p-Nilpotent Maximal Subgroups in Finite Groups

Abstract

Abstract 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 $$\textrm{PSL}_n(q)$$ 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 $$\textrm{PSL}_2({r^a})$$ PSL 2 ( r a ) for certain values of the prime r and the parameter a .