# 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.

• 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. Oct 9, 2012 at 18:20
• ... or by looking here: mathoverflow.net/questions/38/… Oct 9, 2012 at 19:41
• Just leave this here: springerlink.com/content/k52852552w7570v4/?MUD=MP Oct 9, 2012 at 22:34
• Dear Yuri: I'm going to close as duplicate. Please flag the question if you disagree, and I can reopen. Oct 11, 2012 at 4:50