characteristic subgroup

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

A few properties of characteristic subgroups:

• If $H\mathrm{char}G$ then $H$ is a normal subgroup of $G$.

• If $G$ has only one subgroup of a given cardinality then that subgroup is characteristic.

• If $K\mathrm{char}H$ and $H\mathrm{⊴}G$ then $K\mathrm{⊴}G$. (Contrast with normality of subgroups is not transitive.)

• If $K\mathrm{char}H$ and $H\mathrm{char}G$ then $K\mathrm{char}G$.

Proofs of these properties:

• Consider $H\mathrm{char}G$ 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 (http://planetmath.org/GroupHomomorphism) 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 $H\mathrm{char}G$.

• Take $K\mathrm{char}H$ and $H\mathrm{⊴}G$, 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 $K\mathrm{char}H$ and $H\mathrm{char}G$, and let $\varphi$ be an automorphism of $G$. Since $H\mathrm{char}G$, $\varphi [H]=H$, so ${\varphi }_H$, the restriction of $\varphi$ to $H$ is an automorphism of $H$. Since $K\mathrm{char}H$, so ${\varphi }_H[K]=K$. But ${\varphi }_H$ is just a restriction of $\varphi$, so $\varphi [K]=K$. Hence $K\mathrm{char}G$.