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Familiar Exercise: Suppose $G$ is a finite group and $T$ is an automorphism of $G$ which sends more than three quarters of elements of $G$ onto their inverses, then prove that $G$ is abelian.

The group of Quarternions is an example of a group which sends exactly $\displaystyle \frac{3}{4}$ elements onto their inverses.

Is there a finite group with an automorphism $T$ which sends exactly $\displaystyle\frac{4}{5}$ of elements of $G$ onto their inverses?

Similarly can we find groups with Automorphism $T$ which sends exactly $\displaystyle \frac{n}{n+1}$ of elements of $G$ onto their inverses.

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Perhaps of interest, the only finite groups which admit these "3/4 automorphisms" are precisely those with center of index 4. –  user641 Aug 17 '10 at 10:50

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Prove that if $G$ is abelian and $\phi$ is an automorphism of $G$ then $\{g\in G : \phi(g)=g^{-1} \}$ is a subgroup of $G$.

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I don't think the latter part of your question is very interesting; if you do my exercise you'll see why :-) –  Robin Chapman Aug 13 '10 at 20:11
    
@Chandru: As you know, the order of a subgroup has to divide the order of the group, so. ... :) –  user1119 Aug 13 '10 at 20:15

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