nLab inner automorphism

An inner automorphism $\phi:G\to G$ of a group $G$ is any automorphism $\phi_g$ of the form $h\mapsto g h g^{-1}$. The inner automorphisms form a subgroup $Inn(G)$, called the inner automorphism group of $G$, of the entire automorphism group $Aut(G)$; it is the image of the natural map $G\to Aut(G)$ given by $g\mapsto\phi_g$. 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.