5 problems

Let $p$ be a prime, and suppose that $G=P\times A$ is a finite group, where $P\in\operatorname{Syl}_p(G)$ is non-abelian and $A$ is an abelian $p'$-group. Let $\Gamma_K$ denote the…

Let $G$ be a finite non-abelian nilpotent group, and let $H$ be a group. For a finite non-abelian group $K$, let $\Gamma_K$ be the graph whose vertices are the non-central elements…

Let $G$ and $H$ be non-abelian finite groups. For a finite non-abelian group $K$, let $\Gamma_K$ be the graph whose vertices are the non-central elements of $K$, with two vertices…

Class 1 conjecture. The relative non-commuting graph $\Gamma_{S,R}$ is of class $1$.

Order-determination conjecture. If