9 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 $\mathcal{L}$ be the list of nonabelian simple groups not yet covered by the general recognition results: $L_n(q)$ with $8\leq n\leq26$ and $n$ not prime; $U_n(q)$ with…

Let $G$ be a finite simple group other than $A_6$, and let $\Gamma(G)$ be its Gruenberg–Kegel graph, whose vertices are the prime divisors of $|G|$, with distinct primes adjacent w…

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

Conjecture. If $K$ is the solvable radical of $G$, then

Let $G$ be a finite group isospectral to a finite nonabelian simple group $L$. Suppose that $G$ is nonsolvable and has a normal series … where $K$ is the solvable radical of $G$,…

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

Let $S$ be a finite nonabelian simple group. A finite group $G$ is isospectral to $S$ if they have the same sets of element orders, $(G)=(S)$. A group is almost simple with socle…

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