On two questions from the Kourovka Notebook concerning maximal subgroups

Let \(p\) be a prime number.

When \(p\) is odd, we study finite groups in which every maximal subgroup is either non-abelian simple or \(p\)-nilpotent, as well as those in which every maximal subgroup is either non-abelian simple or \(p\)-decomposable.

We prove that every non-simple, non-solvable group satisfying these criteria is \(p\)-nilpotent, and \(p\)-decomposable, respectively.

This answers two open questions posed by V.S.

Monakhov and I.N.

Tyutyanov in the Kourovka Notebook.

Additionally, if \(p=2\), we improve the main result of Monakhov and Tyutyanov by providing a complete classification of non-solvable groups whose maximal subgroups are either non-abelian simple or \(2\)-nilpotent.