p-Nilpotent Maximal Subgroups in Finite Groups

Let p be a prime number and suppose that every maximal subgroup of a finite group is either p-nilpotent or has prime index. Such a group need not be p-solvable, and we study its structure by proving that only one nonabelian simple group of order divisible by p, which belongs to the family \(\textrm{PSL}_n(q)\) ${\text{PSL}}_n(q)$ , can be involved in it. For \(p=2\) $p=2$ , we specify more, and in fact, such a simple group must be isomorphic to \(\textrm{PSL}_2({r^a})\) ${\text{PSL}}_2(r^a)$ for certain values of the prime r and the parameter a.