This book comprises two parts, each devoted to the revision of the proof of a theorem about finite groups. The two theorems are among the results taken as foundation material in the revision of the classification of finite simple groups undertaken by D. Gorenstein, R. Lyons and R. Solomon.

The famous theorem of W. Feit and J. G. Thompson states that every finite group of odd order is solvable. It is the most important among the initial theorems in the classification of finite simple groups. The proof of this theorem divides into two parts. The first consists of the study of the maximal subgroups of a minimal counterexample to the theorem. This part, which is of considerable difficulty, has been revised by H. Bender and G. Glauberman; their work has appeared as a book in this series. The second part of the proof of the Feit-Thompson Theorem uses character theory to show that the existence of a simple group of odd order is impossible. In Part I of this book, we give a revision of this portion of the proof. Thus, with the book of Bender and Glauberman, a complete proof of the theorem is provided.