is f(f(x)) = f(x) f(x) , f is automorphism f^2 is identity

let G be a finite group

if sigma is automorphism from G to G such that

given that is the identity automorphism

prove that G is abelian ( hint prove that every element of G can be written as apply sigma to such expression

as you can see I end up with what I am asking about, did I made something wrong?

I work in it in another way

since sigma^2 is the identity

let

so if x image is x* then x* image is x

Re: is f(f(x)) = f(x) f(x) , f is automorphism f^2 is identity

Quote:

Originally Posted by

**Amer** let G be a finite group

if sigma is automorphism from G to G such that

given that

is the identity automorphism

prove that G is abelian ( hint prove that every element of G can be written as

apply sigma to such expression

as you can see I end up with what I am asking about, did I made something wrong?

I work in it in another way

since sigma^2 is the identity

let

so if x image is x* then x* image is x

I think you are making the mistake of interpreting and not , no?

Re: is f(f(x)) = f(x) f(x) , f is automorphism f^2 is identity

Re: is f(f(x)) = f(x) f(x) , f is automorphism f^2 is identity

the multiplication in Aut(G) is functional composition, not the "element-wise" multiplication of G.

Re: is f(f(x)) = f(x) f(x) , f is automorphism f^2 is identity

Re: is f(f(x)) = f(x) f(x) , f is automorphism f^2 is identity

Quote:

Originally Posted by

**Amer** so what I did wrong in this

the question gave me a hint which said show that every element in G can be written as

so I suppose that

then i apply sigma to both sides

No - every element can be written as , but the may vary. Basically, for every there exists an such that .

Re: is f(f(x)) = f(x) f(x) , f is automorphism f^2 is identity

Re: is f(f(x)) = f(x) f(x) , f is automorphism f^2 is identity

Swlabr, NonCommAlg Thanks very much it is very clear now