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)

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- Jul 31st 2008, 04:44 AMmathemanyakAutoMorphism
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 AMNonCommAlg
let $\displaystyle A=\{a^{-1}f(a): \ a \in G \},$ and suppose that $\displaystyle a^{-1}f(a)=b^{-1}f(b),$ for some $\displaystyle a,b \in G.$ then $\displaystyle f(ab^{-1})=ab^{-1},$ and thus $\displaystyle ab^{-1}=e,$ i.e.

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

__think about this:__do we really need $\displaystyle f$ to be an automorphism or being homomorphism is enough?