On the Hughes conjecture for some finite p-groups

IF 1 3区数学 Q1 MATHEMATICS

Annali di Matematica Pura ed Applicata Pub Date : 2024-01-31 DOI:10.1007/s10231-023-01421-z

Mandeep Singh, Rohit Garg

引用次数: 0

Abstract

Let G be a group, p a prime and \(H_p(G)\) the subgroup of G generated by the elements of order different from p. In 1957, D. R. Hughes conjectured that either \(H_p(G)=1\), \(H_p(G)=G\), or \([G:H_p(G)]=p\). In this paper, we prove this conjecture for finite extraspecial p-groups (where \(p>2\)), finite minimal non-abelian p-groups and finite non-abelian p-groups having cyclic maximal subgroup. Moreover, we give some sufficient conditions for 2-generated finite non-abelian p-groups which guarantee the existence of the Hughes conjecture.

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关于某些有限 p 群的休斯猜想

让 G 是一个群，p 是一个素数，\(H_p(G)\) 是由与 p 不同阶的元素产生的 G 的子群。1957 年，D. R. Hughes 猜想，要么 \(H_p(G)=1\)，\(H_p(G)=G\)，要么 \([G:H_p(G)]=p\)。在本文中，我们证明了有限外特殊 p 群（其中 \(p>2\)）、有限最小非标注 p 群和具有循环最大子群的有限非标注 p 群的这一猜想。此外，我们还给出了保证休斯猜想存在的 2 代有限非阿贝尔 p 群的一些充分条件。

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

Annali di Matematica Pura ed Applicata 数学-数学

CiteScore

2.10

自引率

10.00%

发文量

审稿时长

>12 weeks

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