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

IF 1.1 3区 数学 Q1 MATHEMATICS
Baoyu Zhang, Quanfu Yan, Zhencai Shen
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引用次数: 0

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.

论有限群中子群的 p 弱残差规范
设 G 是有限群,p 是素数。我们定义 \(N^{\mathcal {N}_p*}(G)\) 为 G 的所有双发电机子群的 p-nilpotent 残差的归一化的交集,这些子群的 p-nilpotent 残差都是 nilpotent。我们证明了 \(N^{\mathcal {N}_p}(G)=N^{\mathcal {N}_p*}(G)\).利用本文的方法,我们将能够对 Shen 等人 (Mediterr J Math 19:191, 2022) 中的一个未决问题给出肯定的答案,这也表明类似的结论在许多形式中都成立。我们还证明,当且仅当 G 的每个三发电机子群 H 满足 \(H=N^{\mathcal {N}_p}(H)\) 时,\(G=N^{\mathcal {N}_p}(G)\) 才成立。为此,我们引入并研究了 IO-\(N^{\mathcal {N}_p}\)-群,即 G 群,使得 \(G\ne N^{\mathcal {N}_p}(G),\) 但是 G 的每个适当子群和每个适当商都等于它的 p-nilpotent norm。此外,还给出了 p-nilpotent norm 和 p-nilpotent hypernorm \(N^{\mathcal {N}_p}_infty (G)\) 的新结果。
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来源期刊
CiteScore
1.80
自引率
0.00%
发文量
261
审稿时长
6-12 weeks
期刊介绍: The Mediterranean Journal of Mathematics (MedJM) is a publication issued by the Department of Mathematics of the University of Bari. The new journal replaces the Conferenze del Seminario di Matematica dell’Università di Bari which has been in publication from 1954 until 2003. The Mediterranean Journal of Mathematics aims to publish original and high-quality peer-reviewed papers containing significant results across all fields of mathematics. The submitted papers should be of medium length (not to exceed 20 printed pages), well-written and appealing to a broad mathematical audience. In particular, the Mediterranean Journal of Mathematics intends to offer mathematicians from the Mediterranean countries a particular opportunity to circulate the results of their researches in a common journal. Through such a new cultural and scientific stimulus the journal aims to contribute to further integration amongst Mediterranean universities, though it is open to contribution from mathematicians across the world.
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