## A characteristic subgroup of a normal subgroup is normal

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

1. Prove that if $H$ is characteristic in $K$ and $K$ is normal in $G$, then $H$ is normal in $G$.
2. Prove that if $H$ is characteristic in $K$ and $K$ is characteristic in $G$, then $H$ is characteristic in $G$. Use this to prove that the Klein 4-group $V_4$ is characteristic in $S_4$.
3. Give an example to show that if $H$ is normal in $K$ and $K$ is characteristic in $G$ then $H$ need not be normal in $G$.

1. Let $\varphi_g$ denote the automorphism of $G$ which conjugates by $g$. Since $K$ is normal in $G$, we have $\varphi_g[K] = gKg^{-1} = K$; thus $\varphi_g$ is an automorphism of $K$. Since $H$ is characteristic in $K$, we thus have $gHg^{-1} = \varphi_g[H] = H$ for all $g \in G$. Hence $H$ is normal in $G$.
2. Let $\varphi$ be an automorphism of $G$. Since $K$ is characteristic in $G$, we have $\varphi[K] = K$; thus $\varphi$ is an automorphism of $K$. Since $H$ is characteristic in $K$, we have $\varphi[H] = H$; thus $H$ is characteristic in $G$.

Now $V_4$ is the unique subgroup of order 4 in $A_4$, and $A_4$ is the unique subgroup of order 12 in $S_4$. Thus $V_4 \leq A_4$ and $A_4 \leq S_4$ are characteristic; hence $V_4$ is characteristic in $S_4$.

3. Recall the subgroup lattice of $A_4$. We have $V_4 \leq A_4$ characteristic since $V_4$ is the unique subgroup of order 4, and $Z_2 \leq V_4$ is normal because $V_4$ is abelian. However, letting $Z_2 = \langle (1\ 2)(3\ 4) \rangle$ and $\varphi(\sigma) = (1\ 2\ 3) \sigma (1\ 3\ 2)$, note that $\varphi[\langle (1\ 2)(3\ 4) \rangle] = \langle \varphi((1\ 2)(3\ 4)) \rangle = \langle (1\ 4)(2\ 3) \rangle$. Thus $Z_2$ is not characteristic in $A_4$.