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On Local Nilpotency of the normal subgroups of a Group

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Algebra Colloquium 2016年第3期23卷 531-540页

作者： Zhirang Zhang Jiachao Li School of Applied Mathematics Chengdu University of Information Technology Chengdu 610225 China

A group G is said to have property μ whenever N is a non-locally nitpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property v satisfy property μ, where a group G is said to have property v if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP（G） is the Hirsch-Plotkin radical of G, and φf （G） is the intersection of all the maximal subgroups of finite index in G （here φf（G） = G if no such maximal subgroups exist）. It is shown that a group G has property μ if and only if HP（G/φf（G）） = HP（G）/φf（G） and that the class of groups with property v is a proper subclass of that of groups with property it. Also, the structure of the normal subgroups of a group： nilpotency with the minimal condition, is investigated.

关键词： normal subgroups nilpotent groups locally nilpotent groups Frattini sub-group Hirsch-Plotkin radical

Finite Groups Whose Nontrivial normal subgroups Have Order Two

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Journal of Mathematical Research and Exposition 2011年第4期31卷 675-680页

作者： Shou Hong QIAO Da Chang GUO School of Applied Mathematics Guangdong University of Technology Guangdong 510006 P. R. China

In this paper, we investigate the structure of the groups whose nontrivial normal subgroups have order two. Some properties of this kind of groups are obtained.

关键词： finite groups normal subgroups soluble groups insoluble groups simple groups

Finite Groups Whose Nontrivial normal subgroups Have the Same Order

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Journal of Mathematical Research and Exposition 2008年第4期28卷 807-812页

作者：张勤海曹建基 Department of Mathematics Shanxi Normal University

In this paper, we study finite groups all of whose nontrivial normal subgroups have the same order. In the solvable case, the groups are determined. In the insolvable case, some characterizations are given.

关键词： normal subgroups simple groups minimal non-abelian groups.

On Coprime G-conjugacy Class Sizes in a normal Subgroup

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Acta Mathematica Sinica,English Series 2014年第9期30卷 1588-1594页

作者： Xian He ZHAO Hai Peng QU Gui Yun CHEN College of Mathematics and Information Science He 'nan Normal University Xinxiang 453007 P. R. China Department of Mathematics Shanxi Normal University Linfen 041004 P. R. China School of Mathematics and Statistics Southwest University Chongqing 400715 P. R. China

Let N be a normal subgroup of a group G. Suppose that the positive integers m 〉 n are two longest non-central G-conjugacy class sizes of N with （m, n） = 1. The purpose of this paper is to determine the structure of... 详细信息

关键词： normal subgroups G-conjugacyclass sizes Frobenius group

On the normal Subgroup with Exactly Two G-Conjugacy Class Sizes

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Chinese Annals of Mathematics,Series B 2009年第4期30卷 427-432页

作者： Xianhe ZHAO Xiuyun GUO College of Mathematics and Information Science Henan Normal University Xinxiang 453007 Henan China Department of Mathematics Shanghai University Shanghai 200444 China

Let G be a finite group with a non-central Sylow r-subgroup R, Z（G） the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either i or m. Particularly, it is shown that N is Abelian if N ∩ Z（G）=1 and the G-conjugacy class size of every element of N is either 1 or m.

关键词： normal subgroups Conjugacy class sizes Nilpotent groups