内p-群与外p-群的结构和性质

Finite groups with many subgroups of prime power order

本文以p-群和内∑-群研究成果为基础,以它们的研究方法为依托,采用反证法、分析法,得到若干成果,丰富了研究内∑-群这一领域的成果.文章首先以可解次单群的结构和性质,来引出文章所讨论的任一真子群为素数方幂阶的有限群的结构和性质,给出来一个有限群满足这一性质的充分必要条件,得到了若干结论,并且指出了任一真子群为素数方幂阶的有限群和有限次单群、CP-群之间的包含关系.最后,进一步拓宽这一性质,引出外p-群的定义,给出了一个外p-群的必要条件.

This paper studies the results based on p-groups and inner ∑_ groups. The methods applied in the study of this paper include reversed proof, analysis proof. Firstly it uses the quality and structure of the sub-simplegroups, then introduces the quality and structure of the finite groups with many subgroups of prime power order and necessary and sufficient condition. It also points out the relation of the finite groups with many subgroups of prime power order, sub-simplegroups, CP- groups. In the end, the paper gives the external p- groups and the condition of this groups.