# On the supersolubility of a finite group with NS-supplemented Sylow subgroups

## TL;DR

This paper investigates the conditions under which a finite group is supersoluble, focusing on the role of NS-supplemented Sylow subgroups and maximal subgroups, and establishes new criteria for supersolubility.

## Contribution

It introduces the concept of NS-supplemented Sylow subgroups and proves their role in ensuring a group's supersolubility, extending previous results in group theory.

## Key findings

- Groups with NS-supplemented non-cyclic Sylow subgroups are supersoluble.
- Solubility is established for groups with NS-supplemented maximal subgroups.

## Abstract

A subgroup $A$ of a group~$G$ is said to be {\sl NS-supplemented} in $G$, if there exists a subgroup~$B$ of $G$ such that $G=AB$ and whenever $X$~is a normal subgroup of~$A$ and $p\in \pi(B)$, there exists a Sylow $p$-subgroup~$B_p$ of~$B$ such that $XB_p=B_pX$. In this paper, we proved the supersolubility of a group with NS-supplemented non-cyclic Sylow subgroups. The solubility of a group with NS-supplemented maximal subgroups is obtained.