9 problems

Properness conjecture for perfect objects.

Let $X$ be a proper scheme of dimension $d$ over a field $k$, and let $L_1,\ldots,L_d$ be line bundles on $X$. Let $\langle L_1,\ldots,L_d\rangle$ denote the thick subcategory of…

Two-generation conjecture. If $n \neq 4$, then $\operatorname{\mathfrak{sl}}_n(\mathbf{F}_2)$ is $2$-generated.

FF-subgroup conjecture. All nontrivial subgroups of a finite simple group are FF-subgroups.

The $(2,3)$-generation conjecture. All finite simple classical groups, with some small exceptions in low rank and low characteristic, are $(2,3)$-generated.

Generation-bound conjecture. The bounds in Theorem 5 should extend in the obvious way to the corresponding finite simple groups of Lie type; in particular, $5$ conjugates should be…

Burness's conjecture. There does not exist a finite group $G$ satisfying $s(G)=1$.

Counterexample question. Does there exist a $2$-generated group with no noncyclic proper quotients that is not $\frac{3}{2}$-generated?

Let $G$ be a transitive permutation group of degree $n$, and let $d(G)$ denote the minimal size of a generating set for $G$. Set … The authors conjecture that the best asymptotic b…