# Math Help - Algebra, Problems For Fun (37)

1. ## Algebra, Problems For Fun (37)

Let $G$ be a finite non-abelian group and $f \in \text{Aut}(G).$ Let $S=\{x \in G: \ f(x)=x^{-1} \}.$ Prove that: $|S|\leq \frac{3}{4}|G|.$

2. What about this approach: Suppose $|S|>\dfrac{3|G|}{4}$

Though $S$ is not necessarily a subgroup, it does contain the identity and inverses by the automorphism properties, i.e. if $e$ is the identity and $x\in S$, $f(e)=e;\, f(x^{-1})=f(x)^{-1}=(x^{-1})^{-1}=x$. Then in particular $f(f(x))=x$, and the set of fixed points of $f\circ f$ does form a subgroup. Since this subgroup contains more than half of the elements it follows that $f(f(x))=x$ for all elements of the group.

If we can somehow show that in fact $S=G$, we are done.