Some permutations over ${\mathbb F}_p$ concerning primitive roots

Abstract: Let $p$ be an odd prime and let ${\mathbb F}_p$ denote the finite field with $p$ elements. Suppose that $g$ is a primitive root of ${\mathbb F}_p$. Define the permutation $\tau_g:\,{\mathcal H}_p\to{\mathcal H}_p$ by $$ \tau_g(b):=\begin{cases} g^b,&\text{if }g^b\in{\mathcal H}_p,\ -g^b,&\text{if }g^b\not\in{\mathcal H}_p,\ \end{cases} $$ for each $b\in{\mathcal H}_p$, where ${\mathcal H}_p={1,2,\ldots,(p-1)/2}$ is viewed as a subset of ${\mathbb F}_p$. In this paper, we investigate the sign of $\tau_g$. For example, if $p\equiv 5\pmod{8}$, then $$ (-1)^{|\tau_g|}=(-1)^{\frac{1}{4}(h(-4p)+2)} $$ for every primitive root $g$, where $h(-4p)$ is the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-4p})$.