nLab inner automorphism (changes)

Showing changes from revision #1 to #2: Added | Removed | Changed

An inner automorphism $\phi:G\to G$ of a group $G$ is any automorphism $\phi_g$ of the form $h\mapsto {\mathrm{ghg}}^{-1}gh{g}^{-1}$ h\mapsto ghg^{-1} g h g^{-1} . The inner automorphisms make form a subgroup$Inn(G)$subgroup of the group of automorphisms$\mathrm{<span><del\; class="diffmod">\; Aut</del><ins\; class="diffmod">\; Inn</ins></span>}(G)$ Aut(G) Inn(G) , which called is the image of the natural map$G\to Aut(G)$inner automorphism group , of$\mathrm{<span><del\; class="diffmod">\; g</del><ins\; class="diffmod">\; G</ins></span>}\mapsto {\varphi }_g$ g\mapsto\phi_g G . , The of the entireautomorphism 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 map. Similarly, the monoidal center due Drinfel’d and Majid, in the case when the monoidal category is Picard is a 2-categorical kernel (an observation due L. Breen).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 categorical analogues of the inner automorphism group were studied by Roberts and Schreiber.