p-Nilpotent Maximal Subgroups in Finite Groups

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 $$\textrm{PSL}_n(q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>PSL</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , can be involved in it. For $$p=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , we specify more, and in fact, such a simple group must be isomorphic to $$\textrm{PSL}_2({r^a})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>PSL</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for certain values of the prime r and the parameter a .