Group automorphism not a subgroup?


by wheezyg
Tags: automorphism, subgroup
wheezyg
wheezyg is offline
#1
Sep17-10, 05:24 PM
P: 5
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
Phys.Org News Partner Science news on Phys.org
Going nuts? Turkey looks to pistachios to heat new eco-city
Space-tested fluid flow concept advances infectious disease diagnoses
SpaceX launches supplies to space station (Update)
boboYO
boboYO is offline
#2
Sep17-10, 08:56 PM
P: 104
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
arkajad is offline
#3
Sep18-10, 01:58 AM
P: 1,412
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.


Register to reply

Related Discussions
the order of two automorphism group Linear & Abstract Algebra 0
Automorphism Group Linear & Abstract Algebra 20
Cyclic Automorphism Group Linear & Abstract Algebra 1
Automorphism Group Linear & Abstract Algebra 13
group of automorphism of S3 Calculus & Beyond Homework 3