The conjugacy diameters of non-abelian finite $ p $-groups with cyclic maximal subgroups

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Fawaz Aseeri, J. Kaspczyk
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引用次数: 0

Abstract

Let $ G $ be a group. A subset $ S $ of $ G $ is said to normally generate $ G $ if $ G $ is the normal closure of $ S $ in $ G. $ In this case, any element of $ G $ can be written as a product of conjugates of elements of $ S $ and their inverses. If $ g\in G $ and $ S $ is a normally generating subset of $ G, $ then we write $ \| g\|_{S} $ for the length of a shortest word in $ \mbox{Conj}_{G}(S^{\pm 1}): = \{h^{-1}sh | h\in G, s\in S \, \mbox{or} \, s{^{-1}}\in S \} $ needed to express $ g. $ For any normally generating subset $ S $ of $ G, $ we write $ \|G\|_{S} = \mbox{sup}\{\|g\|_{S} \, |\, \, g\in G\}. $ Moreover, we write $ \Delta(G) $ for the supremum of all $ \|G\|_{S}, $ where $ S $ is a finite normally generating subset of $ G, $ and we call $ \Delta(G) $ the conjugacy diameter of $ G. $ In this paper, we derive the conjugacy diameters of the semidihedral $ 2 $-groups, the generalized quaternion groups and the modular $ p $-groups. This is a natural step after the determination of the conjugacy diameters of dihedral groups.
具有循环最大子群的非阿贝尔有限 $ p $ 群的共轭直径
让 $ G $ 是一个群。如果 $ G $ 是 $ S $ 在 $ G $ 中的正常闭包, 那么 $ G $ 的一个子集 $ S $ 就被称为正常生成 $ G $.如果 $ g\in G $ 和 $ S $ 是 $ G 的正常生成子集,那么我们可以写 $ \| g\|_{S} $ 表示 $ \mbox{Conj}_{G}(S^{\pm 1}) 中最短单词的长度: = \{h^{-1}sh | h\in G, s\in S \, \mbox{or}\对于 $ G 的任何正常生成子集 $ S $, $ 我们写 $\|G\|_{S} = \mbox{sup}\{\|g\|_{S} \, |\, \, g\in G\}.$ 此外,我们把所有 $\|G\|_{S} 的上集写成 $\Delta(G)$,其中 $ S $ 是 $ G 的有限常生成子集,$ 我们称 $ \Delta(G) $ 为 $ G 的共轭直径。 $ 在本文中,我们推导了半二面体 $ 2 $ 群、广义四元数群和模数 $ p $ 群的共轭直径。这是在确定了二面群的共轭直径之后的一个自然步骤。
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
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.
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