Finite supersolvable groups and Hall normally embedded subgroups of prime power order

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-07-03 DOI:10.1007/s11587-024-00873-6

Weicheng Zheng, Wei Meng

引用次数: 0

Abstract

Let G be a finite group. A group G is called a T group if its every subnormal subgroup is normal. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of \(H^G\), where \(H^G\) is the normal closure of H in G. Using the notion of Hall normally embedded subgroups, we characterize supersolvable groups and solvable T-group. First, we prove that if every cyclic subgroup of G of order prime or 4 is Hall normally embedded in G, then G is supersolvable with a well defined structure. Second, we prove that an A-group G is supersolvable if and only if its Sylow subgroups are products of cyclic Hall normally embedded subgroups of G. Final, we show that G is a solvable T-group if and only if every p-subgroup of G is Hall normally embedded in G, for all primes \(p\in \pi (G)\).

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有限可超溶群和霍尔素幂级数常内含子群

设 G 是一个有限群。如果一个群 G 的每个子正常子群都是正常的，那么这个群就叫做 T 群。如果 H 是 \(H^G\) 的霍尔子群，其中 \(H^G\) 是 H 在 G 中的常闭，那么 G 的一个子群 H 称为霍尔常嵌于 G。首先，我们证明，如果 G 的每个素数或 4 阶循环子群都是霍尔常嵌于 G 的，那么 G 是具有定义明确的结构的可超溶群。其次，我们证明了当且仅当一个 A 群 G 的 Sylow 子群是 G 的循环霍尔常内含子群的乘积时，G 是可解的。最后，我们证明了当且仅当 G 的每个 p 子群都是霍尔常内含于 G 时，对于所有素数 \(p\in \pi (G)\)，G 是可解的 T 群。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.