# Thread: Automorphism - question

1. ## Automorphism - question

G - Finite Group
T - Automorphism of G
For all x in G, T(x) = x IFF x=1
Also, T(T(x))=x for all x in G.

Prove G is abelian.

Struggling a lot with this. I suspect $\displaystyle T(x) = x^{-1}$.
And once I establish this, proving G is abelian is simple.

But not going anywhere with the hunch of $\displaystyle T(x) = x^{-1}$.

Any pointers please?

2. Originally Posted by aman_cc
G - Finite Group
T - Automorphism of G
For all x in G, T(x) = x IFF x=1
Also, T(T(x))=x for all x in G.

Prove G is abelian.

Struggling a lot with this. I suspect $\displaystyle T(x) = x^{-1}$.
And once I establish this, proving G is abelian is simple.

But not going anywhere with the hunch of $\displaystyle T(x) = x^{-1}$.

Any pointers please?
define the function $\displaystyle f: G \longrightarrow G$ by $\displaystyle f(g)=g^{-1}T(g).$ now if $\displaystyle f(g_1)=f(g_2),$ then $\displaystyle T(g_1g_2^{-1})=g_1g_2^{-1}$ and thus $\displaystyle g_1=g_2,$ because we're given that $\displaystyle T(x)=x \Longrightarrow x = 1.$ so $\displaystyle f$ is injective and thus

surjective, because $\displaystyle G$ is finite. now let $\displaystyle x \in G.$ then $\displaystyle x=f(g)=g^{-1}T(g),$ for some $\displaystyle g \in G.$ thus $\displaystyle T(x)=T(g^{-1}T(g))=T(g^{-1})T(T(g))=(T(g))^{-1}g=(g^{-1}T(g))^{-1}=x^{-1}.$ hence $\displaystyle G$ is abelian.

3. Originally Posted by NonCommAlg
define the function $\displaystyle f: G \longrightarrow G$ by $\displaystyle f(g)=g^{-1}T(g).$ now if $\displaystyle f(g_1)=f(g_2),$ then $\displaystyle T(g_1g_2^{-1})=g_1g_2^{-1}$ and thus $\displaystyle g_1=g_2,$ because we're given that $\displaystyle T(x)=x \Longrightarrow x = 1.$ so $\displaystyle f$ is injective and thus

surjective, because $\displaystyle G$ is finite. now let $\displaystyle x \in G.$ then $\displaystyle x=f(g)=g^{-1}T(g),$ for some $\displaystyle g \in G.$ thus $\displaystyle T(x)=T(g^{-1}T(g))=T(g^{-1})T(T(g))=(T(g))^{-1}g=(g^{-1}T(g))^{-1}=x^{-1}.$ hence $\displaystyle G$ is abelian.
Thanks !
I could never have guessed it - to define $\displaystyle f(g)=g^{-1}T(g)$ and then do it !!!