Bbb C\setminus\{0\}$ is cyclic?

Answer 1

By definition, each cyclic group is (isomorphic to a group) of the form $\{a^i\mid i\in\Bbb Z\}$ under concatenation, where $a$ is a symbol with

$$a^ra^s=a^{r+s}$$

for all $r,s\in\Bbb Z$ and $a^0=e$ (and, if the group is finite, there is some $n\in\Bbb N$ such that $a^n=e$).

Consider $G=\Bbb C\setminus \{0\}$. We have $2,3\in G$, but no nontrivial power of $2$ is a nontrivial power of $3$ by the Fundamental Theorem of Arithmetic; that is,

$$2^x\neq 3^y$$

for all $x,y\in\Bbb Z\setminus \{0\}$.

Hence $G$ is not cyclic.

Answer 2

It's not cyclic because every cyclic group is countable and $\mathbb{C}^{\times}$ is uncountable.