nLab
inner automorphism

An inner automorphism ϕ:GG\phi:G\to G of a group GG is any automorphism ϕ g\phi_g of the form hghg 1h\mapsto g h g^{-1}. The inner automorphisms form a subgroup Inn(G)Inn(G), called the inner automorphism group of GG, of the entire automorphism group Aut(G)Aut(G); it is the image of the natural map GAut(G)G\to Aut(G) given by gϕ gg\mapsto\phi_g. The center of a group GG 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 22-category-theoretic kernel (an observation due to L. Breen).

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

Revised on September 9, 2009 00:25:19 by Toby Bartels (71.104.230.172)