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

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.