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Homomorphism more than 3/4 the inverse

Let $G$ be a finite group that admits an anti-automorphism $u: G \to G$ and let $S$ be the set of fixed points of $u$.

I am looking for references/results that deal with the structure of $G$ when $S$ is big, i.e., the ratio $$r=\\#(S)/\\#(G)$$ is big. For example, if $r=1$ (i.e., $S=G$) then (obviously) $G$ is abelian. One may check that if $r>1/2$ then $G$ is also abelian if we assume additionally that order of $G$ is an odd number. I've heard that if $r>3/4$ then one may conclude (without any additional assumptions) that $G$ is abelian but cannot find a reference.

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    $\begingroup$ This is the same as asking about the ratio of the number of elements sent by an automorphism to their inverse. The r>3/4 thing is a classic exercise from Herstein and a solution can be found by googling. $\endgroup$
    – Benjamin Steinberg
    Oct 9, 2012 at 18:20
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    $\begingroup$ ... or by looking here: mathoverflow.net/questions/38/… $\endgroup$
    – Anton Klyachko
    Oct 9, 2012 at 19:41
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    $\begingroup$ Just leave this here: springerlink.com/content/k52852552w7570v4/?MUD=MP $\endgroup$
    – Steve D
    Oct 9, 2012 at 22:34
  • $\begingroup$ Dear Yuri: I'm going to close as duplicate. Please flag the question if you disagree, and I can reopen. $\endgroup$
    – S. Carnahan ♦
    Oct 11, 2012 at 4:50

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