On the Norms of p-Nilpotent Residuals of Subgroups in a Finite Group

Abstract

Let G be a finite group and p be a prime. We define $N^{\mathcal {N}_p*}(G)$ to be the intersection of the normalizers of the p-nilpotent residuals of all two-generator subgroups of G whose p-nilpotent residuals are nilpotent. We show that $N^{\mathcal {N}_p}(G)=N^{\mathcal {N}_p*}(G)$. Using the method in the present paper, we will be able to give an affirmative answer to an open problem in Shen et al. (Mediterr J Math 19:191, 2022), which also indicates that similar conclusions hold for many formations. It is also proved that $G=N^{\mathcal {N}_p}(G)$ if and only if every three-generator subgroup H of G satisfies $H=N^{\mathcal {N}_p}(H)$. To this end, we introduce and investigate the IO-$N^{\mathcal {N}_p}$-groups, i.e., the groups G such that $G\ne N^{\mathcal {N}_p}(G),$ but each proper subgroup and each proper quotient of G equals its p-nilpotent norm. Moreover, new results in terms of the p-nilpotent norm and the p-nilpotent hypernorm $N^{\mathcal {N}_p}_\infty (G)$ are given.