有关亚循环群的几个结论摘要：群的构造及其性质是群论研究的重要内容。亚循环群,即循环群被循环群的扩张,是特殊的二元生成群,奥特斯曾研究并给出了有限亚循环群构造；威尔森曾证明了有限群G恒为亚循环群时之阶所具备的条件徐明耀、樊悍、黄平安、陈贵云等国内群论学者,也都先后对亚循环群的各类性质做过研究,得到了许多具有重要价值的结论。而有关循环群综合分析的研究成果不常见。本文以“亚循环群”为主题,综合分析了亚循环群的自同构群,自同群是亚循环群的群以及亚循孙一群等的相关结论通过比较循环群、亚环群、超可解群和可解群之间的联系与区别,利用它们间的关系以及利用群对群结构的影响,并引入弱拟正规等概念来扩充对亚循群的研究,得到了有关亚循环群的性质及其充分条件方面的若干结论。所得结论富了研究亚循环群这一领域的成果。关键词：亚循环群；群对群结构；可解群Abstract:Thestructureandpropertiesofgroupsareimportantcontentsofgrouptheory.Metacyclicgroup,cyclicgroupiscyclicgroupexpansion,isaspecialtwoyuangenerationgroup,atampswasstudiedandpresentedthefinitemetacyclicgroupstructure;WilsonhadprovedthatafinitegroupGconstantforthemetacyclicgroupoforderconditionsXuMingyao,HuangPingan,ChenGui,fanHancloudandotherdomesticgroupscholarsalsohavethenatureofvarioustypesofmetacyclicgroupshavedonetheresearch,andobtainedalotofvaluableconclusions.However,theresearchresultsaboutthecyclicgroupanalysisarenotcommon.Inthispaper,"themetacyclicgroup"asthetheme,acomprehensiveanalysisofthemetacyclicgroupautomorphismgroup,fromthesamegroupconclusionmetacyclicgroupgroupandsubgrouponthesunbycomparingthecyclicgroup,subringgroup,supersolvablegroupandsolvablerelationandthedifferencebetweengroups,therelationshipbetweenamongthem,theinfluenceonthestructureofthegroup,andtheintroductionoftheconceptofweaklyquasinormalexpansionbasedonsubgroupresearch,wegetsomeconclusionsaboutthenatureofthemetacyclicgroupanditssufficientconditions.Theresultsofthisstudyenrichtheresultsofthestudyonthesubcirculationgroup.Keywords:subcyclicgroup;grouppairstructure;solvablegroup目录第1章引言.........................................................21.1本文研究背景................................................21.2国内外研究现状...............................................21.3本文研究方法....................................................3二相关理论概述.....................................................42.1群论的发展及其相关内容......................................42.2群论相关内容................................................4三循环群的斜态射研究理论...........................................53.1理论研究现状...............................................53.2斜态射的重要性研究..........................................6四循环群时的斜态射的分类问题.......................................74.1西洛性质....................................................74.2幂零群......................................................84.3p群........................................................84.4扩张........................................................94.5转移........................................................94.6超可解群...................................................10结论...............................................................10参考文献...........................................................11第1章引言1.1本文研究背景当代科学技术发展的一大特点是,在几乎所有的领域,数学与计算机技术被广泛的应用。近代数学的思想方法、观点和结论正在深入地渗透进自然科学和社会科学的众多的理论分支,这是...