nLab
inner automorphism

An inner automorphism $\varphi :G\to G$ of a group $G$ is any automorphism ${\varphi }_g$ of the form $h\mapsto gh{g}^{-1}$. The inner automorphisms form a subgroup $\mathrm{Inn}(G)$, called the inner automorphism group of $G$, of the entire automorphism group $\mathrm{Aut}(G)$; it is the image of the natural map $G\to \mathrm{Aut}(G)$ given by $g\mapsto {\varphi }_g$. The center of a group $G$ is precisely the kernel of this natural map. Similarly, the monoidal center due to Drinfel’d and Majid, in the case when the monoidal category is Picard, is a $2$-category-theoretic kernel (an observation due to L. Breen).

Higher analogues of the inner automorphism group were studied by Roberts and Schreiber.