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 $${\tau }_g(b):=\{\begin{array}{ll}{g}^b,& \text{if }{g}^b\in {\mathcal{H}}_p,\\ -g^b,& \text{if }{g}^b\notin {\mathcal{H}}_p,\end{array}$$ for each $b\in {\mathcal{H}}_p$, where ${\mathcal{H}}_p=\{1,2,\dots ,(p-1)/2\}$ is viewed as a subset of ${\mathbb{F}}_p$. In this paper, we investigate the sign of ${\tau }_g$. For example, if $p\equiv 5(\mathrm{mod}8)$, 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 $\mathbb{Q}(\sqrt{-4p})$.