Lemma of Characterictic Subgroups

Question

Let G is group and K is subgroup of G then

"K is characteristic in G $\Leftrightarrow $ $\forall k\in K$ and $\forall \phi \in Aut(G)$ , $\phi (k)\in K$.

Here is my solution: $\Rightarrow :$ Let K is characteristic in G. Then for all $\phi \in Aut(G)$ we have $\phi (K)=K$ (definition of characteristic subgroup) .

Since $k\in K$ , $\phi (k)\in \phi (K)=K$.

But I could not show the $(\Leftarrow ):$ part.

Any idea will be appreciated.

Answer 1

Let $\phi\in \text{Aut}(G)$.
Since $\phi(k)\in K$ for all $k\in K$, we have $\phi(K)\subseteq K$.

Note that $\phi^{-1}\in \text{Aut}(G)$.
By the assumption, we have $\phi^{-1}(k)\in K$ for all $k\in K$, which implies that $k\in \phi(K)$ for all $k\in K$. Thus we have $K\subseteq \phi(K)$.