Characteristic Subgroup

Let $G$ be a group and let $H\le G$ be a subgroup . The subgroup $H$ is characteristic in $G$ (written $H \operatorname{char} G$) if for every automorphism $\varphi:G\to G$ (i.e. a bijective group homomorphism ), one has $\varphi(H)=H.$

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).