If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$

Let $(G,\cdot)$ be a group and let $\mathfrak{S}(G)$ be the set of all bijective mappings from $G$ to $G$.

Show that: If $G$ is non-abelian, then $Inn(G):=\{\kappa_a \vert a\in G\}$ is not a normal subgroup of $(\mathfrak{S}(G),\circ)$

First of all, this is not homework, but we had it in university as an exercise and didn't get a solution from our lecturer. So I am just looking for a correct proof, since we have easter holidays and it is not possible to ask him.

Thanks in advance!

Solution 1:

Since $G$ is not abelian, there exists $g \in G \setminus Z(G)$, and hence there exists $h \in G$ with $h^{-1}gh \ne g$. So the inner automorphism $\kappa_h$ does not fix $g$. Let $x$ be the transposition $(1,g) \in {\rm Sym}(G)$. Then $(1,g)^{-1}\kappa_h(1,g)$ does not fix $1$ and so cannot be lie in ${\rm Aut}(G)$.