On the weak norm of $\mathscr{U}_p$-residuals of all subgroups of a finite group

TL;DR

This paper investigates the structure of finite groups through the properties of the weak norm of subgroups relative to certain group classes, extending previous research in group theory.

Contribution

It characterizes finite groups using the weak norm of subgroups with respect to $\u$ and $$, providing new insights into their structure.

Findings

Characterization of finite groups via weak norms

Extension of previous group-theoretic results

Structural insights for $p$-supersolvable and supersolvable groups

Abstract

Let $\mathscr{F}$ be a formation and $G$ a finite group. The weak norm of a subgroup $H$ in $G$ with respect to $\mathscr{F}$ is defined by $N_{\mathscr{F}}(G,H)=\underset{T\leq H}{\bigcap}N_G(T^{\mathscr{F}})$. In particular, $N_{\mathscr{F}}(G)=N_{\mathscr{F}}(G,G)$. Let $N^i_{\mathscr{F}}(G)$,$i\geq 1$, be a upper series of $G$ by setting $N^0_{\mathscr{F}}(G)=1$, $N^{i+1}_{\mathscr{F}}(G)/N^i_{\mathscr{F}}(G)=N_{\mathscr{F}}(G/N^i_{\mathscr{F}}(G))$ and denoted by $N^{\infty}_{\mathscr{F}}(G)$ the terminal term of the series. In this paper, for the case $\mathscr{F}\in\{\mathscr{U}_p,\mathscr{U}\}$, where $\mathscr{U}_p$($\mathscr{U}$,respectively) is the class of all finite $p$-supersolvable groups(supersolvable groups,respectively), we characterize the structure of some given finite groups by the properties of weak norm of some subgroups in $G$ with respect to $\mathscr{F}$. Some of our main results may regard as a continuation of many nice previous work.