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

IF 1.8 3区 数学 Q1 MATHEMATICS
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
AIMS Mathematics
AIMS Mathematics Mathematics-General Mathematics
CiteScore
3.40
自引率
13.60%
发文量
769
审稿时长
90 days
期刊介绍: AIMS Mathematics is an international Open Access journal devoted to publishing peer-reviewed, high quality, original papers in all fields of mathematics. We publish the following article types: original research articles, reviews, editorials, letters, and conference reports.
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