On a problem from the Kourovka Notebook

TL;DR

This paper proves a theorem addressing a problem from the Kourovka Notebook, showing that under certain subgroup permutability conditions, a finite group is p-nilpotent.

Contribution

It provides a new sufficient condition involving subgroup $S$-propermutability for a group to be p-nilpotent, solving a specific problem from the Kourovka Notebook.

Findings

Establishes conditions under which a group is p-nilpotent based on subgroup properties.

Extends known results on subgroup permutability and group structure.

Addresses a previously open problem in finite group theory.

Abstract

In this manuscript, a solution to Problem 18.91(b) in the Kourovka Notebook is given by proving the following theorem. Let $P$ be a Sylow $p$-subgroup of a group $G$ with $|P| = p^n$. Suppose that there is an integer $k$ such that $1 < k < n$ and every subgroup of $P$ of order $p^k$ is $S$-propermutable in $G$, and also, in the case that $p=2$, $k = 1$ and $P$ is non-abelian, every cyclic subgroup of $P$ of order $4$ is $S$-propermutable in $G$. Then $G$ is $p$-nilpotent.