# Group automorphism not a subgroup?

by wheezyg
Tags: automorphism, subgroup
 P: 1,412 You have a group homomorphism $$\rho:\,G\rightarrow \mbox{Aut}(G),\; g\rightarrow\rho_g$$ given by $$\rho_g(h)=ghg^{-1}$$ Thus $$G$$ has an image, possibly with a non-trivial kernel, in $$\mbox{Aut}(G)$$ - these are called "inner automorphisms". But, in general, there can be also "outer automorphisms" - automorphisms of G that can not be implemented by any element of G.