Cyclic subgroup is characteristic in dihedral group

Statement
Let $$G$$ be a dihedral group defined as follows:

$$G = \langle a,x \mid a^n = x^2 = e, xax^{-1} = a^{-1} \rangle$$

or the infinite dihedral group:

$$G = \langle a,x \mid x^2 = e, xax^{-1} = a^{-1} \rangle$$.

Then, for $$n \ge 3$$ or for the infinite dihedral group, the cyclic subgroup $$\langle a \rangle$$ is a characteristic subgroup of $$G$$.

Stronger facts

 * Cyclic subgroup is isomorph-free in dihedral group
 * Cyclic subgroup is prehomomorph-contained in dihedral group

Proof
Let $$H = \langle a \rangle$$. Then, $$H$$ has index two in $$G$$. Any element of $$G$$ outside $$H$$ is of the form $$a^kx$$. Further, $$(a^kx)^2 = a^kxa^kx^{-1} = a^ka^{-k} = e$$. Thus, every element of $$G$$ outside of $$H$$ has order two.

Thus, if $$n \ge 3$$, $$H$$ can be defined as the unique cyclic subgroup generated by an element of order $$n$$.