$G$ is a finite group and $T$ an automorphism of $G$ such that $T$ sends more than $\frac{3}{4}|G|$ elements to their inverse.

Asked Sep 08 '13 at 06:30

Active Dec 27 '13 at 16:04

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If $G$ is a finite group and $T$ an automorphism of $G$ such that $T$ sends more than $\frac{3}{4}|G|$ elements to their inverse. Then we have show that $$T_x=x^{-1}$$ for all $ x \in G$ and $G$ is abelian.

I construct $ S= \{ x \in G : T_x=x^{ -1} \}$ and for $ b \in G $, $ S_b= \{ xb:x\in S \}$ then $| S \cup S_b|\geq (1/2)|G|$. Let $ a \in S\cap S_b $ then $ a=tb $ for some $t\in S $. I dont know how to solve it by this construction or any other method. Give me any kind of help.

edited Dec 27 '13 at 16:04

asked Sep 08 '13 at 06:30

Andy

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$G$ is a finite group and $T$ an automorphism of $G$ such that $T$ sends more than $\frac{3}{4}|G|$ elements to their inverse.

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