【作者】 张艳；

【导师】 班桂宁；

【作者基本信息】 广西大学 ， 基础数学， 2012， 硕士

【摘要】 本文主要研究了中心商的阶为p5以及部分中心循环且中心商的阶为p6的有限非循环p-群.班桂宁等人已经给出了中心循环且满足中心商的阶是p5的群也是LA-群,部分推广了Davitt在1980年关于LA-猜想的重要结论,即：满足中心商的阶小于等于p4的有限非循环p-群G是LA-群.因此本文的主要内容及成果如下：(1)本文首先利用群的扩张理论和自由群的方法证明了中心非循环且中心商的阶是p5的有限非循环p-群是存在的.其次结合群的自同构的特性证明了这些群均是LA-群,由此完全地推广了Davitt在1980年关于LA-猜想的重要结论.(2)基于1980年Rodney James的论文,在阶为p6的第十七到二十三族群和四十一族群的基础上,利用群的扩张理论给出很多中心循环且中心商的阶为p6的有限新p-群,这些群将要么是新的LA-群要么是LA-猜想的一个反例,这为以后进一步推广本文的重要结论打下基础.更多还原

【Abstract】 This thesis mainly studies the p-groups whose central quotients have orders p5and also discusses some finite non-cyclic p-groups having cyclic centre and central quotients of orders p6. Professor Ban and other authors show that groups having cyclic centre and satisfying central quotients of orders p5are also LA-groups, partly promote Davitt’the important conclusion about LA-conjecture in1980, that is, the non-cyclic p-groups G whose central quotients are of order less than p5must be LA-groups. So the main contents and results of this thesis constructed as follows:(1) In the thesis we first determine all p-groups satisfying that the centers are non-cyclic and central quotients of orders p5, and then show their existence by extension theory of group and the method of free group. Next we prove these p-groups are LA-groups by the characteristic of their automorphisms, and also completely generalize the Davitt’s important result for LA-conjecture in1980.(2) In order to generalize the important conclusions of this thesis in the future, we study p-groups which centers are cyclic and central quotients of orders p6, then a number of new p-groups of cyclic centre and central quotients of orders p6are obtained by extension method based on the groups in inoclinism families seventeen to twenty three and forty one in Rodney James’ paper in1980, which are either LA-groups or some counter examples of LA-conjecture.更多还原