[Math] Are all abelian subgroups of a dihedral group cyclic

Question

Are all abelian subgroups of a dihedral group cyclic?

Attempt: I have counter-examples for n=1,2 so I know that it isn't true for n<3. Is it true for n≥3? How do you know this?

Answer 1

For $n$ equal to a multiple of $4$, $n=4k$ with $k \ge 1$, the Dihedral group $D_n$ contains a copy of the non-cyclic abelian group $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$.

One can see this geometrically. Picture a regular $4k$-sided polygon $P$, centered at the origin of the plane, having one opposite pair of sides parallel to the $x$-axis, and another opposite pair of sides parallel to the $y$-axis. The symmetry group of $P$ is $D_{4k}$, and it consists of rotations about the origin through angles which are multiples of $2\pi/4k$ plus reflections across lines that bisect opposite side pairs or that connect opposite vertex pairs. Both the $x$ and $y$-axes bisect opposite side pairs, so reflections across those two axes are in the symmetry group, and they generate a subgroup isomorphic to $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$.