Cyclic subgroup is characteristic in dihedral group

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup satisfying a particular subgroup property (namely, Characteristic subgroup (?)) in a particular group or type of group (namely, 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 $n\geq 3$ or for the infinite dihedral group, the cyclic subgroup $\langle a\rangle$ is a characteristic subgroup of $G$.

Related facts

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 $a^{k}x$. Further, $(a^{k}x)^{2}=a^{k}xa^{k}x^{-1}=a^{k}a^{-k}=e$. Thus, every element of $G$ outside of $H$ has order two.

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