## If a subgroup is unique of a given order then it is characteristic

If $H$ is the unique subgroup of order $|H|$ in $G$, prove that $H$ is characteristic.

Let $\varphi \in \mathsf{Aut}(G)$ and let $H \leq G$; suppose $H$ is the unique subgroup of order $|H|$. Now $\varphi|_H$ is an injection of $H$ onto a subgroup of order $|H|$ in $G$; hence $\varphi[H] = H$. Thus $H$ is characteristic.