Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 2 / 2C (Groups, Rings and Modules)

A group automorphism is a bijective group homomorphism from $G$ to itself. The multiplication in $\mathrm{Aut}(G)$ is composition: $(\alpha \beta )(g)=\alpha (\beta (g))$. Composition is associative, the identity map is the identity element, and each inverse bijection is again a homomorphism, so these maps form the automorphism group.

For conjugation by $h$, multiplication is preserved becauseConjugation by $h^{-1}$ is its inverse, so $\psi (h)$ is an group automorphism. MoreoverHence $\psi :G\to \mathrm{Aut}(G)$ is a homomorphism, with image the inner automorphisms. Its group homomorphism kernel is the group center of $G$, since $\psi (h)$ is the identity precisely when $hg=gh$ for every $g$.