On finite groups with certain complemented p-subgroups

Abstract

Given a prime power $p^d$ where p is a prime and d is a positive integer, we aim to classify two classes of finite groups based on their complementary properties of certain p-subgroups. The first class of groups are those in which $p^{2d}$ divides the order of the groups and all subgroups of order $p^d$ are complemented. The second class of groups are those having a normal elementary abelian Sylow p-subgroup P, with $p^d$ less than $\left|P\right|$, and all subgroups of order $p^d$ are complemented.

KEYWORDS:

• Complemented subgroup
• finite group

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D10

Acknowledgments

The author gratefully acknowledge the support of China Scholarship Council (CSC).