show that if $\lvert \{g\in G | \varphi(g) = g\}\rvert \gt \frac{\lvert G\rvert}{2} $ then $\varphi = Id$

Question

Let $G$ be a finite group, and $\varphi \in Aut(G)$

show that if $\lvert \{g\in G | \varphi(g) = g\}\rvert \gt \frac{\lvert G\rvert}{2} $

then $\varphi = Id$

Answer 1

Hint: let $H=\{g\in G:\varphi(g)=g\}$.

Then $1\in H$. Suppose $x,y\in H$; then $$ \varphi(xy^{-1})=\varphi(x)\varphi(y)^{-1}=xy^{-1} $$ So $H$ is a subgroup of $G$.