Characteristic subgroup

Definition

Let $G$ be a group and let $H$ be a subgroup of $G$. The subgroup $H$ is characteristic in $G$ when $\alpha(H) = H$ for each automorphism $\alpha$ of $G$.

Remarks

It is equivalent to ask $\alpha(H) \subseteq H$ for each automorphism $\alpha$, because then also $\alpha^{-1}(H) \subseteq H$. The centre and the commutator subgroup of a group are characteristic. A characteristic subgroup of a normal subgroup of $G$ is normal in $G$.

Explanations

L1 human

A characteristic subgroup is a subgroup that no symmetry of the group can move. Usually such a subgroup has a description that uses only the group operation, like "the elements that commute with everything". Any renaming of the elements that keeps the products correct must keep such a subgroup in place.