Finite groups that admit an anti-automorphism with many fixed points - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-20T19:30:30Z http://mathoverflow.net/feeds/question/109245 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109245/finite-groups-that-admit-an-anti-automorphism-with-many-fixed-points Finite groups that admit an anti-automorphism with many fixed points Yuri Zarhin 2012-10-09T18:07:59Z 2012-10-10T04:50:11Z <blockquote> <p><strong>Possible Duplicate:</strong><br> <a href="http://mathoverflow.net/questions/38/homomorphism-more-than-3-4-the-inverse" rel="nofollow">Homomorphism more than 3/4 the inverse</a> </p> </blockquote> <p>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 $u$. </p> <p>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.</p>