5 problems

Let $L\in\mathcal{L}$, where $\mathcal{L}$ is the specified list of nonabelian simple groups, let $S$ be a finite simple group of Lie type over a field whose characteristic differs…

Let $G$ be a simple group and let $H$ be a group. Write $|G|$ for the order of $G$ and let $\omega(G)$ denote the set of element orders of $G$. Shi's conjecture. $H\cong G$ if and…

Cross-characteristic composition-factor conjecture. The group $S$ cannot be a group of Lie type in characteristic coprime to $q$.

Recognition conjecture for classical groups. Every finite group isospectral to $L$ is an almost simple group with socle isomorphic to $L$.

Recognition-by-spectrum conjecture. Every finite group isospectral to $L$ is isomorphic to some group $G$ satisfying