Some permutations over $\pmb {\mathbb {F}}_p$ 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 $\tau _g:\,{\mathcal {H}}_p\rightarrow {\mathcal {H}}_p$ by

$$\begin{aligned} \tau _g(b):={\left\{ \begin{array}{ll} g^b&{}\text {if }g^b\in {\mathcal {H}}_p,\\ -g^b&{}\text {if }g^b\not \in {\mathcal {H}}_p,\\ \end{array}\right. } \end{aligned}$$

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 $p\equiv 5\ (\mathrm{{mod}}\ 8)$, then

$$\begin{aligned} (-1)^{|\tau _g|}=(-1)^{\frac{1}{4}(h(-4p)+2)} \end{aligned}$$

for every primitive root g, where $h(-4p)$ is the class number of the ring of integers of the imaginary quadratic field ${\mathbb {Q}}(\sqrt{-4p})$.