# Finite groups that admit an anti-automorphism with many fixed points [duplicate]

Possible Duplicate:
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 was marked as an exact duplicate of an existing 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: mathoverflow.net/questions/38/… – Anton Klyachko Oct 9 '12 at 19:41
Just leave this here: springerlink.com/content/k52852552w7570v4/?MUD=MP – 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