4 problems

A group is simple if it has no nontrivial proper normal subgroups, and finitely presented if it admits a presentation with finitely many generators and finitely many defining relat…

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.

Let $(G_n)$ be any sequence of finite simple groups such that $|G_n|$ tends to infinity with $n$. Let $\mathbb{P}_2(G)$ denote the probability that two randomly chosen elements gen…