有限p-群的结构与p'-自同构

Investigating the structure of finite groups, and the connection between the structure of finite groups and their automorphism groups are important topics in the theory of finite group. Because the structure of a finite group depends on its Sylow-subgroups embedded in it，this project is based on the study of finite p-groups structure to explore the structure of a finite group G that satisfies the following conditions and the connection between the structure of the finite group G and the structure of its Sylow-subgroups..1、For any non-normal subgroup H of G, the normal closures of H contains(is contained in) its normalizer..2、For any normal subgroup N of G, G' mod (G' meet N) is cyclic or N mod (N meet Z(G)) is cyclic..3、Any abelian non-normal subgroup of G is self-centered or its normalizer is equal to the centralizer..Furthermore, we use the technique of the action of groups on groups to discuss the maximal normal subgroup which can act trivially on another finite group. Such a study will further advance Blackburn's theorem.

研究有限群的结构,以及有限群与其自同构之间的相互关系是有限群论中的重要课题。由于有限群的结构依赖于它的Sylow-子群在其中的嵌入方式。本项目正是从研究有限p-群的结构出发探讨满足下列条件的有限群G的结构以及它与其Sylow-子群的结构之间的关系。.1、G中任意非正规子群H，均有H的正规闭包包含(包含于)H的正规化子。.2、G中任意正规子群N，均有G'模(G'交N)循环或N模(N交Z(G))循环。.3、G中任意非正规交换子群均自中心化或者其正规化子等于中心化子。.进一步从有限群作用的角度来探讨其能平凡作用的最大正规子群。这样的研究将进一步推进Blackburn的定理。

群的结构的研究是有限群论中的根本问题，且有限群的结构依赖于它的Sylow-子群在其中的嵌入方式。本项目首先从有限p-群的结构出发，完成了满足下列条件的有限p-群G的结构的研究工作。. 1、G中任意非正规子群H，均有H的正规闭包包含H的正规化子。. 2、G中任意正规子群N，均G'模(G'交N)循环或N模(N交Z(G))循环。. 3、G中恰有1个正规子群既不包含导群G’又不含于中心Z(G)。.也完成了非正规循环子群的正规化子是极大子群的有限群的分类工作。. 另外，群论在图的对称性的研究中有着重要应用。本项目主要研究了双循环群上的三度双凯莱图的对称性，得到了这类图的完全分类。