摘要
本文的主要目的是结合有限p-群的理论来研究局部幂零p-群。论文的主要结果在第二,三,四章。
在第二章里,我们研究了可除阿贝尔p-群存在p-自同构群。文献[5]得到了关于可除阿贝尔p-群的p-自同构群的一个重要结果:即可除阿贝尔p-群存在p-自同构群的充要条件是其秩大于或者等于p-1。在本章,我们不仅得到了秩为p-1或p的可除阿贝尔群的p-自同构群的结构,而且还给出了其p-自同构群详细作用方式。这些结果有助于我们理解(?)ernikov,p-群的结构。
在第三章里,我们研究了特征子群Ω_n(G)和(?)_n(G)与局部幂零p-群G的结构关系。探讨了一类局部幂零p-群G满足|G:(?)_n(G)|＜∞,所得结果是Baumslag.G[10]和Cernikov.S.N[11,12]所得结论的推广。同时也给出了局部幂p-群G满足|Ω_1(G)|≤p~2或|Ω_2(G)|=p~3的详细的结构。
在第四章里,我们把有限正则p-群的理论推广到无限的局部幂零p-群。研究了一类无限正则p-群G满足|G:(?)_n(G)|＜∞,进一步,还研究了一类无限的非正则p-群,但是其真商群或无限真子群是正则的群。
The aim of this thesis is to study locally nilpotent p- groups by using some theory of finite p- groups.The main results of this thesis are in Chapter 2,3,4.
In chapter 2,we study the p-automorphism of divisible abelian groups of rank p-1 and p.In[5],it showed that a divisible abelian groups D has p-automorphism iff the of rank D is greater than p-1.Here we get the order of the p-automorphism groups of the divisible abelian groups of rank p-1 and p.We also investigate how p-automorphism groups act on the divisible abelian groups.This research can help us to understand the structure of(?)ernikov p-groups.
In chapter 3,we show that the characteristic subgroupsΩ_n(G) and(?)_n(G) have strong influence on the structure of locally nilpotent p-group G.We study the structure of locally nilpotent p-groups G with |G:(?)_n(G)|＜∞,which are generalization of Baumslag.G[10]and Cernikov.S.N[11,12].Furthermore,we investigate locally nilpotent p-groups G with |Ω_1(G)|≤p~2 or |Ω_2(G)|=p~3.
In chapter 4,we try to generalize the theory of finite regular p-groups in locally nilpotent p-groups.Infinite regular p-groups with |G:(?)_n(G)|＜∞and some infinite irregular p-groups whose proper factor groups or proper infinite subgroups are regular are investigated in this chapter.
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