Exercise 3, § 5

Let $\Gamma$ be the group of automorphisms of a group G and $\Delta$ the group of inner automorphisms of G; show that $\Delta$ is a normal subgroup of $\Gamma$. For an automorphism $\sigma$ of G to be permutable with all the inner automorphisms of G, it is necessary and sufficient that, for all $x \in G$, $x^{-1}\sigma(x)$ belong to the centre of G; deduce that, if the centre of G is reduced to the identity element, so is the centralizer of $\Delta$ in $\Gamma$.