Characteristic Subgroup

Let $G$ be a group and let $H\le G$ be a subgroup. The subgroup $H$ is characteristic in $G$ if, for every automorphism $\varphi:G\to G$ , one has

\varphi(H)=H.

Equivalent characterizations

Equivalently, $H$ is invariant under the action of the automorphism group $\mathrm{Aut}(G)$ on the underlying set of $G$ . Every characteristic subgroup is normal, since conjugations are inner automorphisms.

Examples

The center $Z(G)$ is characteristic in $G$ .
The commutator subgroup $[G,G]$ is characteristic in $G$ .
In a cyclic group of order $n$ , the unique subgroup of each divisor $d\mid n$ is characteristic.
(Non-example) In $(\mathbb{Z}/2\mathbb{Z})^2$ , any subgroup of order $2$ is not characteristic (automorphisms permute the three such subgroups).