4 problems

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 $[n]^{\pm}$ be the signed set, and let … and … be long cycles on $[n]^{\pm}$. Set $\pi=p\circ s$. Chen–Reidys cycle conjecture. The elements $n$ and $s_n$ belong to the same cy…

Given two rows, indexed $r$ and $s$, of a Latin square, let $\sigma_{s,r}=\sigma_s\sigma_r^{-1}$ be the permutation transforming $\sigma_r$ to $\sigma_s$. A relative cycle structur…