9.19 (1984)

Let $n(X)$ denote the minimum of the indices of proper subgroups of a group $X$. A subgroup $A$ of a finite group $G$ is called wide if $A$ is a maximal element by inclusion of the set $\{X \mid X$ is a proper subgroup of $G$ and $n(X) = n(G)\}$.
$\qquad$ a) Find all wide subgroups in finite projective special linear, symplectic, orthogonal, and unitary groups.
$\qquad$ b) Prove, without using CFSG, that $n(F_1) = |F_1 : 2F_2|$, where $F_1$ and $F_2$ are the Fischer simple groups and $2F_2$ is an extension of a group of order 2 by $F_2$.