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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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marked as duplicate by Benjamin Steinberg, S. Carnahan Oct 11 '12 at 4:50

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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. – Benjamin Steinberg Oct 9 '12 at 18:20
... or by looking here:… – Anton Klyachko Oct 9 '12 at 19:41
Just leave this here: – Steve D Oct 9 '12 at 22:34
Dear Yuri: I'm going to close as duplicate. Please flag the question if you disagree, and I can reopen. – S. Carnahan Oct 11 '12 at 4:50

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