$H$ char $K$ and $K$ char $G$ then $H$ char $G$

Let $G$ be a group and $H \leq G$ and $K \leq G$

Prove that $H$ is characteristic in $K$ and $K$ is characteristic in $G$ then $H$ characteristic in $G$

$H$ char $K$ $\Rightarrow$ $\forall \sigma \in Aut(K)$, $\sigma(H)=H$. $\forall k \in K , kHk^{-1}=H$

$K$ char $G$ $\Rightarrow$ $\forall \sigma \in Aut(G)$, $\sigma(K)=K$. $\forall g \in G , gKg^{-1}=K$

this is what I know. but I do not know how would i deduce that $H$ characteristic in $G$ . Would someone help me out with that. Thank you

1 Answer

If $\sigma\in\text{Aut}(G)$ then $\sigma(K)=K$ because $K\text{ char }G$, so in particular $\sigma$ restricts to an automorphism of $K$. You can now use this to see that $\sigma(H)=H$.