# A note on the supersolvability of a finite group with prime index of some subgroups

## TL;DR

This paper establishes a characterization of supersoluble finite groups by showing that such a group is supersoluble if and only if it contains a supersoluble subgroup of prime index for every prime dividing its order.

## Contribution

It provides a new necessary and sufficient condition for supersolvability based on the existence of specific prime index subgroups.

## Key findings

- A finite group is supersoluble iff it has a supersoluble subgroup of prime index for each prime dividing its order.
- The result offers a practical criterion for verifying supersolvability in finite groups.
- The characterization simplifies the analysis of group structure through subgroup properties.

## Abstract

In this paper, we proved that a group $G$ is supersoluble if and only if for any prime $p\in \pi (G)$ there exists a supersoluble subgroup of index $p$.