On finite groups with certain complemented $p$-subgroups

TL;DR

This paper classifies finite groups with specific complemented p-subgroups, focusing on groups where all subgroups of a certain order are complemented, especially under conditions involving normal elementary abelian Sylow p-subgroups.

Contribution

It provides a classification of finite groups with complemented p-subgroups under particular divisibility and normality conditions, extending understanding of subgroup complementarity.

Findings

Classified groups with all subgroups of order p^d complemented when p^{2d} divides |G|

Characterized groups with normal elementary abelian Sylow p-subgroups where all subgroups of order p^d are complemented

Identified structural properties related to subgroup complementarity in finite groups

Abstract

Given a prime power $p^d$ with $p$ a prime and $d$ a positive integer, we classify the finite groups $G$ with $p^{2d}$ dividing $|G|$ in which all subgroups of order $p^d$ are complemented and the finite groups $G$ having a normal elementary abelian Sylow $p$-subgroup $P$ such that $p^d<|P|$ in which all subgroups of order $p^d$ are complemented.