Q:

Define an involution to show that there is an element of order $2$ in $(\mathbb{Z}/p\mathbb{Z})^{\times}$

I am working on the following exercise:

Let $p$ be an odd prime with $p \equiv 1 \pmod 4$.Define an involution on $(\mathbb{Z}/p\mathbb{Z})^{\times}$ which shows that there is an element of order $2$ in $(\mathbb{Z}/p\mathbb{Z})^{\times}$.

I know what an involution is, but I do not know how to even start with this exercise.  Could you help me? 

A:

Consider the involution $x\mapsto x^{p-2}=x^{-1}.$ 
If there is $y$ such that $y\ne1$ but $y=y^{-1}$ (i.e., $y^2=1$), then we are done. 
Otherwise, pair each element $x$ in $(\mathbb Z/p\mathbb Z)^\times$ with $x^{-1}$.  
Note that $1$ is self-inverse.   
Therefore, since the number ($p-1$) of elements is even, 
there must be another element $y\ne1$ that is without a partner so it is its own inverse $y=y^{-1}$; 
that means   $y$ is an element of order $2$ in $(\mathbb Z/p\mathbb Z)^\times$.

