Group automorphism not a subgroup?

wheezyg

I was wondering if anyone could shed some light on this... I thought Aut(G) was always a subgroup of G but I dont think I can prove it. This is leading me to second guess this intuition. Could I get some reading reccomendations from anyone on this? Thx

boboYO

Your statement 'Aut(G) always a subgroup of G' doesn't really make sense. Elements of Aut(G) are isomorphisms from G to G. So they aren't even the same type of object as elements of G.


Do you mean to say that Aut(G) is always a group under composition? This is easy to prove as the composition of automorphisms is an automorphism ,and the inverse of an automorphism is an automorphism, so Aut(G) has group structure


Or perhaps you mean to ask, 'are all automorphisms inner?'

arkajad

You have a group homomorphism [tex]\rho:\,G\rightarrow \mbox{Aut}(G),\; g\rightarrow\rho_g[/tex] given by

[tex]\rho_g(h)=ghg^{-1}[/tex]

Thus [tex]G[/tex] has an image, possibly with a non-trivial kernel, in [tex]\mbox{Aut}(G)[/tex] - 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.