If $(G,*)$ is a group, then $H$ is a characteristic subgroup of $G$ (written $H \Char G$ if every automorphism of $G$ maps $H$ to itself. That is, if $f\in{\rm Aut}(G)$ and $h\in H$ then $f(h)\in H$

A few properties of characteristic subgroups:

• If $H\Char G$ then $H$ is a normal subgroup of $G$
• If $G$ has only one subgroup of a given cardinality then that subgroup is characteristic.
• If $K\Char H$ and $H\trianglelefteq G$ then $K\trianglelefteq G$ (Contrast with normality of subgroups is not transitive.)
• If $K\Char H$ and $H\Char G$ then $K\Char G$

Proofs of these properties:

• Consider $H\Char G$ under the inner automorphisms of $G$ Since every automorphism preserves $H$ in particular every inner automorphism preserves $H$ and therefore $g*h*g^{-1}\in H$ for any $g\in G$ and $h\in H$ This is precisely the definition of a normal subgroup.
• Suppose $H$ is the only subgroup of $G$ of order $n$ In general, homomorphisms take subgroups to subgroups, and of course isomorphisms take subgroups to subgroups of the same order. But since there is only one subgroup of $G$ of order $n$ any automorphism must take $H$ to $H$ and so $H\Char G$
• Take $K\Char H$ and $H\trianglelefteq G$ and consider the inner automorphisms of $G$ (automorphisms of the form $h\mapsto g*h*g^{-1}$ for some $g\in G$ . These all preserve $H$ and so are automorphisms of $H$ But any automorphism of $H$ preserves $K$ so for any $g\in G$ and $k\in K$ $g*k*g^{-1}\in K$
• Let $K\Char H$ and $H\Char G$ and let $\phi$ be an automorphism of $G$ Since $H\Char G$ $\phi[H]=H$ so $\phi_H$ the restriction of $\phi$ to $H$ is an automorphism of $H$ Since $K\Char H$ so $\phi_H[K]=K$ But $\phi_H$ is just a restriction of $\phi$ so $\phi[K]=K$ Hence $K\Char G$