About 30 years ago, Walter Feit and John G. Thompson [8] proved the Odd Order Theorem, which states that all finite groups of odd order are solvable. In the words of Daniel Gorenstein [15, p. 14], “it is not possible to overemphasize the importance of the Feit-Thompson Theorem for simple group theory.” Their proof consists of a set of preliminary results followed by three parts-local analysis, characters, and generators and relations- corresponding to Chapters IV, V, and VI of their paper (denoted by FT here). Local analysis of a finite group G means the study of the structure of, and the interaction between, the centralizers and normalizers of nonidentity p-subgroups of G. Here Sylow's Theorem is the first main tool. The main purpose of this book is to present a new version of the local analysis of a minimal counterexample G to the Feit-Thompson Theorem, that is, of Chapter IV and its preliminaries. We also include a remarkably short and elegant revision of Chapter VI by Thomas Peterfalvi in Appendix C.

What we would ideally like to prove, but cannot, is that each maximal subgroup M of G has a nonidentity proper normal subgroup M0 such that

1. (1) CM0(α) = 1, for all elements α ∈ M − M0,

2. (2) for all elements g ∈ G ∈ M,

3. (3) M0 is nilpotent,

4. (4) M/M0 is cyclic,

and such that the totality of these subgroups M0, with M ranging over all of the maximal subgroups of G, forms a partition of G:

1. (5) each nonidentity element of G lies in exactly one of the subgroups M0.