p群的研究一直是有限群研究的重要领域。在有限单群分类完成后，自本世纪初开始，p群的研究迎来了一个新的活跃期。本项目的研究内容为p群国际上的领头人Mann提出的几个公开问题。这些问题来源于群论大师Burnside、Hall、Higman、Glauberman等人的工作。涉及到关于p群个数的PORC猜想、导列长给定时p群阶的下界问题、较大的正规交换子群存在性的判断问题等。这些问题虽然是在上个世纪就提出来的，但是至今都未能得到解决。由于它们都是p群研究的前沿问题和难点问题，在国际上相关的研究依然十分活跃。本项目旨在对这些问题展开攻关，探索一些新的研究方法，取得一些开拓性成果,为p群领域的研究做出新的贡献。

The research of p-groups has been an important area in the research of finite groups. After the completion of the classification of the finite simple groups, since the beginning of this century, the research of p-groups has ushered in a new period of activity. This project contains several open questions raised by Mann, an international leader in the field of p-groups. These questions come from the work of masters of the group theory Burnside, Hall, Higman, Glauberman et al. It involves Higman's PORC conjecture, the problem of the lower boundary of the order of p-groups given the length of whose derived group series, and the problem of the existence of large normal abelian subgroup. Although these problems were raised in last century, they have not been solved. Since they belong to the frontier problems and difficult problems for p-groups, the research related to them is still active. This project aims to solve these problems, explore some new research methods, achieve some pioneering results, and make new contributions to the research in finite p-groups.