10 problems

Let $H$ be a finite group, and let $\mathcal{M}(H)$ denote the set of maximal subgroups of $H$. Wall's conjecture. The bound … holds for every finite group $H$. The conjecture woul…

Aschbacher–Guralnick conjecture. One has

Maximal-subgroup generating density conjecture. For every finite group $G$,

One-outside-element conjecture. There exists a generating set of $G$ of size $d(G)$ having only one element outside $M$.

Maximal-subgroup classification conjecture. Exactly one of the following holds:

Let $\Gamma$ be a hyperbolic triangle group. A subgroup has infinite index if its index in $\Gamma$ is infinite, and two subgroups are conjugate when one is obtained from the other…

Let $\Gamma={\rm PSL}_2(\mathbb Z)$ be the modular group, and let $M_h$ be a subgroup obtained from Neumann's construction. A subgroup of $\Gamma$ is maximal nonparabolic if it is…

Let $\Gamma={\rm PSL}_2(\mathbb Z)$ be the modular group, and let $M_h$ denote a subgroup arising from Neumann's construction. A maximal nonparabolic subgroup is a maximal subgroup…

Let $J_1$ denote the first Janko sporadic simple group, and let $L_2(11)$ denote the group $\operatorname{PSL}_2(11)$. Existence conjecture. It was conjectured that $J_1$ contains…

Removal conjecture. The term $r^{1/2}p^{3r^{1/2}}$ can be removed from this bound, so the number of conjugacy classes of maximal subgroups is at most a constant multiple of…