【作者】 刘琳；

【导师】 钟祥贵； 张博儒；

【作者基本信息】 广西师范大学 ， 数学， 2023， 硕士

【摘要】 在有限群中,自中心化子群是一类特殊的子群,TI-子群是正规子群的一个重要推广,它们都对有限群的结构有很强的作用.本文的研究内容主要是围绕TI-子群和自中心化子群进行展开的,共分为五章,具体内容如下.在第一章中,我们主要阐述了 TI-子群和自中心化子群方面的研究进展.在第二章中,我们主要介绍了本文中需要用到的基本定义以及相关的基本引理.在第三章中,我们主要讨论了 TI-子群对有限群结构的影响.在3.1节中,我们刻画了每个非亚循环子群皆是TI-子群或次正规子群的有限群,并证明了这类群是可解群.在3.2节中,Beltrán证明了:设群A互素作用在群G上.若G的任意A-不变子群是幂零子群或TI-子群,则G是可解群且G的任意非幂零A-不变子群皆正规于G.我们结合次正规子群,进一步证明了:设群A互素作用在群G上.若G的极大A-不变子群是幂零子群或TI-子群或次正规子群,则G是可解群且G的任意非幂零极大A-不变子群皆正规于G.在第四章中,我们主要讨论了自中心化子群对有限群结构的影响.在4.1节和4.2节中,我们分别刻画了每个非素数幂阶自中心化子群和每个非p-幂零自中心化子群皆为TI-子群或次正规子群的有限群.在4.3节中,我们引入弱c-正规子群,主要讨论了自中心化子群皆弱c-正规的有限群,得到该类群具有商群遗传性且得到部分特殊子群是弱c-正规自中心化子群的有限群p-幂零和可解的一些充分条件.在第五章中,我们对本文所研究内容进行总结,并提出可进一步研究的方向.更多还原

【Abstract】 The self-centralizing subgroup is a special class of subgroups in finite group theory,and the TI-subgroups is an important generalization of normal subgroups.Both selfcentralizing subgroups and TI-subgroups have a very influence on the structure of finite groups.In this paper,we mainly focus on TI-subgroups and self-centralizing subgroups.There are total five chapters and the specific contents are as follows.In chapter 1,we mainly discuss the research progress of TI-subgroups and selfcentralizing subgroups.In chapter 2,we mainly introduce the basic definitions and the relevant basic lemmas used in this article.In chapter 3,we mainly discuss the effect of TI-subgroups on the structure of finite groups.In section 3.1,we describe that every non-metacyclic subgroup is TI-subgroups or subnormal subgroups,and prove that such groups are solvable.In section 3.2,Beltran proves that suppose that a finite group A acts coprimely on a group G and if every A-invariant subgroup of G is a TI-subgroup or nilpotent,then G is soluble and every non-nilpotent A-invariant subgroup is normal in G.By combining subnormal subgroups,we further prove that suppose that group A acts coprimely on group G,and if every maximal A-invariant subgroup of G is nilpotent or TI-subgroup or subnormal,then G is soluble and all non-nilpotent maximal A-invariant subgroup of G are normal in G.In chapter 4,we mainly discuss the effect of self-centralizing subgroups on the structure of finite groups.In section 4.1 and 4.2,we characterize that all self-centralizing subgroups of non-prime-power order and self-centralizing subgroups of non-p-nilpotent are TI-subgroups or subnormal subgroups.In section 4.3,we introduce weakly c-normal subgroups,and discuss the finite groups in which all self-centralizing subgroups are weakly c-normal subgroups,and obtain some sufficient conditions for the finite groups p-nilpotent whose special subgroups are all weakly c-normal.In chapter 5,we summarize the content of this paper and propose the direction of further research.更多还原