What is the intuition for recognizing cyclic groups?

Question

Compare these two sets -

1. $\{ \pi^{n}: n \in \mathbb{Z} \}$ --- Cyclic
2. $\{ 2^n 3^m: m,n \in \mathbb{Z} \}$ --- Not cyclic.

In the first case $\{ \pi^{n}: n \in \mathbb{Z} \}$ could be expanded to $\{ \ldots \pi^{-1} \pi^{0}, \pi^{1} \ldots \}$

In the same way, $\{ 2^n 3^m: m,n \in \mathbb{Z} \}$ could be expanded to $\{ \ldots 6^{-1}, 6^{0}, 6^{1} \ldots \}$

I see both sets looking the same, yet one is cyclic and the other is not. I have seen the claims' proofs for both sets which makes sense, but intuitively I can't understand whats the reason. Could someone help me?

Thanks!

Answer 1

A cyclic group $G$ is precisely a group that can be written with exactly one generator; that is, in terms of presentations, either

$$G\cong\langle a\mid a^n\rangle$$

(in which case, $G\cong \Bbb Z_n$) for $n\in \Bbb N$, or

$$G\cong\langle a\mid \varnothing \rangle$$

(in which case $G\cong \Bbb Z$).

Nothing more; nothing less.

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Your example 2 is a little off: it is, in fact, not cyclic, as it has at least two generators (for example, $2$ and $3$), but equals $$\{\dots, 1/6, 1/4,1/3,1/2,1, 2,3,4,6,\dots\}.$$

Answer 2

Adding to the previous answer, the intuition for this definition comes from finite cyclic groups. These are precisely the groups that consist of one cycle that can be iterated through by multiplying by one element. For example the group $\{1, i, -1, -i\}$ can be iterated through by multiplication by $i$. In the infinite case this is no longer true, but we still use the term for a group generated by a single element.