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I was helping a friend prepare for his intro abstract final and he mentioned the professor had once asked the question name a group and an automorphism that takes 3/4 of the elements of the group to their own inverses (D4, identity). I tried to figure out how to approach this question in general but can't see how.

Further questions:

Can we construct the group?

Does anything change if we allow infinite groups?

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Maybe edit the title of your question: a/b is a rational number $\le 1$. And if we allow infinite groups, how do you define a fraction of the elements? Unless there is a natural density function, you might be in trouble, for example: $(\mathbb{Z}/2)^\mathbb{N} \times \mathbb{Z}^\mathbb{N}$ –  David Roberts Jan 3 '12 at 23:14

1 Answer 1

up vote 15 down vote accepted

This may be a well-known chestnut? (well-known to those that know it well, that is)

The fraction can never be between 3/4 and 1. To prove this, suppose $\phi\colon G\to G$ is an automorphism of $G$ that sends more than 3/4 of the elements of $G$ to their inverses. Let $S=\lbrace g\in G\colon \phi(g)=g^{-1}\rbrace$.

Notice that if $g$, $h$ and $gh$ all lie in $S$ then on the one hand, $\phi(gh)=(gh)^{-1}=h^{-1}g^{-1}$. On the other hand, $\phi(gh)=\phi(g)\phi(h)=g^{-1}h^{-1}$, so that $g^{-1}$ and $h^{-1}$ commute. It follows that $g$ and $h$ commute.

Fix $g\in S$ and consider $A=\lbrace h\colon h\in S\text{ and } gh\in S\rbrace$. There are less than $|G|/4$ $h$'s for which the first condition fails and less than $|G|/4$ $h$'s for which the second condition fails, so that $|A|>|G|/2$. By the above, it follows that the centralizer of $g$ (i.e. the set of $h$'s that commute with $g$) is a superset of $A$. Since the centralizer is a subgroup, by Lagrange's theorem the centralizer of $g$ must be all of $G$. That is $g$ lies in the center of the group. Now we have that more than $1/2$ of the group lies in the center (which is again a subgroup), so that $G$ is Abelian.

Now since $S$ is more than half of the group, the subgroup generated by $S$ must be all of $G$, so that every element of $G$ is a product of elements of $S$. Now $g\in G$, write $g=s_1\ldots s_n$. Then $\phi(g)=\phi(s_1)\ldots\phi(s_n)=s_1^{-1}\ldots s_n^{-1}=s_n^{-1}\ldots s_1^{-1}=g^{-1}$, so that $\phi(g)=g^{-1}$ for all $g\in G$.

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I believe the full set of fractions is known, see MacHale and Liebeck, "Groups With Automorphisms Inverting Most Elements". –  Steve D Jan 4 '12 at 0:49
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This answer shows that the ratio $|S|/|G|$ can never lie strictly between $3/4$ and $1$, but the ratio $3/4$ itself is actually possible. Namely, the dihedral group of order $8$ has an automorphism that sends precisely $6$ elements to their respective inverses. –  Florian Eisele Jan 4 '12 at 0:53
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Yes, in fact the paper I mentioned above shows the only possibilities between 1/2 and 1 are fractions of the form $(n+1)/2n$, for integers $n$. Every such fraction is realized, and if a group realizes one fraction, it cannot realize another. –  Steve D Jan 4 '12 at 0:55
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Also groups realizing $3/4$ are precisely those with center of index $4$. –  Steve D Jan 4 '12 at 0:56
    
Yes, this is a well-known chestnut: mathoverflow.net/questions/48 . –  Anton Klyachko Dec 15 '12 at 6:33

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