7 problems

Let $q=p^n$ be a prime power and let $\mathbb{F}_q$ be the finite field with $q$ elements. A mapping $g$ has fixed point $0$ and a cycle of length $q-1$ if its permutation action c…

Let $\Gamma$ be a finite group. An orthomorphism of $\Gamma$ is a bijection $\varphi:\Gamma\to\Gamma$ such that the map $g\mapsto g^{-1}\varphi(g)$ is also bijective. Hall–Paige co…

Let $\Gamma$ be an Abelian group of order $n$ such that $|I(\Gamma)|\neq 1$, where $I(\Gamma)$ denotes the set of involutions. An orthomorphism is a bijection of $\Gamma$ for which…

Non-abelian Friedlander–Gordon–Tannenbaum conjecture. If $k\geq 3$ and $k$ divides $n-1$, then there exists an orthomorphism of $G$ that fixes the identity element and permutes the…

Let $q$ range over prime powers, and let an irregular orthomorphism be an orthomorphism whose translate $T_g\theta=\theta(x+g)-\theta(g)$ is non-cyclotomic for every…

Let $q$ range over prime powers, and let an orthomorphism polynomial over … . Its reduced degree is the degree of its unique representative of degree at most $q-1$. Reduced-degree…

Let $\mathbb{F}_q$ be a finite field, and let an orthomorphism $\theta$ be called irregular if $T_g[\theta]$ is non-cyclotomic for every $g\in\mathbb{F}_q$. Existence conjecture. I…