Groups $A \leq B \trianglelefteq G$. Prove that $A \trianglelefteq G$

Question

Groups $A \leq B \leq G$ are given, and for all automorphisms $\kappa \in \text{Aut}(B)$, $\kappa(A) = A$.

We are also given $B \trianglelefteq G$.

I proved that $A \trianglelefteq B$ by using the above property of an isomorphism.

$$\phi(a) = bab^{-1}, b \in B, \forall a \in A$$ And got stuck.

Thus, how do I show $A \trianglelefteq G$.

Answer 1

Let $g\in G, a\in A$. We have to show that $gag^{-1}\in A$. First define the map $\varphi:B\to B$ by $\varphi(b)=gbg^{-1}$. It is well defined because $B\trianglelefteq G$, and it is an automorphism of $B$. By assumption $\varphi(A)=A$, so in particular $\varphi(a)\in A$.