On HC-subgroups of a finite group

TL;DR

This paper explores the structure of finite groups by analyzing subgroups of prime power order that satisfy a specific normality condition called $\\mathscr{H}C$-subgroups, revealing new insights into their subgroup configurations.

Contribution

It introduces the concept of $\\mathscr{H}C$-subgroups and investigates their impact on the overall structure of finite groups, extending existing subgroup theory.

Findings

Characterization of finite groups with $\\mathscr{H}C$-subgroups of prime power order

Conditions under which such subgroups influence group structure

New structural theorems relating $\\mathscr{H}C$-subgroups to group properties

Abstract

A subgroup $H$ of a finite group $G$ is said to be an $\mathscr{H}C$-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^g \cap N_T(H)\leq H$ for all $g\in G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that certain subgroups of $G$ of arbitrary prime power order are $\mathscr{H}C$-subgroups of $G$.