An inner automorphism of a group is any automorphism of the form . The inner automorphisms form a subgroup , called the inner automorphism group of , of the entire automorphism group ; it is the image of the natural map given by . The center of a group 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 -category-theoretic kernel (an observation due to L. Breen).
Higher analogues of the inner automorphism group were studied by Roberts and Schreiber.