This is a translation from the Japanese of the second volume (chapters four through six) of my book "Gunron" (Iwanami Shoten, 1978). After discussing the concept of commutators in the fourth chapter, we tum to a discussion of the methods and theorems pertaining to finite groups. The last chapter is intended as an introduction to the recent progress in the theory of simple groups. Forihe translation, I have kept the main body of the text unchanged, however I have added a few comments in the last chapter in order to inform the readers of the most recent progress. I would like to express my appreciation to Kazuko Suzuki for her devoted help in translating this book. Finally, it gives me great pleasure to acknowl­ edge my indebtedness to my wife, Naoko, for her constant support and understanding, and for converting the long and often illegible manuscript into a beautifully typed one. To her, I express my sincere thanks and appreciation. July, 1985 Michio Suzuki Contents (Part II) List of Notation IX Chapter 4 Commutators § 1. Commutator Subgroups §2. Nilpotent Groups 13 37 §3. Commutator Calculations 54 §4. Finite p-Groups . . .