# characteristic subgroup

If $(G,*)$ is a group, then $H$ is a characteristic subgroup of $G$ (written $H\operatorname{\,char\,}G$) 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 $H\operatorname{\,char\,}G$ then $H$ is a normal subgroup of $G$.

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

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

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

Proofs of these properties:

• Consider $H\operatorname{\,char\,}G$ 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 $H\operatorname{\,char\,}G$.

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

Title characteristic subgroup CharacteristicSubgroup 2013-03-22 12:50:56 2013-03-22 12:50:56 yark (2760) yark (2760) 13 yark (2760) Definition msc 20A05 FullyInvariantSubgroup NormalSubgroup SubnormalSubgroup characteristic