Why fully characteristic subgroup is normal?

Question

Let's take a group $G$. We say that the subgroup $H \leq G $ is fully characteristic if $$\forall \phi \in \mathrm{End} (G) : \phi(H) \subseteq H.$$

Is this the fully characteristic subgroup normal?

A first thought is to apply the Theorem :

$$\forall g \in G, \forall h \in H :ghg^{-1}\in H. $$ But how could this help as? I have definitely stuck.

Update: Could we find a normal subgroup $H\trianglelefteq G$ such that $H$ is not strictly characteristic? (so the reverse statement is not valid)

Thank you.

P.S.: I apologize for not having any other progress, but I don't know how to continue.

Answer 1

Hint: the map $\phi_g\colon G\to G$, $\phi_g(x)=gxg^{-1}$ is an endomorphism of $G$ (actually an automorphism).

Answer 2

Consider these:

• A subgroup is fully characteristic iff it is invariant under all endomorphisms

• A subgroup is normal iff it is invariant under all inner automorphisms

• Every inner automorphism is an endomorphism