Prove that $Tx=x^{-1}~\forall~x\in G.$

Question

Let $G$ be a finite group and suppose the automorphism $T$ sends more than $\dfrac{3}{4}th$ of the elements of $G$ onto their inverses. Prove that $Tx=x^{-1}~\forall~x\in G.$

Answer 1

Cool problem. You should also be able to show that $G$ is Abelian. Groupprops has this: http://groupprops.subwiki.org/wiki/Automorphism_sends_more_than_three-fourths_of_elements_to_inverses_implies_abelian

Here's another related problem on Groupprops. This time, the endomorphism maps $\frac{3}{4}$ of all elements in $G$ to their squares. The proof is very similar. http://groupprops.subwiki.org/wiki/Endomorphism_sends_more_than_three-fourths_of_elements_to_squares_implies_abelian