Finite Groups Whose Maximal Subgroups are 2-Nilpotent or Normal

Abstract

We describe the structure of those finite groups whose maximal subgroups are either 2-nilpotent or normal. Among other properties, we prove that if such a group G does not have any non-trivial quotient that is a 2-group, then G is solvable. Also, if G is a solvable group satisfying the above conditions, then the 2-length of G is less than or equal to 2. If, on the contrary, G is not solvable, then G has exactly one non-abelian principal factor and the unique simple group involved is one of the groups \(\textrm{PSL}_2(p^{2^a})\), where p is an odd prime and \(a\ge 1\), or p is a prime satisfying \(p\equiv \pm 1\) \((\textrm{mod}~ 8)\) and \(a=0\).