How to show ${D}_3$ is not cyclic?

Question

I understand how to show if a group involving $\mathbb{Z}$ is cyclic but not in the case of dihedral groups. I am specifically interested in showing (or knowing) that $D_3$ is not cyclic. How do I go about it?

Answer 1

Here are three separate reasons:

1. $D_3$ has three elements of order $2$. A cyclic group has at most one element of order $2$.

2. $D_3$ is not even abelian. Every cyclic group is abelian.

3. Every element in $D_3$ has order at most $3$ but $D_3$ has order $6$.

The same argument works for $D_n$.

Answer 2

It is in some sense half-cylic, because the two operations you need to systematically be able to construct all elements are:

1. One reflection.
2. One rotation.

The rotation "part" will be a cyclic group.

If you can show that no combination of the rotations will accomplish the reflection then you are done.

Answer 3

Let's write $D_3$ like following figure

Here $r_0$ is unit elements of $D_3$. In the table, we can calculate order of all elements. They're at most $3$. But order of $D_3$, $|D_3|=6\neq 3$.That is $D_3$ is not cyclic.

Moreover, we know that all cyclic groups are Abelian. But, in the table easily shown that non-Abelian. Thus $D_3$ is not cyclic.