Some Special Classes Of Finite P-Groups

Finite p-groups play an important role in group theory. After the classification of finite simple groups is finally completed, the study of finite p-groups becomes more and more active. Many leading group theorists, for example, G. Glauberman, Z. Janko etc, turn their attention to the study of finite p-groups. There are many topics in which lots of questions are worthy studying. One of these is to classify some special classes of finite p-groups. In this thesis, we classify three classes of finite p-groups.Firstly, we study minimal non-class-two p-groups. We give two necessary and sufficient conditions, and we get a complete classification of such groups. A. Mann conjectured most of finite p-groups is class 2. So our group list will play an important role when we use the minimal counterexample method to study p-groups. Secondly, we reclassify finite two-generated p-groups whose derived group is cyclic (p is odd). Since we bring the properties of regular p-groups into the work, our computation is clearer and simpler than the work of R. J. Miech. In fact, our group list maybe is the first readable list. Finally, in order to finish D. S. Passman's work about finite p-groups with chn(G) = 1, we investigate finite 2-groups in which every subgroup is cyclic or normal and give a classification for this kind of meta-Hamilton 2-groups.