Herstein section 2.8. Automorphisms

[abhishekkgp]

Q. Let \(\displaystyle G\) be a finite group, \(\displaystyle T\) an automorphism of \(\displaystyle G\) with the property that \(\displaystyle T(x)=x \iff x=e\). Suppose further that \(\displaystyle T^2=I\). Prove that \(\displaystyle G\) is abelian.

ATTEMPT:
Using the "counting principle" it can be shown that each \(\displaystyle g \in G\) can be written as \(\displaystyle g=x^{-1} T(x)\) for some \(\displaystyle x \in G\). Also for a given \(\displaystyle g \in G\), this \(\displaystyle x\) is unique.
I can't see what to do next. Can some please help.
Thank you.

 

[daon2]

Took me a bit, but here is the idea:

You are correct about \(\displaystyle G\) consisting of the (uniquely defined) elements \(\displaystyle g_a = a^{-1}T(a)\), so I will assume you know that, and that \(\displaystyle g_a = g_b \iff a=b\). So let \(\displaystyle g_a \in G\). You can show directly that \(\displaystyle T(g_a) = T^{-1}(g_a) = g_a^{-1}\).

The first equal sign comes from the fact that \(\displaystyle T^{-1} = T\) and the second comes from applying \(\displaystyle T\) to \(\displaystyle g_a\) and playing around a bit.

So \(\displaystyle T\) is simply the automorphism sending each element to its inverse. That is, \(\displaystyle T(a) = a^{-1}\) for every \(\displaystyle a \in G\).

The rest should be pretty straight forward.

 

[abhishekkgp]

Took me a bit, but here is the idea:

You are correct about \(\displaystyle G\) consisting of the (uniquely defined) elements \(\displaystyle g_a = a^{-1}T(a)\), so I will assume you know that, and that \(\displaystyle g_a = g_b \iff a=b\). So let \(\displaystyle g_a \in G\). You can show directly that \(\displaystyle T(g_a) = T^{-1}(g_a) = g_a^{-1}\).

The first equal sign comes from the fact that \(\displaystyle T^{-1} = T\) and the second comes from applying \(\displaystyle T\) to \(\displaystyle g_a\) and playing around a bit.

So \(\displaystyle T\) is simply the automorphism sending each element to its inverse. That is, \(\displaystyle T(a) = a^{-1}\) for every \(\displaystyle a \in G\).

The rest should be pretty straight forward.

Thank you daon. That solved the question. How do i give you reputation points?

 

[daon2]

Thank you daon. That solved the question. How do i give you reputation points?

This board doesn't have that feature, but you're welcome