AutoMorphism

• Jul 31st 2008, 04:44 AM
mathemanyak
AutoMorphism
Let g be a finite group and let f in Aut(G) with all g in G, f(g)=g iff g=e
Show that all g in G, there exist a in G s.t. g=a^(-1).f(a)
• Jul 31st 2008, 05:28 AM
NonCommAlg
Quote:

Originally Posted by mathemanyak
Let g be a finite group and let f in Aut(G) with all g in G, f(g)=g iff g=e
Show that all g in G, there exist a in G s.t. g=a^(-1).f(a)

let $A=\{a^{-1}f(a): \ a \in G \},$ and suppose that $a^{-1}f(a)=b^{-1}f(b),$ for some $a,b \in G.$ then $f(ab^{-1})=ab^{-1},$ and thus $ab^{-1}=e,$ i.e.

$a=b.$ thus $A=G. \ \ \ \square$

think about this: do we really need $f$ to be an automorphism or being homomorphism is enough?